Computer-implemented system and method for determining constraint-based options in combinatorial applications using cell-to-symbol tally technique

Abstract

The present invention discloses a computer-implemented system (102) and method (600) for determining constraint-based options in combinatorial applications using a cell-to-symbol tally technique The process begins by generating (604) options tables based on the obtained data associated with a partially filled components. The process is followed by generating (606) a list of potential candidates for each empty component based on rules associated with the combinatorial applications. The process further includes identifying (608) critical factors based on a number of empty components and distribution of the candidates. The process further includes predicting (610) valid candidates from the critical factors for first pre-determined number of components (key cells). The process further includes filtering (612) invalid candidates from critical factors based on cell-to-symbol tally technique. The process further includes filling (614) the list of the potential candidates in second pre-determined number of components (rest cells) by the cell-to-symbol tally technique.

Description

COMPUTER-IMPLEMENTED SYSTEM AND METHOD FOR DETERMINING CONSTRAINT-BASED OPTIONS IN COMBINATORIAL APPLICATIONS USING CELL-TO-SYMBOL TALLY TECHNIQUEEARLIEST PRIORITY DATE:This Application claims priority from a Provisional patent application filed in India having Patent Application No. 202441073666, filed on 30 September 2024 and titled “SYSTEM AND METHOD FOR DETERMINING CONSTRAINT-BASED OPTIONS IN COMBINATORIAL APPLICATIONS USING CELL-TO- SYMBOL TALLY”.FIELD OF INVENTIONEmbodiments of the present invention relate to computational problem-solving systems and more particularly relate to a computer-implemented system and a method for determining one or more constraint-based options in combinatorial applications using a cell-to-symbol tally technique.BACKGROUNDCombinatorial optimization problems, particularly those classified as nondeterministic polynomial (NP)-complete, have long posed significant challenges in computer science and mathematics. The combinatorial optimization problems, which include well-known puzzles like Sudoku, are characterized by their computational complexity and the exponential growth of potential solutions as problem size increases.Traditional methods to solving the combinatorial optimization problems often rely on brute-force methods, backtracking algorithms, or heuristic techniques. Whileeffective for small-scale instances, the traditional methods frequently struggle with larger, more complex problems due to the rapid expansion of the solution space.The sudoku, a popular logic-based number placement puzzle, serves as a prime example of these challenges. As the grid size increases beyond the standard 9x9, computational resources required to solve the puzzle grow exponentially. This scalability issue has limited the practical application of existing algorithms to larger, more complex Sudoku variants.Current methods for solving high-rank Sudoku puzzles often employ techniques such as depth-first search, constraint propagation, and various heuristics. However, the current methods become prohibitively time-consuming as the puzzle size increases, sometimes requiring hours or even days to solve extremely large grids.Additionally, most current Sudoku solving techniques are tailored specifically to the standard 9x9 puzzle format. The Sudoku solving techniques lack the flexibility to handle arbitrary Nthrank Sudoku puzzles with different grid sizes and sub-grid dimensions. This limits the applicability of the Sudoku solving techniques to the wide range of the Sudoku variants that exist.Another limitation of existing approaches is their reliance on brute-force search methods, which may be computationally expensive and inefficient, especially for more challenging Sudoku puzzles. The existing approaches may fail to leverage the inherent structure, and patterns present in the Sudoku puzzles, leading to suboptimal performance.In real-world applications, similar combinatorial problems arise in resource allocation, task scheduling, traffic management, and the like. The real-world applications require managing complex one or more constraint-based options and optimizing dynamic systems, similar to the challenges found in Sudoku. Existing solutions struggle with scalability and flexibility, leading to inefficiencies and suboptimal performance.Therefore, there is a need for an improved computer-implemented system and method for determining one or more constraint-based options in one or more combinatorial applications using a cell-to-symbol tally technique, in order to address the aforementioned issues.SUMMARYThis summary is provided to introduce a selection of concepts, in a simple manner, which is further described in the detailed description of the disclosure. This summary is neither intended to identify key or essential inventive concepts of the subject matter nor to determine the scope of the disclosure.In order to overcome the above deficiencies of the prior art, the present disclosure is to solve the technical problem by providing a computer-implemented method for determining one or more constraint-based options in one or more combinatorial applications using a cell-to-symbol tally technique.The computer-implemented method comprises obtaining, by one or more hardware processors, data associated with one or more partially filled components, from one or more data sources. The one or more partially filled components comprise at least one of: one or more cells in one or more grids, one or more intersections, one or more partially allocated network grids, and one or more partially completed task schedules. The computer-implemented method further comprises generating, by the one or more hardware processors, one or more options tables based on the obtained data associated with the one or more partially filled components.The computer-implemented method further comprises generating, by the one or more hardware processors, a list of one or more potential candidates for each empty component in one or more components based on one or more rules associated with the one or more combinatorial applications. The list of the one or more potential candidates comprises at least one of: a list of potential symbols, a list of potential traffic signal timings, a list of potential resources, and a list of potential tasks.The computer-implemented method further comprises identifying, by the one or more hardware processors, one or more critical factors based on at least one of: a number of empty components and distribution of the one or more candidates. The one or more critical factors comprise at least one of: two or more member sets being an intersection set of two adjacent valid components, one or more critical intersections allow for smoother traffic flow across an entire network, one or more critical nodes in a network for efficient usage of network bandwidth, and one or more critical tasks allow an entire workflow to be completed efficiently.The computer-implemented method further comprises predicting, by the one or more hardware processors, one or more valid candidates from the one or more critical factors for a first pre-determined number of components. The computer- implemented method further comprises filtering, by the one or more hardware processors, one or more invalid candidates from the one or more critical factors based on the cell-to-symbol tally technique. The computer-implemented method further comprises filling, by the one or more hardware processors, the list of the one or more potential candidates in a second pre-determined number of components by the cell-to-symbol tally technique.In an embodiment, for filling the list of the one or more potential candidates in the second pre-determined number of components by the cell-to-symbol tally technique, comprises: (a) selecting, by the one or more hardware processors, the one or more critical factors along each row, column, and block separately; (b) determining, by the one or more hardware processors, whether the selected one or more critical factors are equal to a number of the one or more potential candidates within the one or more critical factors; and (c) upon determining that the selected one or more critical factors are equal to the number of the one or more potential candidates within the one or more critical factors, predicting, by the one or more hardware processors, the one or more components of the one or more critical factors that are bound to the number of the one or more potential candidates, to remove the number of the one or more potential candidates from one or more remaining critical factors.In another embodiment, the computer-implemented method further comprises: (a) generating, by the one or more hardware processors, the one or more options tables based on the data associated with the one or more components of the one or more combinatorial applications; (b) upon determining the presence of the single critical factor, determining, by the one or more hardware processors, for one or more errors by identifying one or more identical critical factors in same at least one of: row, column, and block; (c) upon no errors, inserting, by the one or more hardware processors, the one or more potential candidates into the one or more components; (d) updating, by the one or more hardware processors, the one or more options tables upon inserting the one or more potential candidates into the one or more components; (e) determining, by the one or more hardware processors, whether at least one of: the one or more components are complete and one or more missing candidate errors; (f) determining, by the one or more hardware processors, a single hidden candidate when at least one of: the one or more components are incomplete and free candidate errors; (g) upon determining the single hidden candidate, determining, by the one or more hardware processors, for one or more errors by identifying two or more hidden candidates for the one or more components; (h) filtering, by the one or more hardware processors, the one or more invalid candidates from the one or more critical factors to generate a critical factor with a single candidate; and (i) predicting, by the one or more hardware processors, the one or more components of the one or more critical factors that are bound to the number of the one or more potential candidates, to remove the number of the one or more potential candidates from one or more remaining critical factors, based on the cell-to-symbol tally technique.In yet another embodiment, upon predicting the one or more components of the one or more critical factors that are not bound to the number of the one or more potential candidates, the computer implemented method further comprising: (a) predicting, by the one or more hardware processors, the one or more candidates for the one or more components; and (b) determining, by the one or more hardware processors, whether the one or more candidates inserted into the one or more components, areviolating the one or more rules associated with the one or more combinatorial applications, by correcting the one or more errors based on an error rectification technique.In yet another embodiment, the computer-implemented method further comprises: (a) upon inserting the one or more candidates into the one or more components, generating, by the one or more hardware processors, one or more unique identities being assigned to the one or more components; and (b) replacing, by the one or more hardware processors, the one or more candidates by the one or more unique identifiers being assigned to the one or more components when one or more changes made to the one or more options table based on the error rectification technique, wherein the one or more unique identifiers are configured to identify error causing predictions accurately.In yet another embodiment, the error rectification technique is configured to at least one of identify one or more error causing candidates, rectify the one or more errors, break one or more error loops, and provide one or more insights for one or more future predictions.In yet another embodiment, the one or more constraint-based options comprise at least one of one or more numerals, one or more network resources, one or more time slots, and one or more traffic signal timings. The one or more combinatorial applications comprise at least one of traffic signal optimization, resource allocation in network, scheduling and task management, and Sudoku puzzle.In an aspect, a computer-implemented system for determining one or more constraint-based options in one or more combinatorial applications using a cell-to- symbol tally technique, is disclosed. The computer-implemented system comprises one or more hardware processors and a memory unit. The memory unit is coupled to the one or more hardware processors. The memory unit comprises a set of program instructions in form of a plurality of subsystems, configured to be executed by the one or more hardware processors.The plurality of subsystems comprises a data obtaining subsystem configured to obtain data associated with one or more partially filled components, from one or more data sources. The one or more partially filled components comprise at least one of: one or more cells in one or more grids, one or more intersections, one or more partially allocated network grids, and one or more partially completed task schedules.The plurality of subsystems further comprises an options table generating subsystem configured to: (a) generate one or more options tables based on the obtained data associated with the one or more partially filled components; and (b) generate a list of one or more potential candidates for each empty component in one or more components based on one or more rules associated with the one or more combinatorial applications. The list of the one or more potential candidates comprises at least one of: a list of potential symbols, a list of potential traffic signal timings, a list of potential resources, and a list of potential tasks.The plurality of subsystems further comprises a factors identifying subsystem configured to: (a) identify one or more critical factors based on at least one of: a number of empty components and distribution of the one or more candidates, wherein the one or more critical factors comprise at least one of: two or more member sets being an intersection set of two adjacent valid components, one or more critical intersections allow for smoother traffic flow across an entire network, one or more critical nodes in a network for efficient usage of network bandwidth, and one or more critical tasks allow an entire workflow to be completed efficiently; and (b) predict one or more valid candidates from the one or more critical factors for a first pre-determined