Network protection heuristic algorithm design of space division multiplexing elastic optical network

Abstract

The invention discloses a network protection heuristic algorithm for a space division multiplexing elastic optical network, which is used for improving the flexibility and the stability of the network. Under different network faults, the algorithm integrates an optimal adaptation spectrum allocation algorithm and a K shortest path algorithm, and efficiently allocates a working path, a protection path and corresponding spectrum resources for a service request. Through the algorithm, the problem of low efficiency of an integer linear programming model on a large-scale problem is effectively solved, and optimal balance between solution quality and calculation time is realized, so that a network can continuously provide stable and reliable services in the face of potential network faults. The method has an important reference value for designing and deploying a flexible and reliable space division multiplexing elastic optical network by a network operator.

Description

Technical Field

[0001] The present invention belongs to the field of optical network networking planning, and particularly relates to the design of a heuristic algorithm for network protection in a space-division multiplexing elastic optical network. Background Art

[0002] With the acceleration of the global informatization process and the popularization and development of cutting-edge technologies such as 5G technology, telemedicine, and autonomous vehicles, the demand for network services is experiencing an unprecedented surge. This growth in demand directly drives the urgent need for high-performance network transmission links. The large consumption of Internet content, the migration of enterprise-level applications to cloud platforms, and the wide deployment of Internet of Things devices have all significantly increased the dependence on network transmission capabilities and the number of links. This not only forces large-scale expansion and upgrade of existing network infrastructure but also promotes in-depth research on more efficient and reliable network transmission technologies. The main challenge faced by network operators and service providers is how to maintain a high-quality service level while expanding network capacity to cope with the huge data traffic demands from various fields. Such changes in the market and technology have set higher standards for network design and management, driving the necessity of technological innovation and network optimization and becoming a key factor in meeting future challenges.

[0003] Therefore, in the context of the rapid growth of current network demand, traditional optical network technologies face limitations in bandwidth and scalability and are difficult to meet the high-speed and large-capacity requirements of future data transmission. To address this challenge, the technology of space-division multiplexing elastic optical network has been developed, which represents a major innovation in optical transmission technology. According to Shannon's law, the transmission rate of a single optical fiber has a fixed upper limit, and exceeding this limit will inevitably lead to communication errors and data loss. The space-division multiplexing elastic optical network uses multi-core fiber technology in a single optical cable to parallelly transmit multiple independent optical signal channels, greatly increasing the transmission capacity and efficiency of the network. Moreover, through space-division multiplexing technology (Space Division Multiplexing), multi-dimensional manipulation of optical signals is achieved, which includes the utilization of independent cores and modes, and each core can be regarded as an independent transmission channel. The application of this technology significantly improves the total throughput of the optical fiber while maintaining compatibility with existing optical network devices, facilitating the smooth upgrade of the network. At the same time, the space-division multiplexing elastic optical network combines advanced elastic optical network technology (Elastic Optical Network), which enables the network to support efficient spectrum utilization, optimizes the allocation of spectrum resources, and reduces the problem of spectrum fragmentation. Therefore, the space-division multiplexing elastic optical network provides a powerful tool for network operators to cope with the challenges of future data traffic surges and also promotes the development of the entire communication industry.

[0004] While improving the transmission rate of the network, the security and reliability of the network also deserve attention. Although high-speed data transmission improves efficiency, it also amplifies potential security risks and stability challenges. Network failures and data loss may cause more serious consequences in a high-speed network environment. Therefore, as the network capacity and speed increase, strengthening network protection technology becomes particularly important. For this reason, it is crucial to design efficient network protection technology to ensure that the network is not only fast but also stable and secure. By introducing technologies such as Shared Backup Path Protection, it is possible to quickly switch to a pre-set protection path in the event of a link failure, thus ensuring the continuity of data transmission and the uninterrupted network service. Such a protection strategy ensures that while improving network performance, it also greatly enhances the network's security and overall reliability, bringing confidence and guarantee to users and service providers. However, in real network failures, the types of network failures are very complex, such as fiber optic failures, node failures, etc. These failures will make the shared spectrum conditions in the shared protection path technology very complex, and the difficulty of routing and spectrum allocation in the protection path also increases significantly.

