Method and device for determining independent set on alternating group graph, equipment and storage medium

Abstract

This invention discloses a method, apparatus, device, and storage medium for determining independent sets on an alternating group graph, comprising: determining an n-dimensional alternating group graph AG. n All vertices of AG; according to the vertex labeling rules, AG n The process involves recursively partitioning the data until it is divided into several AG4s. All vertices of all obtained AG4s are placed into set T. From set T, all vertices of one AG4 are removed and placed into set X. From set X, a vertex x is removed and placed into set S, ensuring that vertex x is not adjacent to any vertex in set S. All neighboring vertices of vertex x in the current AG4 and other AG4s in set T are obtained, and all corresponding neighboring vertices are deleted from sets X and T. This process is repeated until set T is empty. The independent set S obtained through this invention can be applied to interconnected networks to help optimize data transmission and resource allocation strategies, thereby improving network performance and efficiency.

Description

Technical Field

[0001] This invention belongs to the field of computer network technology, and specifically relates to a method, apparatus, device and storage medium for determining independent sets on an alternating group graph. Background Technology

[0002] High-performance parallel computers are playing an increasingly important role in scientific research, education, petroleum, meteorology, and other related fields. As the performance of high-performance parallel computers continues to improve, the number of processors they possess is becoming increasingly large. A network formed by connecting several processors in a specific way is called an interconnection network. An interconnection network can be represented by a simple graph G = (V, E), where V(G) represents the set of vertices and E(G) represents the set of edges. The vertices in graph G represent the processor nodes in the interconnection network, while the edges represent the connections between processor nodes.

[0003] Alternating Group Chart AG n Alternating group graphs are a typical interconnected network topology. They possess many superior properties, such as vertex transitivity, edge transitivity, strong hierarchy, high connectivity, small network diameter, and small average distance. Due to these advantages, alternating group graphs have attracted increasing attention.

[0004] For a subset S of vertices in a graph, if there is no edge between any two vertices (i.e., they are not adjacent), then set S is called an independent set of the graph. An independent set is a subset of the vertex set of an alternating group graph where no two vertices are adjacent to each other. In other words, an independent set is the set of non-adjacent vertices in the graph.

[0005] like Figure 1 In G, S = {V1, V5, V7} is an independent set of G, where vertices v1, v5, and v7 are not adjacent to each other.

[0006] Currently in alternating group map AG n There is currently no solution for independent sets. Summary of the Invention

[0007] To address the aforementioned technical problems, this invention proposes a method, apparatus, device, and storage medium for determining independent sets on alternating group graphs.

[0008] To achieve the above objectives, the technical solution of the present invention is as follows:

[0009] In a first aspect, the present invention discloses a method for determining independent sets on an alternating group graph, comprising:

[0010] Step S1: Determine the n-dimensional alternating group graph AG nAll vertices of n, n≥4;

[0011] Step S2: According to the vertex labeling rules, transform the n-dimensional alternating group graph AG n The recursive partitioning is performed step by step until it is divided into several alternating group graphs AG4;

[0012] Step S3: Put all vertices of all alternating group graphs AG4 obtained in step S2 into set T;

[0013] Step S4: Take all vertices of an alternating group graph AG4 from set T and put them into set X;

[0014] Step S5: Take a vertex x from set X and put it into set S, and vertex x is not adjacent to any vertex in set S;

[0015] Get all neighboring vertices of vertex x in the current alternating group graph AG4 and other alternating group graphs AG4 in set T, and delete all corresponding neighboring vertices in set X and set T;

[0016] Step S6: Repeat step S5 until set X is empty, then continue to execute step S7;

[0017] Step S7: Repeat steps S4-S6 until all alternating group graphs AG4 in set T have been evaluated and set T is empty. The final set S is the n-dimensional alternating group graph AG. n An independent set.