number of components.The plurality of subsystems further comprises a cell-to-symbol tally subsystem configured to: (a) filter one or more invalid candidates from the one or more critical factors based on the cell-to-symbol tally technique; and (b) fill the list of the one or more potential candidates in a second pre-determined number of components by the cell-to-symbol tally technique.To further clarify the advantages and features of the present invention, a more particular description of the invention will follow by reference to specific embodiments thereof, which are illustrated in the appended figures. It is to be appreciated that these figures depict only typical embodiments of the invention and are therefore not to be considered limiting in scope. The invention will be described and explained with additional specificity and detail with the appended figures.BRIEF DESCRIPTION OF THE DRAWINGSThe disclosure will be described and explained with additional specificity and detail with the accompanying figures in which:FIG. 1 illustrates an exemplary block diagram representation of a network architecture depicting a computer-implemented system for determining one or more constraint-based options in one or more combinatorial applications using a cell-to- symbol tally technique, in accordance with an embodiment of the present disclosure;FIG. 2 illustrates a detailed view of the computer-implemented system as shown in FIG. 1 for determining the one or more constraint-based options in the one or more combinatorial applications using the cell-to-symbol tally procedure, in accordance with an embodiment of the present disclosure;FIG. 3A illustrates an exemplary first flow chart depicting a process for solving arbitrary Nthrank Sudoku puzzles, in accordance with an embodiment of the present disclosure;FIG. 3B illustrates an exemplary second flow chart depicting the process for solving the arbitrary Nthrank Sudoku puzzles, in accordance with an embodiment of the present disclosure;FIG. 4 illustrates an exemplary flow diagram depicting a sudoku generating process, in accordance with an embodiment of the present disclosure;[0001JFIG. 5 illustrates an exemplary graphical plot depicting relation between a rank of the Sudoku puzzle and a ratio of initial symbols to grid size, in accordance with an embodiment of the present disclosure; andFIG. 6 illustrates a flow chart illustrating a computer-implemented method for determining the one or more constraint-based options in the one or more combinatorial applications using the cell-to-symbol tally technique, in accordance with an embodiment of the present disclosure.Further, those skilled in the art will appreciate that elements in the figures are illustrated for simplicity and may not have necessarily been drawn to scale. Furthermore, in terms of the method steps, equipment, and parameters used herein may have been represented in the figures by conventional symbols, and the figures may show only those specific details that are pertinent to understanding the embodiments of the present disclosure so as not to obscure the figures with details that will be readily apparent to those skilled in the art having the benefit of the description herein.DETAILED DESCRIPTION OF THE PRESENT INVENTIONFor the purpose of promoting an understanding of the principles of the disclosure, reference will now be made to the embodiment illustrated in the figures and specific language will be used to describe them. It will nevertheless be understood that no limitation of the scope of the disclosure is thereby intended. Such alterations and further modifications in the illustrated system, and such further applications of the principles of the disclosure as would normally occur to those skilled in the art are to be construed as being within the scope of the present disclosure.The terms "comprises", "comprising", or any other variations thereof, are intended to cover a non-exclusive inclusion, such that a process or method that comprises a list of steps does not include only those steps but may include other steps not expressly listed or inherent to such a process or method. Similarly, one or morecomponents preceded by "comprises... a" does not, without more constraints, preclude the existence of other components or additional components. Appearances of the phrase "in an embodiment", "in another embodiment" and similar language throughout this specification may, but not necessarily do, all refer to the same embodiment.Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by those skilled in the art to which this disclosure belongs. The system, methods, and examples provided herein are only illustrative and not intended to be limiting.In the following specification and the claims, reference will be made to a number of terms, which shall be defined to have the following meanings. The singular forms “a”, “an”, and “the” include plural references unless the context clearly dictates otherwise.A computer system (standalone, client or server computer system) configured by an application may constitute a “module” that is configured and operated to perform certain operations. In one embodiment, the “module” may be implemented mechanically or electronically, so a module may comprise dedicated circuitry or logic that is permanently configured (within a special -purpose processor) to perform certain operations. In another embodiment, a “module” may also comprise programmable logic or circuitry (as encompassed within a general -purpose processor or other programmable processor) that is temporarily configured by software to perform certain operations.Accordingly, the term “module” should be understood to encompass a tangible entity, be that an entity that is physically constructed permanently configured (hardwired) or temporarily configured (programmed) to operate in a certain manner and / or to perform certain operations described herein.Referring now to the drawings, and more particularly to FIG. 1 through FIG. 7, where similar reference characters denote corresponding features consistentlythroughout the figures, there are shown preferred embodiments, and these embodiments are described in the context of the following exemplary system and / or method.FIG. 1 illustrates an exemplary block diagram representation of a network architecture 100 depicting a computer-implemented system 102 for determining one or more constraint-based options in one or more combinatorial applications using a cell-to-symbol tally technique, in accordance with an embodiment of the present disclosure.According to an exemplary embodiment of the present disclosure, the network architecture 100 may include the computer-implemented system 102, one or more communication networks 106, one or more databases 104, and one or more communication devices 108. The computer-implemented system 102 may be communicatively coupled to the one or more databases 104, and the one or more communication devices 108 via the one or more communication networks 106. The one or more communication networks 106 may be, but not limited to, a wired communication network and / or a wireless communication network.The wired communication network may comprise, but not limited to, at least one of Ethernet connections, Fiber Optics, Power Line Communications (PLCs), Serial Communications, Coaxial Cables, Quantum Communication, Advanced Fiber Optics, Hybrid Networks, and the like. The wireless communication network may comprise, but not limited to, at least one of wireless fidelity (wi-fi), cellular networks (including 4G (fourth generation) and 5G (fifth generation) technologies), Bluetooth, ZigBee, long-range wide area network (LoRaWAN), satellite communication, radio frequency identification (RFID), 6G (sixth generation) networks, advanced loT protocols, mesh networks, non-terrestrial networks (NTNs), near field communication (NFC), and the like.The one or more communication networks 106 are configured to facilitate seamless data exchange and communication between the computer-implemented system 102 and the one or more databases 104 for real-time data analysis.In an exemplary embodiment, the one or more databases 104 may include, but not limited to, storing, and managing data related to the one or more constraint-based options. The one or more databases 104 serve as a central repository for all relevant data, enabling efficient data retrieval and analysis to support decision-making processes. The one or more databases 104 also facilitates the determination of the one or more constraint-based options in the combinatorial applications, ensuring that the computer-implemented system 102 operates at peak efficiency. Furthermore, the one or more databases 104 may manage user access controls, configuration settings, and system logs, providing a comprehensive solution for data management and a security within the network architecture 100.In an exemplary embodiment, the one or more communication devices 108 may represent various network endpoints, such as, but not limited to, user devices, mobile devices, smartphones, Personal Digital Assistants (PDAs), tablet computers, phablet computers, wearable computing devices, Virtual Reality / Augmented Reality (VR / AR) devices, laptops, desktops, display interface panels, control panels, human machine interface panels, liquid crystal display (LCD) screens, lightemitting diode (LED) screens, and the like. The one or more communication devices 108 are configured to function as an intermediate unit between the computer-implemented system 102 and one or more users. The one or more communication devices 108 are equipped with a user interface that allows the one or more users to interact with the computer-implemented system 102. The user interface may include graphical displays, touchscreens, voice recognition, and other input / output mechanisms that facilitate easy access to data and control functions.The computer-implemented system 102 is configured to determine the one or more constraint-based options in the one or more combinatorial applications using the cell-to-symbol tally technique / procedure. The computer-implemented system 102 is configured to obtain data associated with one or more partially filled components, from one or more data sources (i.e., the one or more databases 104). In an embodiment, the one or more components may include at least one of: one or morecells in one or more grids, one or more intersections, one or more partially allocated network grids, and one or more partially completed task schedules.The computer-implemented system 102 is further configured to generate one or more options tables based on the obtained data associated with the one or more partially filled components. The computer-implemented system 102 is further configured to generate a list of one or more potential candidates for each empty component in one or more components based on one or more rules associated with the one or more combinatorial applications. In an embodiment, the list of the one or more potential candidates comprises at least one of: a list of potential symbols, a list of potential traffic signal timings, a list of potential resources, and a list of potential tasks.The computer-implemented system 102 is further configured to identify one or more critical factors based on at least one of: a number of empty components and distribution of the one or more candidates. In an embodiment, the one or more critical factors may include at least one of: two or more member sets being an intersection set of two adjacent valid components, one or more critical intersections allow for smoother traffic flow across an entire network, one or more critical nodes in a network for efficient usage of network bandwidth, and one or more critical tasks allow an entire workflow to be completed efficiently.The computer-implemented system 102 is further configured to predict one or more valid candidates from the one or more critical factors for a first pre-determined number of components. The computer-implemented system 102 is further configured to filter one or more invalid candidates from the one or more critical factors based on the cell-to-symbol tally technique. Finally, the computer- implemented system 102 is further configured to fill the list of the one or more potential candidates in a second pre-determined number of components by the cell- to-symbol tally technique.The combinatorial applications may comprise, but not constrained to, at least one of a: traffic signal optimization, resource allocation in network, scheduling and taskmanagement, and the like. A process used in solving Sudoku puzzles is applied to real-world applications such as optimizing traffic signal timings, efficiently allocating network resources, managing task schedules, and the like. The computer- implemented system 102 is configured to provide optimal solutions for traffic flow, resource distribution, scheduling, and the like ensuring effectiveness and accuracy.The one or more constraint-based options may comprise, but not limited to, at least one of: numerals, the network resources, time slots, the traffic signal timings, and the like.Just as in the Sudoku puzzle, where key cells reduce the number of possibilities, key intersections (analogous to the key cells) may be identified to streamline traffic flow. The computer-implemented system 102 may dynamically allocate time to traffic signals, minimizing waiting time at the key intersections based on real-time data, much like solving a puzzle by eliminating the options.The computer-implemented system 102 may be employed to identify key nodes (similar to the key cells) that, when optimized, improve the performance of an entire network. The cell-to-symbol tally procedure may dynamically allocate resources by filtering out inefficient configurations and focusing on the most critical areas.The computer-implemented system 102 may be adapted to schedule tasks in such a way that key tasks (key cells) are identified and completed first, allowing the rest of the schedule to fall into place logically, minimizing conflicts and inefficiencies.In an exemplary embodiment, the computer-implemented system 102 leverages a unique approach, referred to as the "cell-to-symbol tally" procedure / technique. The cell-to-symbol tally procedure filters invalid symbols from possible options for each empty cell in the Sudoku puzzle. The cell-to-symbol tally procedure involves selecting sets of cells along each and every row, column, and block separately such that the number of sets selected equals the number of unique symbols within the sets. Once this tally is achieved, it is confirmed that the cells of the sets and thesymbols are bound to each other, thereby enabling the removal of the symbols from the rest of the sets.In addition to the cell-to-symbol tally procedure, the computer-implemented system 102 also incorporates a process for identifying the key cells in the Sudoku puzzle. The key cells are those cells