[0005] In this case, designing an efficient network protection heuristic algorithm to effectively deal with complex network failure types becomes the core task of ensuring the transmission reliability of the space-division multiplexing elastic optical network. This algorithm needs to be able to anticipate and adapt to changing network conditions while maintaining the efficient operation of the network under various failure scenarios. By comprehensively considering the solution accuracy and operation efficiency of the algorithm, the algorithm can optimize network resources, thus ensuring that the network can continuously provide stable and reliable services in the face of potential network failures. Summary of the Invention

[0006] During the data transmission process of the space-division multiplexing elastic optical network, force majeure factors such as failures of the fiber core, site, or optical cable may cause service requests to be interrupted, significantly reducing the stability of data transmission and causing huge losses to daily life and production. To address this problem, the present invention develops a network protection heuristic algorithm for the space-division multiplexing elastic optical network. This algorithm comprehensively uses the best-fit spectrum allocation algorithm and the K-shortest path algorithm to provide effective protection for the network under different network failures. This algorithm not only has a small gap in solution accuracy from the exact solution calculated by integer linear programming but also greatly improves its solution speed.

[0007] The present invention relates to a network protection algorithm for space-division multiplexing elastic optical networks, aiming to improve network performance through a series of innovative algorithm steps. First, we define and analyze the topology graph of the network to understand its structural characteristics in detail. Then, all service requests are sorted in descending order according to the size of the service demand for routing and spectrum resource allocation. To efficiently solve the spectrum allocation problem of service requests in the link, the present invention proposes an optimal fit spectrum allocation algorithm, which can allocate continuous and available spectrum resources for all service requests in their working paths and protection paths, significantly improving the utilization rate of spectrum resources and the solution efficiency of the algorithm. In addition, the K shortest path algorithm calculates the candidate paths of the K working paths and protection paths under the corresponding conditions to optimize the path selection of service requests and select the most suitable working path and protection path for them.

[0008] Finally, by performing data simulation and comparing and analyzing the effects of the algorithm, the specific steps are as follows:

[0009] S1. First, analyze the source points, destination points, and demand sizes of each service request, and sort all service requests in descending order according to the demand size of the service requests.

[0010] S2. Propose an optimal fit spectrum allocation algorithm to allocate continuous and identical available spectrum segments on the working paths and protection paths of all service requests.

[0011] S3. According to the candidate paths of the K working paths calculated by the K shortest path, on the premise of ensuring the reachability of the working optical paths of all service requests, apply the optimal fit spectrum allocation algorithm to allocate spectrum resources with the goal of minimizing the maximum spectrum index used by the working path. Use the rerouting strategy to iteratively optimize its allocation results.

[0012] S4. According to different network fault types, use the K shortest path algorithm to calculate the candidate paths of the K protection paths for all service requests. According to whether there are common links in the working paths, apply the optimal fit spectrum allocation algorithm to perform routing, fiber core, and spectrum allocation for the protection paths of all service requests.

[0013] S5. Perform data simulation, compare and analyze the solution accuracy and solution efficiency of the algorithm under different network faults in different network topologies and different service request matrices, and verify the effectiveness of the algorithm.

[0014] In S2, in order to reduce spectrum fragmentation and improve search efficiency, so as to find the best spectrum selection scheme for the working path and standby path of the request, an optimal fit spectrum allocation algorithm is designed and proposed. The steps of this algorithm are as follows:

[0015] A1. Spectrum Continuity Check: According to the spectrum requirements of the service request, check whether there is continuous spectrum on each core of each link in the selected path, and output the corresponding (flag2, info2) based on a certain core c on a certain link e of the input. Here, flag2 is a boolean variable, flag2 = 1 indicates that there is continuous spectrum meeting the service requirements, and flag2 = 0 indicates that there is no continuous spectrum meeting the service requirements. info2 is an integer variable used to mark the maximum spectrum index traversed during this search process, that is, the next smallest available starting spectrum index in the link during this traversal.

[0016] A2. Select Single-Mode Optical Fiber for Transmission in the Link: According to the output of step A1, select an available core for each link in all the input paths and output the corresponding (flag3, info3). If there are multiple available cores in a link, the core with a smaller index is preferentially selected. Here, flag3 is a boolean variable, flag3 = 1 indicates that there is at least one available core among all cores C, and flag3 = 0 indicates that all cores in C are unavailable. info3 is an integer variable defined as the minimum value of info2 obtained from all the links traversed in the selected path, that is, the next traversal can start from the spectrum index info3 without having to search from the beginning, so as to reduce duplicate and redundant calculations.