[0018] Based on the above technical solution, the following improvements can be made:

[0019] As a preferred embodiment, step S2 includes the following:

[0020] First, the n-dimensional alternating group graph AG n Vertex set V(AG) n ) = A n A n for <n>The set of all even permutations of the vertex, where each vertex is labeled with a number of length n: p1p2...p n For example, n≥4;

[0021] Secondly, according to the nth position of the vertex label p n AG of n-dimensional alternating group graph n Each vertex is assigned to an alternating group graph AG of n. n-1 In the diagram, there are n alternating group graphs AG. n-1 Each by It means, and The last digit of each vertex label is i, and l≤i≤n;

[0022] Then, according to the (n-1)th vertex label p n-1 AG of each n-1 dimensional alternating group graph n-1 Each vertex is assigned to an n-1 alternating group graph AG n-2 In, and each AG n-2 The (n-1)th position of the vertex label is the same;

[0023] Repeat the above recursive steps until each vertex of each alternating group graph AG5 is assigned to one of the five alternating group graphs AG4, and the fifth bit of the vertex label of each AG4 is the same, thus achieving partitioning.

[0024] As a preferred option, step S3 specifically involves:

[0025] Step S3: Place all vertices of the alternating group graph AG4 obtained in step S2 into set T with vertex labels.

[0026] As a preferred option, in step S5...

[0027] pass and The operation obtains all neighboring vertices of vertex x in the current alternating group graph AG4 and other alternating group graphs AG4 in set T;

[0028] in: This represents the permutation obtained by rotating the first, second, and i-th elements of vertex x by one position clockwise.

[0029] This represents the permutation obtained by rotating the first, second, and i-th elements of vertex x counterclockwise by one position.

[0030] 3≤i≤n.

[0031] Secondly, the present invention discloses an apparatus for determining independent sets on an alternating group graph, comprising:

[0032] The vertex determination module is used to determine the n-dimensional alternating group graph AG. n All vertices of n, n≥4;

[0033] The alternating group graph partitioning module is used to partition an n-dimensional alternating group graph AG according to the vertex labeling rules. n The recursive partitioning is performed step by step until it is divided into several alternating group graphs AG4;

[0034] The first set determination module is used to put all vertices of all alternating group graphs AG4 obtained by the alternating group graph partitioning module into set T;

[0035] The second set determination module is used to extract all vertices of an alternating group graph AG4 from set T and put them into set X;

[0036] The independent set determination module is used to take a vertex x from set X and put it into set S, wherein vertex x is not adjacent to any vertex in set S;

[0037] Get all neighboring vertices of vertex x in the current alternating group graph AG4 and other alternating group graphs AG4 in set T, and delete all corresponding neighboring vertices in set X and set T;

[0038] The first repeated execution module is used to repeatedly execute the methods in the independent set determination module until the set X is empty, and then continue to execute the methods in the second repeated execution module.

[0039] The second repeated execution module is used to repeatedly execute the methods in the second set determination module, the independent set determination module, and the first repeated execution module in sequence until all alternating group graphs AG4 in set T have been evaluated and set T is empty. The final set S is the n-dimensional alternating group graph AG4. n An independent set.

[0040] As a preferred approach, the alternating group graph partitioning module is used to perform the following methods:

[0041] First, the n-dimensional alternating group graph AG n Vertex set V(AG) n ) = A n A n for <n>The set of all even permutations of the vertex, where each vertex is labeled with a number of length n: p1p2...p n For example, n≥4;

[0042] Secondly, according to the nth position of the vertex label p n AG of n-dimensional alternating group graph n Each vertex is assigned to an alternating group graph AG of n. n-1 In the diagram, there are n alternating group graphs AG. n-1 Each by It means, and The last digit of each vertex label is i, and l≤i≤n;

[0043] Then, according to the (n-1)th vertex label p n-1 AG of each n-1 dimensional alternating group graph n-1 Each vertex is assigned to an n-1 alternating group graph AG n-2 In, and each AG n-2 The (n-1)th position of the vertex label is the same;

[0044] Repeat the above recursive steps until each vertex of each alternating group graph AG5 is assigned to one of the five alternating group graphs AG4, and the fifth bit of the vertex label of each AG4 is the same, thus achieving partitioning.

[0045] As a preferred embodiment, the first set determination module is used to put all vertices of all alternating group graphs AG4 obtained by the alternating group graph partitioning module into set T with vertex labels.