that cannot be solved by the cell-to-symbol tally procedure alone and require additional steps for resolution of the key cells. The computer-implemented system 102 predicts the valid symbols for the key cells, thereby facilitating the complete solution of the Sudoku puzzle.Furthermore, the computer-implemented system 102 includes an error checking procedure / technique and an error rectification procedure / technique. The error checking procedure and error rectification procedure ensure that the inserted symbols will not violate Sudoku constraints and, in the event of a violation, enable the correction of any errors.Though few components and a plurality of subsystems 114 are disclosed in FIG. 1, there may be additional components and subsystems which is not shown, such as, but not limited to, ports, routers, repeaters, firewall devices, network devices, the one or more databases 104, network attached storage devices, assets, machinery, instruments, facility equipment, emergency management devices, image capturing devices, any other devices, and combination thereof. The person skilled in the art should not be limiting the components / subsystems shown in FIG. 1. Although FIG. 1 illustrates the computer-implemented system 102, and the one or more communication devices 108 connected to the one or more databases 104, one skilled in the art can envision that the computer-implemented system 102, and the one or more communication devices 108 may be connected to several user devices located at various locations and several databases via the one or more communication networks 106.Those of ordinary skilled in the art will appreciate that the hardware depicted in FIG. 1 may vary for particular implementations. For example, other peripheral devices such as an optical disk drive and the like, local area network (LAN), widearea network (WAN), wireless (e.g., wireless-fidelity (Wi-Fi)) adapter, graphics adapter, disk controller, input / output (I / O) adapter also may be used in addition or place of the hardware depicted. The depicted example is provided for explanation only and is not meant to imply architectural limitations concerning the present disclosure.Those skilled in the art will recognize that, for simplicity and clarity, the full structure and operation of all data processing systems suitable for use with the present disclosure are not being depicted or described herein. Instead, only so much of the computer-implemented system 102 as is unique to the present disclosure or necessary for an understanding of the present disclosure is depicted and described. The remainder of the construction and operation of the computer-implemented system 102 may conform to any of the various current implementations and practices that were known in the art.FIG. 2 illustrates an exemplary block diagram representation 200 of the computer- implemented system 102 as shown in FIG. 1 for determining the one or more constraint-based options in the combinatorial applications using the cell-to-symbol tally procedure, in accordance with an embodiment of the present disclosure.In an exemplary embodiment, the computer-implemented system 102 comprises at least one of one or more hardware processors 110, a memory unit 112, and a storage unit 204. The one or more hardware processors 110, the memory unit 112, and the storage unit 204 are communicatively coupled through a system bus 202 or any similar mechanism. The system bus 202 functions as a central conduit for data transfer and communication between the one or more hardware processors 110, the memory unit 112, and the storage unit 204. The system bus 202 facilitates the efficient exchange of information and instructions, enabling a coordinated operation of the computer-implemented system 102. The system bus 202 may be implemented using various technologies, including but not limited to, parallel buses, serial buses, or high-speed data transfer interfaces such as, but not limited to,at least one of a: universal serial bus (USB), peripheral component interconnect express (PCIe), and similar standards.The memory unit 112 is operatively connected to the one or more hardware processors 110. The memory unit 112 comprises the set of computer-readable instructions in the form of the plurality of subsystems 114. The plurality of subsystems 114 comprises a data obtaining subsystem 206, an options table generating subsystem 208, a factors identifying subsystem 210, a cell-to-symbol tally subsystem 212, and an error rectifying subsystem 214.The one or more hardware processors 110, as used herein, means any type of computational circuit, such as, but not limited to, a microprocessor unit, microcontroller, complex instruction set computing microprocessor unit, reduced instruction set computing microprocessor unit, very long instruction word microprocessor unit, explicitly parallel instruction computing microprocessor unit, graphics processing unit, digital signal processing unit, or any other type of processing circuit. The one or more hardware processors 110 may also include embedded controllers, such as generic or programmable logic devices or arrays, application-specific integrated circuits, single-chip computers, and the like.The memory unit 112 may be the non-transitory volatile memory and the nonvolatile memory. The memory unit 112 may be coupled to communicate with the one or more hardware processors 110, such as being a computer-readable storage medium. The one or more hardware processors 110 may execute machine-readable instructions and / or source code stored in the memory unit 112. A variety of machine-readable instructions may be stored in and accessed from the memory unit 112. The memory unit 112 may include any suitable elements for storing data and machine-readable instructions, such as read-only memory, random access memory, erasable programmable read-only memory, electrically erasable programmable read-only memory, a hard drive, a removable media drive for handling compact disks, digital video disks, diskettes, magnetic tape cartridges, memory cards, and the like. In the present embodiment, the memory unit 112 includes the plurality ofsubsystems 114 stored in the form of machine-readable instructions on any of the above-mentioned storage media and may be in communication with and executed by the one or more hardware processors 110.The storage unit 204 may be a cloud storage or the one or more databases 104 such as those shown in FIG. 1. The storage unit 204 may store, but not limited to, recommended course of action sequences dynamically generated by the computer- implemented system 102. These action sequences comprise options table generating, key cell identification, error rectification, and the like. The storage unit 204 may be any kind of database such as, but not limited to, relational databases, dedicated databases, dynamic databases, monetized databases, scalable databases, cloud databases, distributed databases, any other databases, graph databases, vector databases, and a combination thereof.The plurality of subsystems 114 includes the data obtaining subsystem 206 that is communicatively connected to the one or more hardware processors 110. The data obtaining subsystem 206 is configured to obtain the data associated with one or more partially filled components, from the one or more data sources (i.e., the one or more databases 104). The one or more partially filled components may include at least one of: the one or more cells in one or more grids, the one or more intersections, the one or more partially allocated network grids, and the one or more partially completed task schedules.The plurality of subsystems 114 further includes the options table generating subsystem 208 that is communicatively connected to the one or more hardware processors 110. The options table generating subsystem 208 is configured to generate the one or more options table based on the obtained data associated with the partially filled Sudoku grid as input. For every empty component (e.g., cell), the options table generating subsystem 208 is configured to generate a list of one or more potential candidates (e.g., possible symbols) based on one or more rules (e.g., rows, columns, and blocks in Sudoku rules) associated with the one or more combinatorial applications. In an embodiment, the list of the one or more potentialcandidates may include at least one of: the list of potential symbols, the list of potential traffic signal timings, the list of potential resources, and the list of potential tasks.Similarly, the one or more partially filled components (e.g., partially filled traffic grid) representing the current state of traffic at different intersections (similar to the partially filled Sudoku grid) is employed by the options table generating subsystem 208. For each intersection (analogous to an empty cell), the options table generating subsystem 208 is configured to generate a list of possible / potential traffic signal timings (or phases), ensuring that signal durations and coordination across the intersections obey traffic rules and constraints (similar to the Sudoku rules of the rows, the columns, and the blocks). The dynamic options table containing all possible signal timings for each intersection, assists in optimizing traffic flow across the network.Similarly, the one or more partially filled components (e.g., a partially allocated network grid) showing the current allocation of resources such as bandwidth or computing power across nodes is employed by the options table generating subsystem 208. For each under-utilized or empty node (like an empty Sudoku cell), the options table generating subsystem 208 is configured to generate a list of possible / potential resource allocations based on network rules (similar to Sudoku’s row, column, and block constraints). The rules may include bandwidth limits, load balancing requirements, or latency constraints. The options table shows possible configurations of the resource allocations for each node, allowing for an efficient and balanced network operation.Similarly, the one or more partially filled components (e.g., a partially completed task schedule) shows assigned time slots and the available tasks that still need to be allocated. The partially completed task schedule is employed by the options table generating subsystem 208. For each unassigned time slot (or open task), the options table generating subsystem 208 is configured to generate a list of possible tasks (analogous to Sudoku symbols) that may fit into that time slot, ensuring theallocation follows scheduling rules (such as time constraints, task dependencies, or worker availability). A complete options table shows valid task-to-time-slot assignments, aiding in the efficient and logical completion of the schedule.The one or more options tables are shown in below tables. Table 1:Table 2:The Table 1 illustrates a Sudoku puzzle board (where possible options of each empty cell to be filled), displaying the current Sudoku grid with filled and empty cells. The Table 2 represents the options table (i.e., a 3-dimensional table) that lists the possible numbers (or symbols) for each empty cell based on Sudoku rules. The options table is updated dynamically as the Sudoku puzzle progresses.The plurality of subsystems 114 further includes the factors identifying subsystem 210 that is communicatively connected to the one or more hardware processors 110. The factors identifying subsystem 210 is configured to identify one or more critical factors. The one or more critical factors may include at least one of: the two or more member sets being the intersection set of two adjacent valid components, the one or more critical intersections allow for smoother traffic flow across the entire network, the one or more critical nodes in the network for efficient usage of the network bandwidth, and the one or more critical tasks allow the entire workflow to be completed efficiently.For every instance (i.e., whenever updates are made to the Sudoku puzzle board and the options table), the factors identifying subsystem 210 needs to identify at least one two-member set, there can be more also, and it depends on a number of empty cells and symbol distribution. The two-member set is an intersection set of two nearest valid cells and the two nearest valid cells must be either in the same column or in the same row. The two nearest valid symbols must be a member of a three- member set or another two-member set, either in the same column or in the same row. The factors identifying subsystem 210 is configured to predict the valid symbol from the two-member set for the cell. During the process of solving the Sudoku puzzle, n number of two-member sets can occur but the factors identifyingsubsystem 210 can predict only for few cells (i.e., the first pre-determined number of components / cells). These few cells are called the key cells, and the rest of the cells (i.e., the second pre-determined number of components / cells) are filled by the cell-to-symbol tally procedure.In the case of the puzzle configured with a unique solution, then the maximum number of the key cells to be filled is N-l, where N is the number of unique symbols and these key cells form a single Hamiltonian path. In the case of multiple solutions, the number of the key cells to be filled is greater than N and the key cells are randomly distributed.In an exemplary embodiment, the factors identifying subsystem 210 is configured to identify the critical intersections (key intersections) that, once optimized, allow for smoother traffic flow across the entire network. The factors identifying subsystem 210 is configured to scan the traffic network (options table) to find the fewest key intersections where traffic signal optimization will have the greatest impact on overall flow. The factors identifying subsystem 210 is configured to analyze traffic data (e.g., vehicle counts and congestion levels) to locate the intersections that create bottlenecks or impact multiple other intersections. The factors identifying subsystem 210 is configured to identify if the Hamiltonian path exists through the key intersections where traffic flows smoothly along a specific sequence of green lights without interruption. The factors identifying subsystem 210 is configured to detect whether a continuous route may be optimized to allow vehicles to travel across multiple intersections with minimal delays. The factors identifying subsystem 210 provides a set of intersections where traffic signal adjustments may maximize traffic efficiency and a sequence of intersections that may be optimized for uninterrupted traffic flow if the Hamiltonian path is present.The factors identifying subsystem 210 is configured to identify the critical nodes in the network (key nodes) that, once allocated resources, enable efficient usage of network bandwidth or processing power. The factors identifying subsystem 210 is configured to scan the network (options table) to