[0017] A3. Find the Range of Available Continuous Spectrum FS: Search for the available continuous spectrum resources for the service request r on each link in the selected path, and repeat steps A1 and A2 until continuous spectrum segments that are available, the same, and meet the service requirements are found on the cores of all links.

[0018] A4. Update Spectrum Usage: Update the spectrum usage of the network according to the output path of the algorithm and the corresponding spectrum allocation results.

[0019] The allocation of the working path for the service request in S3 specifically includes the following steps:

[0020] B1. Service Request Sorting: First, sort all service requests from large to small according to their requirements.

[0021] B2. Determine the Working Path: For each sorted request r, use the K shortest path algorithm to calculate K shortest paths as the working path candidates. Then, select the shortest path with available spectrum among the candidate paths as the working path according to the best-fit spectrum allocation algorithm.

[0022] B3. Spectrum Allocation for the Working Path: For a certain candidate path, apply the best-fit spectrum allocation algorithm for calculation. If there is a set of available core allocations AG rand the corresponding spectrum range F r , then use this candidate path as the working path for this service request, and at the same time allocate all the spectra within the spectrum range F r = [f s , f s + t r - 1] to this candidate path.

[0023] B4. Update and record: After completing the allocation of the working path for the service requirement, update the network spectrum usage T and record the maximum used spectrum index of the working path In addition, it is also necessary to record the allocation situation W of the working path for the service requirement for reference when allocating the protection path.

[0024] B5. Iterative update: Reroute the request r that has the greatest impact on the maximum used spectrum index of the target working path * and check whether the target is improved. If it is improved, update the corresponding information; if not, continue to iterate until the number of iterations reaches the threshold.

[0025] The allocation of the protection path for the service request in S4 is specifically carried out in the following steps:

[0026] C1. Sorting of service requests: First, sort all service requests from largest to smallest according to their requirements.

[0027] C2. Calculation of candidates for protection paths: For each sorted request r, according to the available links and nodes in the network under different network failures, and on the premise of ensuring that there are no common links with its working path, use the K - shortest path algorithm to calculate K shortest paths as candidates for the protection path. Since different types of network failures may affect different links and nodes, the set of available links and nodes when determining the protection path will also be different.

[0028] C3. Allocation of the protection path for the first request: First, select the shortest path among its candidate paths for the service request with the largest demand as the protection path, and apply the best - fit spectrum allocation algorithm to allocate the corresponding spectrum resources according to its demand, update the maximum spectrum index used by the protection path and the total spectrum f bp reserved for the protection path, and add this request to the set R′ of allocated requests.

[0029] C4. Remaining protected path allocation: For the current service request r to be allocated, check whether there are common links / nodes / shared risk link groups between its working path and the working path of the already allocated service request r'∈R'. If so, apply the best-fit spectrum allocation algorithm to find the shortest path with available spectrum segments as the protected path for this request; if not, select the path with the most common links with the protected path of r' from the K candidate paths as its protected path. After the allocation is completed, add this request to the set R' of already allocated requests.

[0030] C5. Spectrum allocation for the protected path: For a certain candidate path, apply the best-fit spectrum allocation algorithm for calculation. If there is a set of available core allocations AG r and the corresponding spectrum range F r , then take this candidate path as the working path of this service request, and at the same time allocate all the spectra in the spectrum range F r =[f s , f s +t r -1] in the network to this candidate path.

[0031] C6. Update and record: After completing the allocation of the protected path for the service requirements, update the network spectrum usage T and record the maximum used spectrum index of the working path and the total spectrum f bp reserved for the working path.