[0046] As a preferred embodiment, the independent set determination module is used to determine the independent set through... and The operation obtains all neighboring vertices ω of vertex x in the current alternating group graph AG4 and other alternating group graphs AG4 in set T.

[0047] in: This represents the permutation obtained by rotating the first, second, and i-th elements of vertex x by one position clockwise.

[0048] This represents the permutation obtained by rotating the first, second, and i-th elements of vertex x counterclockwise by one position.

[0049] 3≤i≤n.

[0050] Thirdly, the present invention discloses a computing device, comprising:

[0051] One or more processors;

[0052] Memory;

[0053] And one or more programs, wherein the one or more programs are stored in memory and configured to be executed by one or more processors, and the one or more programs include instructions for determining independent sets on any of the above-described alternating group graphs.

[0054] Fourthly, the present invention discloses a storage medium storing one or more computer-readable programs, the one or more programs including instructions adapted to be loaded by a memory and executed by any of the methods described above for determining independent sets on an alternating group graph.

[0055] This invention discloses a method, apparatus, device, and storage medium for determining independent sets on an alternating group graph, which has the following beneficial effects:

[0056] First, this invention utilizes the structural properties of alternating group graphs to construct an alternating group graph AG for any integer n ≥ 4. n Let S be an independent set of vertices, in which all vertices in S are not adjacent to each other.

[0057] Secondly, this invention can be applied to interconnected networks to help optimize data transmission and resource allocation strategies, thereby improving network performance and efficiency. When independent cluster nodes are identified, independent network channels and server resources can be allocated to these nodes. Attached Figure Description

[0058] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly introduced below. It should be understood that the following drawings only show some embodiments of the present invention and should not be regarded as a limitation on the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort.

[0059] Figure 1 This is a schematic diagram of an independent set.

[0060] Figure 2 This is a schematic diagram of vertex labeling for a 3D alternating group graph AG3.

[0061] Figure 3 This is a schematic diagram of vertex labeling for a 4-dimensional alternating group graph AG4.

[0062] Figure 4 This is a flowchart illustrating a method for determining independent sets on an alternating group graph, as provided in an embodiment of the present invention.

[0063] Figure 5 A schematic diagram of five AG4s provided in an embodiment of the present invention;

[0064] (a) is A schematic diagram;

[0065] (b) is A schematic diagram;

[0066] (c) is A schematic diagram;

[0067] (d) is A schematic diagram;

[0068] (e) is A schematic diagram.

[0069] Figure 6 This is a schematic diagram of a device for determining independent sets on an alternating group graph, provided in an embodiment of the present invention.

[0070] Figure 7 This is a schematic diagram of a computing device provided in an embodiment of the present invention.

[0071] Wherein: 201-Vertex determination module, 202-Alternating group graph partitioning module, 203-First set determination module, 204-Second set determination module, 205-Independent set determination module, 206-First repeated execution module, 207-Second repeated execution module;

[0072] 301 - Processor, 302 - Memory. Detailed Implementation

[0073] The preferred embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0074] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0075] Using ordinal numbers such as "first," "second," "third," etc. to describe ordinary objects merely indicates different instances of similar objects and is not intended to imply that the objects being described must have a given order in time, space, sequence, or any other way.

[0076] Furthermore, the expression "includes" is an "open-ended" expression, which means only that there is a corresponding component or step, and should not be interpreted as excluding additional components or steps.

[0077] The following is a brief description of some of the technical terms or basic knowledge involved in this invention.

[0078] remember <n>Let be a set {1, 2, ..., n}, and n ≥ 3.

[0079] Let p = p1p2...p n for <n>A permutation of p, that is, for any integer i ≠ j, p i ∈ <n>And p i ≠p j .

[0080] For integers 1 ≤ i ≤ n, p i This represents the i-th element in permutation p. If i > j, then p i <p j Then the element pair (p) is called an element pair. i <p j Let p be an inversion of p. If a permutation has an even number of inversions, it is called an even permutation.

[0081] Taking p = 13425 as an example, the element pairs (3, 2) and (4, 2) are two inversions, so p is an even permutation.