find the nodes where the resourceallocation will have the greatest effect on improving performance across the entire system 102. The key cell identification subsystem 208 is configured to analyse network traffic, bandwidth usage, and node load to locate the most important nodes (e.g., routers or servers) to optimize resource distribution. The factors identifying subsystem 210 is configured to identify the Hamiltonian path of the nodes where the resource allocation may be done in sequence to create an efficient data flow across the network. The factors identifying subsystem 210 is configured to determine whether a sequence of key nodes may be optimized so that resource distribution is maximized without redundancy or delays. The factors identifying subsystem 210 is configured to provide a prioritized list of nodes where the resource allocation may yield the highest network performance. The factors identifying subsystem 210 is configured to provide the sequence of nodes where the resources may be distributed efficiently for maximum impact.The factors identifying subsystem 210 is configured to identify critical tasks or time slots (key tasks) that, once scheduled, allow the entire workflow or schedule to be completed more efficiently. The factors identifying subsystem 210 is configured to scan the task schedule (options table) to find the most important tasks or time slots that must be filled to allow the entire schedule to be completed without conflicts. The factors identifying subsystem 210 is configured to analyse task dependencies, resource availability, and time constraints to locate the tasks that, when scheduled, will optimize the overall workflow. The factors identifying subsystem 210 is configured to identify whether the Hamiltonian path exists where the tasks may be scheduled in a specific sequence without causing delays or conflicts. The factors identifying subsystem 210 is configured to detect if a sequence of tasks or the time slots may be filled in a way that allows the schedule to flow smoothly from start to finish. The factors identifying subsystem 210 provides a prioritized list of tasks that need to be scheduled to unlock the rest of the task management process. The factors identifying subsystem 210 is configured to provide the sequence of tasks or the time slots that must be filled in order to complete the entire schedule efficiently.The plurality of subsystems 114 further includes the cell-to-symbol tally subsystem 212 that is communicatively connected to the one or more hardware processors 110. The cell-to-symbol tally subsystem 212 is configured with the cell-to-symbol tally procedure. The cell-to-symbol tally procedure is a procedure of filtering the invalid symbols from the sets. The cell-to-symbol tally procedure is a perfectly logical and cent percent error proof procedure. Except the key cells, all the cells are filled by the cell-to-symbol tally procedure. The cell-to-symbol tally procedure is a simple procedure of selecting the sets along each and every row, column, and block separately in such a manner that the number of sets selected needs to be equal to a number of unique symbols within the sets. Once the set count tallies with symbol count, it is confirmed that the cells of the sets and the symbols are bound to each other, therefore these symbols are safely removed from the rest of the sets.In other words, for filling the list of the one or more potential candidates in the second pre-determined number of components (i.e., rest cells) by the cell-to-symbol tally technique, the cell-to-symbol tally subsystem 212 is configured to select the one or more critical factors along each row, column, and block separately. The cell- to-symbol tally subsystem 212 is further configured to determine whether the selected one or more critical factors are equal to a number of the one or more potential candidates within the one or more critical factors. Upon determining that the selected one or more critical factors are equal to the number of the one or more potential candidates within the one or more critical factors, the cell-to-symbol tally subsystem 212 is further configured to predict the one or more components of the one or more critical factors that are bound to the number of the one or more potential candidates, to remove the number of the one or more potential candidates from one or more remaining critical factors.In an exemplary embodiment, the cell-to-symbol tally subsystem 212 is configured to filter out invalid signal timing options for each intersection, gradually optimizing the traffic flow. The cell-to-symbol tally subsystem 212 is configured to group the intersections into rows (streets), columns (traffic routes), and blocks (zones), and matches the number of intersections needing timing adjustments with possibletiming configurations. The cell-to-symbol tally subsystem 212 is configured to identify the intersections that may share similar traffic signal timings and groups the intersections, accordingly, ensuring that no conflicts arise from the signal timings at nearby intersections. Once a set of intersections and signal timings tally, the cell-to-symbol tally subsystem 212 is configured to eliminate those timings from other groups, narrowing down the possible configurations for the remaining intersections. The cell-to-symbol tally subsystem 212 is configured to remove incompatible signal timings from adjacent intersections to avoid conflicting signals and ensure smoother traffic flow.The cell-to-symbol tally subsystem 212 is configured to ensure that the number of intersections selected equals the number of possible signal timings, confirming the correct timing sequence for each intersection. The cell-to-symbol tally subsystem 212 is configured to verify that the final signal configuration across all intersections leads to optimal traffic movement without delays or gridlock. The cell-to-symbol tally subsystem 212 is configured to provide fewer possible timing configurations for each intersection as invalid options are filtered out. The cell-to-symbol tally subsystem 212 is configured to provide certain intersections that may have confirmed optimal signal timings that reduce traffic congestion.In an exemplary embodiment, the cell-to-symbol tally subsystem 212 is configured to filter out invalid resource allocation options for each node in the network, gradually leading to a more efficient allocation of resources. The cell-to-symbol tally subsystem 212 is configured to group network nodes (servers, routers) based on their resource needs and matches the number of nodes with the available resources. The cell-to-symbol tally subsystem 212 is configured to cluster the nodes that may share similar resource allocations (e.g., bandwidth or CPU power) without overloading or underutilizing any resource. Once a set of nodes and resource allocations tally, the cell-to-symbol tally subsystem 212 is configured to eliminate those resources from other groups, narrowing down the possible allocations for the remaining nodes.The cell-to-symbol tally subsystem 212 is configured to filter out resource options that conflict with other node groups to ensure an efficient and balanced network. The cell-to-symbol tally subsystem 212 is configured to ensure that the number of nodes selected matches the available resources, confirming the proper resource allocation for each node. The cell-to-symbol tally subsystem 212 is configured to confirm that the resource allocations across all nodes are optimized, without overallocating or creating bottlenecks. The cell-to-symbol tally subsystem 212 is configured to provide fewer possible resource allocation options for each node as invalid possibilities are filtered out. The cell-to-symbol tally subsystem 212 is configured to provide certain nodes that may have confirmed resource allocations that maximize network efficiency.In an exemplary embodiment, the cell-to-symbol tally subsystem 212 is configured to filter out invalid task assignment options for each time slot, gradually optimizing the task schedule. The cell-to-symbol tally subsystem 212 is configured to group the tasks by time slots, ensuring the number of available tasks matches the number of open slots. The cell-to-symbol tally subsystem 212 is configured to organize the tasks that may be scheduled within similar time slots or blocks of time, ensuring no time conflicts or overlaps. Once a set of tasks and time slots tally, the cell-to-symbol tally subsystem 212 is configured to eliminate those tasks from other time slots, narrowing down the possible assignments for the remaining tasks.The cell-to-symbol tally subsystem 212 is configured to remove conflicting task options from other time slots to avoid scheduling conflicts and ensure all tasks are allocated efficiently. The cell-to-symbol tally subsystem 212 is configured to ensure that the number of time slots selected equals the number of unique tasks, confirming the correct task assignment for each slot. The cell-to-symbol tally subsystem 212 is configured to verify that the tasks are scheduled in such a way that all time slots are filled optimally, and no tasks are left unscheduled. The cell-to-symbol tally subsystem 212 is configured to provide newer task assignment options for each time slot as invalid possibilities are filtered out. The cell-to-symbol tally subsystem 212is configured to provide certain time slots that may have confirmed task assignments, leading to a fully optimized schedule.Table 3:Table 4:The Table 3 illustrates the Sudoku puzzle board after filling the empty cells using the cell-to-symbol tally procedure. The Table 3 shows how certain cells in the Sudoku grid are populated based on the logical deductions made by the cell-to- symbol tally procedure. In some aspects, the filled cells represent the unique symbols determined through the cell-to-symbol tally procedure.The Table 4 depicts the options table (where invalid symbols / candidates are filtered by five stage process of cell-to-symbol tally technique) after filtering the invalid symbols using the cell-to-symbol tally procedure. The Table 4 shows the remaining valid symbol options for each cell in the Sudoku puzzle. In some cases, the cells with only one remaining option can be ready for immediate filling, while others may require further analysis or prediction steps. The Table 4 is configured with only one key cell [1,2] = {3,9}, and the valid symbol for this cell is 3. After the cell is updated, the entire Sudoku puzzle is solved by the cell-to-symbol tally procedure.Table 5:The Table 5 illustrates the options table for an 81 x 81 Sudoku grid, showcasing the cell-to-symbol tally procedure's effect on filtering the invalid symbols. The Table 5 is divided into three columns: “Column sets,” “Cell-to-symbol Tally”, and “Sets After Filter.” The "Column sets" represent the initial possible symbols for each cell, while the “Cell-to-symbol Tally” shows the sets identified by the procedure. The “Sets After Filter” column demonstrates the remaining valid options after applying the cell-to-symbol tally procedure. In this case, certain rows are highlighted, indicating that five symbols are bound to five cells, demonstrating the effectiveness of filtering, and reducing the number of possible options for each cell.In an exemplary embodiment, the factors identifying subsystem 210 is configured with the key cell identification procedure. If the cell-to-symbol tally procedure fails to update the options table, the factors identifying subsystem 210 is configured to search for the two-member set and attempt to guess the invalid symbol to create a singleton set. The implemented method is configured to guess the invalid symbol with condition priority, ranging from strongest condition to weaker condition. Prediction errors cannot be a significant concern, as the error rectification procedure will eliminate such errors if occur.The plurality of subsystems 114 further includes the error rectifying subsystem 214 that is communicatively connected to the one or more hardware processors 110. Before inserting a singleton symbol into the Sudoku puzzle board, the error rectifying subsystem 214 with the error check procedure is configured to check for constraint violation, if any violation is found appropriate error flag is raised. Based on the error flag, the error rectifying subsystem 214 with the error rectification procedure is configured to find error causing symbol and roll back data to that point, rectifying the error.The error rectifying subsystem 214 is configured to handle inconsistencies or the errors in traffic signal timing, ensuring that the signal coordination remains validacross intersections. The error rectifying subsystem 214 is configured to continuously verify that traffic signal rules are maintained, such as no conflicting signals in adjacent intersections (e.g., ensuring no two adjacent intersections are green at the same time if they share lanes). The error rectifying subsystem 214 is configured to monitor signal timing updates to ensure compliance with traffic regulations and flow constraints across the network. The error rectifying subsystem 214 is configured to identify any mistakes or contradictions in the signal timing, such as conflicts between timings at intersections or patterns that cause gridlock.The error rectifying subsystem 214 is configured to scan for timing overlaps or excessive delays that may lead to traffic congestion or signal inefficiencies. The error rectifying subsystem 214 is configured to adjust the traffic signal options table to a previous state if an error is found, recalculating optimal timings to ensure that the computer-implemented system 102 stays efficient. The error rectifying subsystem 214 is configured to resolve conflicting signal timings, recompute optimal flow, and update the signal timings accordingly. The traffic signal timing issues are resolved, ensuring smooth traffic flow. The options table is adjusted, with the invalid timings removed to ensure a valid, error-free configuration.In an exemplary embodiment, the error rectifying subsystem 214 is configured to handle the errors in the resource allocation, ensuring that all the nodes in the network receive the correct resources without conflicts or overloads. The error rectifying subsystem 214 is configured to continuously verify that network allocation rules (e.g., bandwidth, processing power, load balancing) are maintained, ensuring no duplication or over-allocation of the resources. The error rectifying subsystem 214 is configured to monitor network resource allocation updates and checks for rule violations such as exceeding bandwidth limits or unbalanced load distribution. The error rectifying subsystem 214 is configured to identify mistakes or contradictions in the resource allocation, such as two nodes receiving conflicting resources or exceeding capacity.The error rectifying subsystem 214 is configured to scan for allocation conflicts