[0032] Advantages of the present invention: The network protection heuristic algorithm for the space-division multiplexing elastic optical network proposed by the present invention aims to enhance the network's ability to cope with diverse and unpredictable network failures. This method formulates corresponding network protection strategies according to different network failures and service requirements by comprehensively applying the best-fit spectrum allocation algorithm and the K shortest path algorithm. The best-fit spectrum allocation algorithm improves the efficiency of searching for available spectra, reduces spectrum fragmentation, and improves spectrum utilization. The setting of the working path and the protected path applies the K shortest path algorithm, which not only greatly improves the solution efficiency but also ensures the solution quality. Through the integration of the above algorithms, the present invention can effectively cope with various network failures that may occur in the space-division multiplexing elastic optical network - core failures, link failures, node failures, and shared risk link group failures. In addition, this method realizes the trade-off between the solution accuracy and the solution efficiency, ensuring that the network can quickly recover in case of any failure, guaranteeing the continuity and reliability of critical data transmission, and thus significantly improving the overall stability of the network. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] Figure 1 It is a schematic diagram of the best-fit spectrum allocation algorithm applied in the present invention.

[0034] Figure 2 Pseudo-code schematic diagram of the optimal adaptation spectrum allocation algorithm applied in the present invention.

[0035] Figure 3 Pseudo-code schematic diagram of the decision-making working path algorithm applied in the present invention.

[0036] Figure 4 Pseudo-code schematic diagram of the decision-making protection path algorithm applied in the present invention.

[0037] Figure 5 Network topology diagram for data simulation of the present invention.

[0038] Figure 6 Schematic diagram of the result comparison of the maximum spectrum index used for the working path and protection path of the present invention.

[0039] Figure 7 Schematic diagram of the result comparison of the total reserved spectrum of the protection path of the present invention. Figure 8 Abstract drawing of the present invention. Detailed implementation manners

[0040] For the convenience of those skilled in the art to understand the technical content of the present invention, the present invention will be further described in detail below with reference to the accompanying drawings and specific examples.

[0041] Figure 1 Schematic diagram of the optimal adaptation spectrum allocation algorithm applied in the present invention. As Figure 1 (a) shows, a simple example is used to illustrate the algorithm for the convenience of readers. A service request r with a demand size of 4 spectra is transmitted from node A to node D in a chain network, where each link has two available cores. In the figure, green indicates that the spectrum has been occupied, and white indicates that the spectrum is not occupied. Check whether there are 4 available, continuous and consistent spectrum segments on the cores of each link of this path. For link AB, the spectra [1, 2, 3, 4] on core 1 and core 2 can be allocated; therefore, for both core 1 and core 2, (flag2 = 1, info2 = 4) is output, where flag2 = 1 indicates that there is a continuous spectrum segment that meets the service demand size, and info2 = 4 indicates the maximum spectrum index under this condition, that is, 4 in the spectra [1, 2, 3, 4]. According to the outputs of core 1 and core 2 of link AB, an available core is selected for the service request. If there are multiple available cores in a link, as shown in link AB, we preferentially select the first optical fiber, and its output is (flag3 = 1, info3 = 1); on the contrary, if there is no available core, as shown in link CD, then we will set the starting spectrum index f of the spectrum FS startSet to 3, which is the next smallest available starting spectrum index, so the output is (flag3 = 0, info3 = 3). Then, we continue to search for spectrum segments that meet the service requirements on all links of the path starting from the 3rd spectrum. Repeat the above steps until a continuous spectrum that is available on the optical fibers of all links and meets the request r for transmission is found. As Figure 1 As shown in (b), there are the same and continuous spectrum segments [3, 4, 5, 6] on core 2 of link AB, core 1 of link BC, and core 1 of link CD.

[0042] Figure 2 It is a schematic diagram of the pseudo-code of the best-fit spectrum allocation algorithm applied in the present invention.

[0043] Figure 3 It is a schematic diagram of the pseudo-code of the decision-making working path algorithm applied in the present invention.

[0044] Figure 4 It is a schematic diagram of the pseudo-code of the decision-making protection path algorithm applied in the present invention.

[0045] Figure 5 It is a network topology diagram for data simulation in the present invention. As Figure 5 shown, the network topology for data simulation in the present invention includes the US national backbone network with 17 nodes and 21 links, and the European optical network with 28 nodes and 34 links.

[0046] Figure 6 It is a schematic diagram of the result comparison of the maximum spectrum index used by the working path and the protection path in the present invention. As the number of requests increases, the gap between the solution obtained by the heuristic algorithm proposed in the present invention and the optimal solution increases from about 0% to 9.1%. However, the heuristic algorithm proposed in the present invention can solve the problem in less than 2 seconds, which is far less than the execution time of the mixed-integer linear programming model. In addition, the heuristic algorithm proposed in the present invention is superior to the previous K shortest path and first-fit algorithms in terms of both the maximum spectrum index usage and the execution time. When the number of requests increases to 500 and 1000, due to memory limitations, it is difficult for the mixed-integer linear programming model to solve, so the optimal solution cannot be found. Compared with the previous K shortest path and first-fit algorithms, the heuristic algorithm proposed in the present invention requires a longer but acceptable execution time; however, it performs better in terms of the maximum spectrum index usage. In addition, for the same request sequence, regardless of the scale of the number of requests, the European optical network uses more spectrum resources than the US national backbone network.

[0047] Figure 7 It is a schematic diagram of the result comparison of the total reserved spectrum of the protection path in the present invention. As Figure 7As shown, the results of the total reserved spectrum for various types of network failures under different network topologies by the mixed-integer linear programming method and the heuristic algorithm proposed in the present invention. The horizontal axis represents the number of transmission requests, and the vertical axis represents the number of backup spectrum slices. The solid line represents the optimal result obtained from the mixed-integer linear programming model, and the dashed line represents the result of the heuristic algorithm proposed in the present invention. Regardless of the network failure type or network topology, as the number of requests increases, the total reserved spectrum resources also increase. In particular, the network needs to reserve the most backup spectrum resources for shared risk link group failures, while the protection path for core failures requires the least amount of spectrum resources. This is because in a certain network, for the same transmission request matrix, shared risk link group failures affect the working paths of the most requests, while core failures have the least impact. At the same time, link failures and node failures have a more significant impact on the working paths of requests than core failures, but less than shared risk link group failures, and the impacts of the two are almost equivalent. However, more spectrum resources are reserved for link failures than for node failures because the number of links in the network is more than the number of nodes, meaning that a single link failure affects more requests. In addition, due to the larger number and more concentrated distribution of network nodes and links in the European optical network, the reserved spectrum quantity is significantly more than that of the US national backbone network. In the European optical network and the US national backbone network, the difference between the feasible solution obtained by the algorithm proposed in the present invention and the optimal solution of the mixed-integer linear programming model is approximately 1.3% to 5.8%

[0048] Those of ordinary skill in the art will realize that the embodiments described herein are for helping the reader understand the principles of the present invention and should be understood that the scope of protection of the present invention is not limited to such specific statements and embodiments. For those skilled in the art, various changes and modifications can be made to the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included within the scope of the claims of the present invention.

Claims

1. A network protection heuristic algorithm design for space-division multiplexed elastic optical networks, characterized in that: Optimize working routing decisions, protection path configuration, and spectrum resource allocation under different network topologies and network fault conditions. This design aims to cope with the diverse faults that the network may encounter and the uncertainty of business needs, thereby significantly improving the overall robustness of the network. This method integrates the best fit spectrum allocation algorithm (Best Fit) and the K shortest path algorithm (K Shortest Path) to efficiently allocate working paths and protection paths and corresponding spectrum resources to business requests. Through this algorithm, the problem of low efficiency of integer linear programming models in large-scale path selection, core utilization, and spectrum allocation problems (Routing, Space, and Spectrum Assignment) is effectively solved, achieving the best balance between solution quality and computing time. Finally, data simulation is used to verify the effectiveness and feasibility of the proposed algorithm.

2. According to claim 1, the network protection heuristic algorithm design of the space-division multiplexing elastic optical network is characterized in that: The protection path of each service request in the network is determined according to the setting of the working path to ensure that there is no common link between the working path and the protection path. The algorithm first selects a path from the K candidate paths calculated by the K shortest path algorithm as the working path of the service request and allocates the corresponding spectrum resources. Then, the routing and spectrum resource allocation of the protection path are determined according to the output of the working path. Finally, the rerouting strategy is used to iteratively optimize the decision of the working path and the protection path.

3. According to claim 2, the network protection heuristic algorithm design of the space-division multiplexing elastic optical network is characterized in that: The network topology is defined as a bidirectional acyclic directed graph G containing |V| nodes and |E| links, where each link consists of |C| single-mode optical fibers. The service requests in the network are sorted in descending order according to their demand size to form a service request sequence R. According to the overall network topology and the source and destination of each request, the K shortest path algorithm is applied to calculate K shortest paths as candidate paths for its working path. The spectrum resources provided by each single-mode optical fiber are |F|=320 spectrum slices (FS).