and resource bottlenecks in the network, ensuring no node is overburdened or underutilized. The error rectifying subsystem 214 is configured to adjust the network options table to correct resource allocation errors, ensuring optimal distribution. The error rectifying subsystem 214 is configured to resolve conflicting allocations, recalculate resource distribution, and update the nodes with a valid allocation plan. Resource allocation conflicts are resolved, ensuring balanced network performance. Adjustments are made to the options table, thereby removing invalid resource configurations.The error rectifying subsystem 214 is configured to manage scheduling errors, ensuring that the tasks are assigned to the time slots without conflicts or delays. The error rectifying subsystem 214 is configured to continuously verify that scheduling rules are maintained, ensuring no duplicate tasks in time slots or overlaps in resource assignments. The error rectifying subsystem 214 is configured to monitor the task scheduling process and check for conflicts such as double-booking of time slots or unallocated tasks. The error rectifying subsystem 214 is configured to identify scheduling mistakes, such as conflicting task assignments or unallocated tasks that violate scheduling rules.The error rectifying subsystem 214 is configured to scan the task schedule for conflicts or gaps, ensuring that all the tasks are assigned properly without overlap or missed deadlines. The error rectifying subsystem 214 is configured to adjust the task schedule to correct assignment errors, ensuring that the schedule remains valid and conflict-free. The error rectifying subsystem 214 is configured to reassign the tasks to the available time slots, resolve the conflicts, and update the schedule accordingly. Task assignment conflicts are resolved, ensuring a valid and efficient schedule. The adjustments made to the options table remove invalid task-time-slot pairings.In an exemplary embodiment, the computer-implemented system 102 is applicable for P vs. NP applications, where P stands for Polynomial Time, referring toproblems that are solved efficiently (i.e., in a reasonable amount of time). NP stands for Nondeterministic Polynomial Time, which refers to the problems for which finding the solution takes an exponential amount of time. The computer- implemented system 102 is configured to provide the solutions for the NP problems swiftly.In another exemplary embodiment, the computer-implemented system 102 is also applied to complex biological problems such as Deoxyribonucleic Acid (DNA) folding, which is crucial for understanding how genes express themselves and how mutations may lead to diseases like cancer. In the context of finding a right medicine for cancer treatment, the computer-implemented system 102 exhibits the numerous ways in which a DNA sequence folds, presenting millions of possible configurations. By using the cell-to-symbol tally procedure, the computer- implemented system 102 is configured to systematically eliminate invalid folding patterns, narrowing down to the most like configurations that cause cancerous behaviour. This would allow researchers to pinpoint specific areas for drug targeting, optimizing treatments, and potentially speeding up the discovery of personalized cancer therapies.In another exemplary embodiment, the computer-implemented system 102 is also configured to solve the Traveling Salesman Problem (TSP) and transportation planning by generating the options table of all possible routes between cities or locations. The computer-implemented system 102 then uses the cell to symbol tally procedure to filter out inefficient or invalid routes. This is done by identifying key points (such as critical cities or distribution hubs) and progressively narrowing down the possible paths. As the computer-implemented system 102 eliminates less optimal routes based on factors such as distance or cost, the computer-implemented system 102 converges on the shortest, most efficient route that connects all locations, thereby optimizing the overall travel or delivery plan.The computer-implemented system's 102 flexible approach, using the cell-to- symbol tally procedure, efficiently solves a wide range of complex, constraint-based problems. From optimizing the transportation routes to the resource allocation and scheduling the tasks, the computer-implemented system's 102 ability to eliminate the invalid options and focus on key solutions makes the computer- implemented system 102 adaptable to various real -world applications. The scalability of the computer-implemented system 102 ensures that the computer- implemented system 102 is configured to handle both small challenges and large- scale challenges with ease.FIG. 3A illustrates an exemplary first flow chart 300 A depicting a process for solving the arbitrary Nthrank Sudoku puzzles, in accordance with an embodiment of the present disclosure.The process begins with an initialization step, where the puzzle board and the options table are created based on the input Sudoku puzzle. The computer- implemented system 102 first attempts to find a single member set in the options table, as shown in step 302. If the single member set is found, the computer- implemented system 102 checks for the errors by looking for the identical sets in same at least one of the: row, column, and block, as shown in step 304. If no error is found, the symbol from the single member set is inserted into the Sudoku puzzle board and the options table is updated accordingly, as shown in step 306.The process then checks if the Sudoku puzzle is complete and if there are any missing symbol errors, as shown in step 308. If the Sudoku puzzle is not complete and there are no errors, the process proceeds to find hidden single symbols, as shown in step 310. If the hidden single symbol is found, the computer-implemented system 102 checks for the errors by looking for more than one hidden symbol for the same cell, as shown in step 312. If no error is found, the computer-implemented system 102 eliminates the invalid symbols and creates the single symbol set, as shown in step 314.Next, the process applies the cell-to-symbol tally procedure. Once the cell-to- symbol tally procedure is achieved, it is confirmed that the cells of the sets and thesymbols are bound to each other, thereby enabling the removal of the symbols from the rest of the sets, as shown in step 316.FIG. 3B illustrates an exemplary second flow chart 300B depicting the process for solving the arbitrary Nthrank Sudoku puzzles, in accordance with an embodiment of the present disclosure.If all the previous steps, as described in FIG. 3A, fail to solve the puzzle, then the second flow chart 300B is considered. The process moves to predict the symbol for the key cell, as shown in step 318. The key cells cannot be solved by the cell-to- symbol tally procedure alone and require additional steps for the resolution of the key cells. The computer-implemented system 102 is configured to provide the prediction of the valid symbols for the key cells, thereby facilitating the complete solution of the Sudoku puzzle.If any errors occur during the process, there is a step to rectify the error, as shown in step 320. The error rectification procedure ensures that the inserted symbols cannot violate the Sudoku constraints, and in the event of the violation, the computer-implemented system 102 enables the correction of any errors. This iterative process, with multiple decision points and error checks throughout, demonstrates a systematic approach to solving the Sudoku puzzles by progressively eliminating invalid options and filling in the cells based on logical deductions.In an embodiment, upon inserting the one or more candidates into the one or more components, the error rectifying subsystem 214 of the computer-implemented system 102 is configured to generate one or more unique identities being assigned to the one or more components, during the prediction. The error rectifying subsystem 214 of the computer-implemented system 102 is further configured to replace the one or more candidates by the one or more unique identifiers being assigned to the one or more components when one or more changes made to the one or more options table based on the error rectification technique. The one or more unique identifiers are configured to identify error causing predictions accurately apart from rollback of the data.In an embodiment, the error rectifying subsystem 214 with the error rectification technique is configured to identify one or more error causing candidates (symbols). The error occurs mainly due to excess options removal by a particular ID. The error rectifying subsystem 214 is configured to analyze the ID that removes maximum options from the options table to determine that particular ID as the wrong prediction ID.In another embodiment, the error rectifying subsystem 214 with the error rectification technique is further configured to rectify the one or more errors. Some times the error rectification procedure falls in an endless loop. In such case, both the symbols are invalid for that particular cells in which recent ID removed the symbol to make it a two symbol set is the error causing ID.In another embodiment, the error rectifying subsystem 214 with the error rectification technique is further configured to break one or more error loops, and provide one or more insights for one or more future predictions. Once the one or more error loops break, the particular set will become three symbol set, in which symbol added to make it three symbol set is the valid symbol and this symbol is stored in metadata table for future prediction.In an exemplary embodiment, for traffic signal optimization, the computer- implemented system 102 solves timing grids to determine optimal signal patterns. For traffic signal optimization, the computer-implemented system 102 measures the efficiency of signal timing adjustments. The computer-implemented system 102 verifies that timing patterns adhere to traffic flow rules.For resource allocation in networks, the computer-implemented system 102 solves resource distribution puzzles to ensure balanced and efficient resource allocation. For resource allocation in networks, the computer-implemented system 102 monitors the performance of resource distribution. The computer-implemented system 102 confirms that the resource distributions are optimal and conflict-free.For task management and scheduling, the computer-implemented system 102 resolves scheduling puzzles to optimize task assignments and avoid conflicts. For task management and scheduling, the computer-implemented system 102 tracks the effectiveness of task assignments and scheduling. The computer-implemented system 102 checks that the task assignments are correctly scheduled without overlaps or conflicts.FIG. 4 illustrates an exemplary flow diagram 400 depicting a sudoku generating process, in accordance with an embodiment of the present disclosure.At step 402, the sudoku generating process includes generating a solution with a spiral shift procedure. The spiral shift procedure is a simple and fastest procedure for generating a solution board of any size. In the spiral shift procedure, fill the left most column of the board with 1 to N numbers randomly, copy the entire column to the next column, shift downward, and copy the symbols outside the boundary to the top empty cells, this will continue till the right most column.In shift condition, if the previous column and present column are within the same stack (column of blocks) shift size must be “block size”, otherwise “just one cell” (as shown in Table 6). In another exemplary embodiment, the spiral shift procedure may be implemented in other ways too.Table 6:At step 404, the sudoku generating process includes a randomizing solution. In this step, shift the columns within the stack randomly, shift the rows within the bandrandomly, and shift the stacks randomly and shift the bands randomly (as shown inTable 7)Table 7:At step 406, the sudoku generating process includes implementing Sudoku properties. Henceforth, a solvable Sudoku puzzle board (as shown in Table 8) is created.Table 8:At step 408, the sudoku generating process includes creating the unique solution puzzle board. Most of the time the sudoku generating process creates multiple solutions puzzle boards. If the Sudoku puzzle is configured with multiple solutions, then the solved puzzle can end up with multiple solution options in the options table, i.e., all the sets are two-member sets, and all the symbols are members of twin sets. The computer-implemented system 102 adds some of the symbols to the Sudoku puzzle board to make all the sets of the options table empty sets.At step 410, the sudoku generating process includes key cell options. Zero key cells are configured with maximum initial symbols and result in the complete implementation of the Sudoku properties. For the zero key cells, the Sudoku puzzle is solved by the cell-to-symbol tally procedure alone. Less than N key cells are 5 configured with optimum initial symbols and result in the partial implementation of the Sudoku properties. Greater than N key cells are configured with the minimum number of the initial symbols. For greater than N key cell, the Sudoku is solvable, but the key cells are distributed randomly.Table 9:0The Table 9 represents data collected from the processes of generation, solving, and verification of the Sudoku puzzles. The computer-implemented system 102tracks the elapsed time, which can vary depending on the performance of machines. All standard Sudoku-solving procedures were used in gathering the data, with the exception of the error rectification procedure. The Sudoku puzzles were generated with maximum key cells and minimum initial symbols option, ensuring a more complex and challenging setup.FIG. 5 illustrates an exemplary graphical plot 500 depicting the relation between a rank of the Sudoku puzzle and a ratio of initial symbols to grid size, in accordance with an embodiment of the present disclosure.In an exemplary embodiment, the graphical plot 500 illustrates the relationship between the rank of the Sudoku puzzles and the ratio of initial symbols to grid size. The graphical plot 500 shows a curved line that starts at a lower percentage for lower ranks and increases steadily as the rank increases. In some aspects, the graphical plot 500 demonstrates that as the rank of the Sudoku puzzle increases, the percentage of the ratio of initial symbols to grid size also increases. The data is collected by generating rank 3 to rank 20 Sudoku puzzles in random order with minimum initial symbols option i.e., a key cell count is greater than N and it is being observed that as the size of the grid increases the percentage of the initial symbols also increase proportionately. This shows that after a certain size, the number of empty cells to be filled can be less than 1 percent.

[0002] Table 10:The world's hardest Sudoku puzzle (shown in Table 10) features 8 key cells, which are crucial for solving the Sudoku puzzle. These 8 key cells form a single Hamiltonian path, meaning the Sudoku puzzle is configured to be solved by sequentially filling in the key cells, following the unique Hamiltonian path. The Hamiltonian path connects the key cells in a way that ensures the entire Sudoku puzzle is solved logically without guesswork.FIG. 6 illustrates a flow chart illustrating a computer-implemented method 600 for determining the one or more constraint-based options in the one or more combinatorial applications using the cell-to-symbol tally technique, in accordance with an embodiment of the present disclosure.At step 602, the data associated with the one or more partially filled components, from the one or more data sources (i.e., the one or more databases 104). In an embodiment, the one or more partially filled components may include at least one of: the one or more cells in one or more grids, the one or more intersections, the one or more partially allocated network grids, and the one or more partially completed task schedules.At step 604, the one or more options tables are generated based on the obtained data associated with the one or more partially filled components.At step 606, the list of one or more potential candidates are generated for each empty component in the one or more components based on the one or more rules associated with the one or more combinatorial applications. The list of the one or more potential candidates may include at least one of: the list of potential symbols, the list of potential traffic signal timings, the list of potential resources, and the list of potential tasks.At step 608, the one or more critical factors are identified based on at least one of: the number of empty components and the distribution of the one or more candidates. The one or more critical factors may include at least one of: the two or more member sets being intersection set of two adjacent valid components, the one or more critical intersections allow for smoother traffic flow across the entire network, the one or more critical nodes in the network for efficient usage of the network bandwidth, and the one or more critical tasks allow the entire workflow to be completed efficiently.At step 610, the one or more valid candidates are predicted from the one or more critical factors for the first pre-determined number of components.At step 612, the one or more invalid candidates are filtered from the one or more critical factors based on the cell-to-symbol tally technique.At step 614, the list of the one or more potential candidates are filled in the second pre-determined number of components by the cell-to-symbol tally technique.Numerous advantages of the present disclosure may be apparent from the discussion above. In accordance with the present disclosure, the computer- implemented system 102 for determining the one or more constraint-based options in the combinatorial applications using the cell-to-symbol tally procedure is disclosed. The computer-implemented system 102 is configured to provide a scalable and efficient solution for solving the Sudoku puzzles of arbitrary sizes, from the standard 9x9 grid to larger Nth rank puzzles.This approach provides a significant advancement in the field of computational methods for solving logic puzzles, particularly the Sudoku puzzles of varying complexity. The computer-implemented system 102 is configured to manage the one or more constraint-based options using advanced techniques inspired by Sudoku solving. The computer-implemented system 102 is further configured to provide a scalable and adaptable approach to optimizing resource allocation, task scheduling, traffic control, and the like. This approach provides an efficient solution for complex, real-time scenarios, meeting the need for improved management systems capable of handling a wide range of constraints and dynamic conditions.The computer-implemented system 102 is further configured to predict the valid symbol for every empty cell, predict the invalid symbol with 100% accuracy, and filter the invalid symbol out. Only a few key cells need to be predicted by guessing or by reasonable methods. Even if a constraint violation error occurs, the computer- implemented system 102 is configured to rectify the constraint violation error. The time complexity to solve the Sudoku puzzle by this method even in the worst scenario is O(N4).While specific language has been used to describe the invention, any limitations arising on account of the same are not intended. As would be apparent to a person skilled in the art, various working modifications may be made to the method in order to implement the inventive concept as taught herein.The written description describes the subject matter herein to enable any person skilled in the art to make and use the embodiments. The scope of the subject matter embodiments is defined by the claims and may include other modifications that occur to those skilled in the art. Such other modifications are intended to be within the scope of the claims if they have similar elements that do not differ from the literal language of the claims or if they include equivalent elements with insubstantial differences from the literal language of the claims.The embodiments herein can comprise hardware and software elements. The embodiments that are implemented in software include but are not limited to, firmware, resident software, microcode, and the like. The functions performed by various modules described herein may be implemented in other modules or combinations of other modules. For the purposes of this description, a computer- usable or computer-readable medium can be any apparatus that can comprise, store, communicate, propagate, or transport the program for use by or in connection with the instruction execution system, apparatus, or device.The medium can be an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system (or apparatus or device) or a propagation medium. Examples of a computer-readable medium include a semiconductor or solid-state memory, magnetic tape, a removable computer diskette, a random-access memory (RAM), a read-only memory (ROM), a rigid magnetic disk and an optical disk. Current examples of optical disks include compact disk-read only memory (CD-ROM), compact disk-read / write (CD-R / W) and DVD.Input / output (VO) devices (including but not limited to keyboards, displays, pointing devices, and the like.) can be coupled to the computer-implemented system 102 either directly or through intervening VO controllers. Network adapters may also be coupled to the computer-implemented system 102 to enable the data processing system to become coupled to other data processing systems or remote printers or storage devices through intervening private or public networks. Modems, cable modem and Ethernet cards are just a few of the currently available types of network adapters.A representative hardware environment for practicing the embodiments may include a hardware configuration of an information handling / computer- implemented system 102 in accordance with the embodiments herein. The computer-implemented system 102 herein comprises at-least one processor or central processing unit (CPU). The CPUs are interconnected via the system bus 202 to various devices such as a random-access memory (RAM), read-only memory(ROM), and an input / output (I / O) adapter. The I / O adapter can connect to peripheral devices, such as disk units and tape drives, or other program storage devices that are readable by the computer-implemented system 102. The computer- implemented system 102 can read the inventive instructions on the program storage devices and follow these instructions to execute the methodology of the embodiments herein.The computer-implemented system 102 further includes a user interface adapter that connects a keyboard, mouse, speaker, microphone, and / or other user interface devices such as a touch screen device (not shown) to the bus to gather user input. Additionally, a communication adapter connects the bus to a data processing network, and a display adapter connects the bus to a display device which may be embodied as an output device such as a monitor, printer, or transmitter, for example.A description of an embodiment with several components in communication with each other does not imply that all such components are required. On the contrary, a variety of optional components are described to illustrate the wide variety of possible embodiments of the invention. When a single device or article is described herein, it will be apparent that more than one device / article (whether or not they cooperate) may be used in place of a single device / article. Similarly, where more than one device or article is described herein (whether or not they cooperate), it will be apparent that a single device / article may be used in place of the more than one device or article, or a different number of devices / articles may be used instead of the shown number of devices or programs. The functionality and / or the features of a device may be alternatively embodied by one or more other devices which are not explicitly described as having such functionality / features. Thus, other embodiments of the invention need not include the device itself.The illustrated steps are set out to explain the exemplary embodiments shown, and it should be anticipated that ongoing technological development will change the manner in which particular functions are performed. These examples are presented herein for purposes of illustration, and not limitation. Further, the boundaries of thefunctional building blocks have been arbitrarily defined herein for the convenience of the description. Alternative boundaries can be defined so long as the specified functions and relationships thereof are appropriately performed. Alternatives (including equivalents, extensions, variations, deviations, and the like, of those described herein) will be apparent to persons skilled in the relevant art(s) based on the teachings contained herein. Such alternatives fall within the scope and spirit of the disclosed embodiments. Also, the words "comprising," "having," "containing," and "including," and other similar forms are intended to be equivalent in meaning and be open-ended in that an item or items following any one of these words is not meant to be an exhaustive listing of such item or items or meant to be limited to only the listed item or items. It must also be noted that as used herein and in the appended claims, the singular forms “a,” “an,” and “the” include plural references unless the context clearly dictates otherwise.Finally, the language used in the specification has been principally selected for readability and instructional purposes, and it may not have been selected to delineate or circumscribe the inventive subject matter. It is therefore intended that the scope of the invention be limited not by this detailed description, but rather by any claims that issue on an application based here on. Accordingly, the embodiments of the present invention are intended to be illustrative, but not limiting, of the scope of the invention, which is set forth in the following claims.