4. According to claim 3, the network protection heuristic algorithm design of the space-division multiplexing elastic optical network is characterized in that: The overall spectrum resource usage of the network is defined as a three-layer dictionary T = {e:c:f:0} (e∈E, c∈C, f∈F), which defines the usage of each spectrum f in each core c on each link e in the network, where 0 represents unused and 1 represents used. The working path allocation for each request is defined as a single-layer dictionary Its content includes the link e that the request working path passes through, the core AG used by the request on the link r [e] and the spectrum segment F allocated for the request r .

5. According to claim 4, the network protection heuristic algorithm design of the space-division multiplexing elastic optical network is characterized in that: K candidate paths are selected in order from smallest to largest path length. For a selected path Spectrum requirements based on service requests r As well as the current network spectrum usage T, the best adaptation spectrum allocation algorithm is applied to output the core allocation AG for the request r And the corresponding spectrum segment F r .

6. The network protection heuristic algorithm design of a space-division multiplexed elastic optical network according to claim 5 is characterized in that: The best adaptive spectrum allocation algorithm includes two-stage selection - link core selection and spectrum segment selection. By iteratively updating the latest search position, the best link core and spectrum resources are allocated to each service request according to its working path, thereby reducing repeated and redundant searches, improving the efficiency of the spectrum allocation algorithm and reducing spectrum fragmentation.

7. The network protection heuristic algorithm design of a space-division multiplexed elastic optical network according to claim 6 is characterized in that: The best adaptive spectrum allocation algorithm is based on the selected candidate path Allocate the best spectrum segment for the service request. If it exists, define the path as the requested working path and output the corresponding core and spectrum allocation plan; if it does not exist, traverse the next path among the K candidate paths and apply the best adaptation spectrum allocation algorithm until an available spectrum segment is found. After the search is completed, update the current network spectrum usage T according to the routing and resource allocation plan of the working path.

8. The network protection heuristic algorithm design of a space-division multiplexed elastic optical network according to claim 7, characterized in that: After obtaining the allocation result W of the service request working path and the updated current network spectrum usage T, corresponding protection paths are allocated to all service requests in the request sequence R according to different network failure types.

9. A network protection heuristic algorithm design for a space-division multiplexed elastic optical network according to claim 8, characterized in that: Apply the K shortest path algorithm to calculate K paths that have no common links with its working path for each request r as candidate paths for its protection path The K shortest path algorithm uses different links and nodes when calculating routes under different network failures. Specifically, for a link core failure, the core on the link in the network cannot be used by the protection path. For a link failure, the link and all the cores on it in the network cannot be used by the protection path. For a node failure, the node, the links that the node passes through, and all the cores on the link in the network cannot be used by the protection path. For a shared risk link group failure, all links in the group, all nodes that the links pass through, and all the cores on the links in the network cannot be used by the protection path.

10. A network protection heuristic algorithm design for a space-division multiplexed elastic optical network according to claim 9, characterized in that: For a certain request service r, K candidate paths are selected in order of path length from small to large. Before selecting a candidate protection path, first determine whether the working path of the request r has a common link / node / shared risk link group with the working paths of all previously allocated service requests. If so, select the shortest path among the K candidate paths as the protection path of the service request, and apply the best adaptation spectrum allocation algorithm to allocate corresponding spectrum resources to the protection path; if not, select the path among the K paths that has the most link overlaps with the allocated request working path as its protection path, and apply the best adaptation spectrum allocation algorithm to allocate corresponding spectrum resources to the protection path. After the search is completed, update the current network spectrum usage T according to the routing and resource allocation scheme of the protection path.

11. The network protection heuristic algorithm design of a space-division multiplexing elastic optical network according to claim 10, characterized in that: Under different network topologies and different numbers of service requests, the effectiveness of the proposed algorithm is verified by comparing the maximum spectrum index used by the working path and the algorithm calculation time. Under different network topologies, different numbers of service requests and different network failures, the effectiveness of the proposed algorithm is verified by comparing the maximum spectrum index used by the protection path, the total number of spectrum reserved for the protection path and the algorithm calculation time. At the same time, it is compared with other classic algorithms to reflect the performance advantages of the algorithm.

Images