Claims

I / WE CLAIM:

1. A computer-implemented method (600) for determining one or more constraint-based options in one or more combinatorial applications using a cell-to- symbol tally technique, the computer-implemented method (600) comprising: obtaining (602), by one or more hardware processors (110), data associated with one or more partially filled components, from one or more data sources, wherein the one or more partially filled components comprise at least one of: one or more cells in one or more grids, one or more intersections, one or more partially allocated network grids, and one or more partially completed task schedules; generating (604), by the one or more hardware processors (110), one or more options tables based on the obtained data associated with the one or more partially filled components; generating (606), by the one or more hardware processors (110), a list of one or more potential candidates for each empty component in one or more components based on one or more rules associated with the one or more combinatorial applications, wherein the list of the one or more potential candidates comprises at least one of: a list of potential symbols, a list of potential traffic signal timings, a list of potential resources, and a list of potential tasks; identifying (608), by the one or more hardware processors (110), one or more critical factors based on at least one of: a number of empty components and distribution of the one or more candidates, wherein the one or more critical factors comprise at least one of: two or more member sets being an intersection set of two adjacent valid components, one or more critical intersections allow for smoother traffic flow across an entire network, one or more critical nodes in a network for efficient usage of network bandwidth, and one or more critical tasks allow an entire workflow to be completed efficiently; predicting (610), by the one or more hardware processors (110), one or more valid candidates from the one or more critical factors for a first pre-determined number of components;filtering (612), by the one or more hardware processors (110), one or more invalid candidates from the one or more critical factors based on the cell-to-symbol tally technique; and filling (614), by the one or more hardware processors (110), the list of the one or more potential candidates in a second pre-determined number of components by the cell-to-symbol tally technique.

2. The computer-implemented method (600) as claimed in claim 1, wherein filling (614) the list of the one or more potential candidates in the second predetermined number of components by the cell-to-symbol tally technique, comprises: selecting, by the one or more hardware processors (110), the one or more critical factors along each row, column, and block separately; determining, by the one or more hardware processors (110), whether the selected one or more critical factors are equal to a number of the one or more potential candidates within the one or more critical factors; and upon determining that the selected one or more critical factors are equal to the number of the one or more potential candidates within the one or more critical factors, predicting, by the one or more hardware processors (110), the one or more components of the one or more critical factors that are bound to the number of the one or more potential candidates, to remove the number of the one or more potential candidates from one or more remaining critical factors.

3. The computer-implemented method (600) as claimed in claim 1, further comprising: generating, by the one or more hardware processors (110), the one or more options tables based on the data associated with the one or more components of the one or more combinatorial applications; determining (302), by the one or more hardware processors (110), whether a presence of a single critical factor in the one or more options tables;upon determining the presence of the single critical factor, determining (304), by the one or more hardware processors (HO), for one or more errors by identifying one or more identical critical factors in same at least one of: row, column, and block; upon no errors, inserting (306), by the one or more hardware processors (110), the one or more potential candidates into the one or more components; updating, by the one or more hardware processors (110), the one or more options tables upon inserting the one or more potential candidates into the one or more components; determining (308), by the one or more hardware processors (110), whether at least one of: the one or more components are complete and one or more missing candidate errors; determining (310), by the one or more hardware processors (110), a single hidden candidate when at least one of: the one or more components are incomplete and free candidate errors; upon determining the single hidden candidate, determining (312), by the one or more hardware processors (110), for one or more errors by identifying two or more hidden candidates for the one or more components; filtering (314), by the one or more hardware processors (110), the one or more invalid candidates from the one or more critical factors to generate a critical factor with a single candidate; and predicting (316), by the one or more hardware processors (110), the one or more components of the one or more critical factors that are bound to the number of the one or more potential candidates, to remove the number of the one or more potential candidates from one or more remaining critical factors, based on the cell- to-symbol tally technique.

4. The computer-implemented method (600) as claimed in claim 3, upon predicting the one or more components of the one or more critical factors that are not bound to the number of the one or more potential candidates, further comprising:predicting (318), by the one or more hardware processors (110), the one or more candidates for the one or more components; and determining (320), by the one or more hardware processors (110), whether the one or more candidates inserted into the one or more components, are violating the one or more rules associated with the one or more combinatorial applications, by correcting the one or more errors based on an error rectification technique.

5. The computer-implemented method (600) as claimed in claim 4, further comprising: upon inserting the one or more candidates into the one or more components, generating, by the one or more hardware processors (110), one or more unique identities being assigned to the one or more components; and replacing, by the one or more hardware processors (110), the one or more candidates by the one or more unique identifiers being assigned to the one or more components when one or more changes made to the one or more options table based on the error rectification technique, wherein the one or more unique identifiers are configured to identify error causing predictions accurately.

6. The computer-implemented method (600) as claimed in claim 5, wherein the error rectification technique is configured to at least one of: identify one or more error causing candidates, rectify the one or more errors, break one or more error loops, and provide one or more insights for one or more future predictions.

7. The computer-implemented method (600) as claimed in claim 1, wherein the one or more constraint-based options comprise at least one of: one or more numerals, one or more network resources, one or more time slots, and one or more traffic signal timings, and wherein the one or more combinatorial applications comprise at least one of: traffic signal optimization, resource allocation in network, scheduling and task management, and Sudoku puzzle.

8. A computer-implemented system (102) for determining one or more constraint-based options in one or more combinatorial applications using a cell-to- symbol tally technique, the computer-implemented system (102) comprising: one or more hardware processors (110); and a memory unit (112) coupled to the one or more hardware processors (110), wherein the memory unit (112) comprises a set of program instructions in form of a plurality of subsystems (114), configured to be executed by the one or more hardware processors (110), wherein the plurality of subsystems (114) comprises: a data obtaining subsystem (206) configured to obtain data associated with one or more partially filled components, from one or more data sources, wherein the one or more partially filled components comprise at least one of: one or more cells in one or more grids, one or more intersections, one or more partially allocated network grids, and one or more partially completed task schedules; an options table generating subsystem (208) configured to: generate one or more options tables based on the obtained data associated with the one or more partially filled components; and generate a list of one or more potential candidates for each empty component in one or more components based on one or more rules associated with the one or more combinatorial applications, wherein the list of the one or more potential candidates comprises at least one of: a list of potential symbols, a list of potential traffic signal timings, a list of potential resources, and a list of potential tasks; a factors identifying subsystem (210) configured to: identify one or more critical factors based on at least one of: a number of empty components and distribution of the one or more candidates, wherein the one or more critical factors comprise at least one of: two or more member sets being an intersection set of two adjacent valid components, one or more critical intersections allow for smoother traffic flow across an entire network, one or morecritical nodes in a network for efficient usage of network bandwidth, and one or more critical tasks allow an entire workflow to be completed efficiently; and predict one or more valid candidates from the one or more critical factors for a first pre-determined number of components; and a cell-to-symbol tally subsystem (212) configured to: filter one or more invalid candidates from the one or more critical factors based on the cell-to-symbol tally technique; and fill the list of the one or more potential candidates in a second pre-determined number of components by the cell-to-symbol tally technique.

9. The computer-implemented system (102) as claimed in claim 8, wherein in filling the list of the one or more potential candidates in the second pre-determined number of components by the cell-to-symbol tally technique, the cell-to-symbol tally subsystem (212) is configured to: select the one or more critical factors along each row, column, and block separately; determine whether the selected one or more critical factors are equal to a number of the one or more potential candidates within the one or more critical factors; and upon determining that the selected one or more critical factors are equal to the number of the one or more potential candidates within the one or more critical factors, predict the one or more components of the one or more critical factors that are bound to the number of the one or more potential candidates, to remove the number of the one or more potential candidates from one or more remaining critical factors.

10. The computer-implemented system (102) as claimed in claim 8, wherein the cell-to-symbol tally subsystem (212) is further configured to:generate the one or more options tables based on the data associated with the one or more components of the one or more combinatorial applications; determine whether a presence of a single critical factor in the one or more options tables; upon determining the presence of the single critical factor, determine for one or more errors by identifying one or more identical critical factors in same at least one of: row, column, and block; upon no errors, insert the one or more potential candidates into the one or more components; update the one or more options tables upon inserting the one or more potential candidates into the one or more components; determine whether at least one of: the one or more components are complete and one or more missing candidate errors; determine a single hidden candidate when at least one of: the one or more components are incomplete and free candidate errors; upon determining the single hidden candidate, determine for one or more errors by identifying two or more hidden candidates for the one or more components; filter the one or more invalid candidates from the one or more critical factors to generate a critical factor with a single candidate; and predict the one or more components of the one or more critical factors that are bound to the number of the one or more potential candidates, to remove the number of the one or more potential candidates from one or more remaining critical factors, based on the cell-to-symbol tally technique.

11. The computer-implemented system (102) as claimed in claim 10, further comprising an error rectifying subsystem (214) configured to: predict the one or more candidates for the one or more components; and determine whether the one or more candidates inserted into the one or more components, are violating the one or more rules associated with the one or morecombinatorial applications, by correcting the one or more errors based on an error rectification technique.

12. The computer-implemented system (102) as claimed in claim 11, wherein the error rectifying subsystem (214) is further configured to: upon inserting the one or more candidates into the one or more components, generate one or more unique identities being assigned to the one or more components; and replace the one or more candidates by the one or more unique identifiers being assigned to the one or more components when one or more changes made to the one or more options table based on the error rectification technique, wherein the one or more unique identifiers are configured to identify error causing predictions accurately.

13. The computer-implemented system (102) as claimed in claim 12, wherein the error rectifying subsystem (214) with the error rectification technique is configured to at least one of identify one or more error causing candidates, rectify the one or more errors, break one or more error loops, and provide one or more insights for one or more future predictions.

14. The computer-implemented system (102) as claimed in claim 8, wherein the one or more constraint-based options comprise at least one of one or more numerals, one or more network resources, one or more time slots, and one or more traffic signal timings, and wherein the one or more combinatorial applications comprise at least one of traffic signal optimization, resource allocation in network, scheduling and task management, and Sudoku puzzle.

Images