[0082] Let A be the name of the person in question. n for <n>The set of all even permutations of the integers, for any integer 3 ≤ i ≤ n. and For defined in A n The two operations on. (or The expression represents the permutation obtained by rotating the 1st, 2nd, and ith elements of p by one position clockwise (or counterclockwise).

[0083] For example, suppose p = 13425, then n-dimensional alternating group diagram AG n Each vertex is labeled with a length of n: p1p2...p n To express.

[0084] n-dimensional alternating group diagram AG n It is a typical interconnected network topology.

[0085] For any integer n ≥ 3, an n-dimensional alternating group graph AG n It is recursively defined as follows:

[0086] (1) As Figure 2 As shown, AG3 is a graph consisting of three vertices labeled 123, 231 and 312, which are connected by three edges: (123, 312), (123, 231) and (231, 312).

[0087] (2) For any integer n≥4, AG n Composed of n AGs n-1 The copy constructor. For any integer 1 ≤ i ≤ n, let AG n The i-th subgraph, then The last digit of each vertex label is i.

[0088] AG n It can be divided into n copies: n-dimensional alternating group diagram AG n Vertex set V(AG) n ) = A n Two vertices p, q∈A n Adjacent if and only if (abbreviated as) )or (abbreviated as) ), where i∈{3,4,...,n}.

[0089] Taking the 4-dimensional alternating group graph AG4 as an example, its vertex labels are as follows: Figure 3 As shown.

[0090] To achieve the objectives of this invention, in some embodiments of a method, apparatus, device, and storage medium for determining independent sets on an alternating group graph, the method for determining independent sets on an alternating group graph includes the following steps:

[0091] Step S101: Determine the n-dimensional alternating group graph AG n All vertices of n, n≥4;

[0092] Step S102: According to the vertex labeling rules, transform the n-dimensional alternating group graph AG n The recursive partitioning is performed step by step until it is divided into several alternating group graphs AG4;

[0093] Step S103: Place all vertices of all alternating group graphs AG4 obtained in step S102 into set T with vertex labels;

[0094] Step S104: Take all vertices of an alternating group graph AG4 from set T and put them into set X;

[0095] Step S105: Take a vertex x from set X and put it into set S, and vertex x is not adjacent to any vertex in set S;

[0096] pass and The operation obtains all neighboring vertices of vertex x in the current alternating group graph AG4 and other alternating group graphs AG4 in set T, and deletes all corresponding neighboring vertices in set X and set T.

[0097] in: This represents the permutation obtained by rotating the first, second, and i-th elements of vertex x by one position clockwise.

[0098] This represents the permutation obtained by rotating the first, second, and i-th elements of vertex x counterclockwise by one position.

[0099] 3≤i≤n;

[0100] Step S106: Repeat step S105 until set X is empty, then continue to execute step S7;

[0101] Step S107: Repeat steps S104-S106 until all alternating group graphs AG4 in set T have been evaluated and set T is empty. The final set S is the n-dimensional alternating group graph AG. n An independent set.

[0102] For alternating group graphs AG n This invention solves for independent sets based on the properties of alternating group graphs.

[0103] An n-dimensional alternation group graph AG n Composed of n AGs n-1 Copy (notation) It is constructed from ) wherein: each AG n-1 The last digit of each vertex label is the same, i.e., the nth vertex.

[0104] Each AG n-1 Composed of n-1 AGs n-2 Copy (notation) It is constructed from ) wherein: each AG n-2 The (n-1)th bit of each vertex label is the same.

[0105] Based on the above alternating group diagram AG n Based on the characteristics, they are similarly divided until each AG5 is composed of 5 copies of AG4 (i.e.: It is constructed from ) and the 5th position of each AG4 is the same.

[0106] Based on the above division method, this invention adopts a recursive approach to divide the alternating group graph AG. n It is gradually decomposed into many alternating group graphs AG4.

[0107] Therefore, step S102 of the present invention includes the following:

[0108] First, the n-dimensional alternating group graph AG n Vertex set V(AG) n ) = A n A n for <n>The set of all even permutations of the vertex, where each vertex is labeled with a number of length n: p1p2...p n For example, n≥4;

[0109] Secondly, according to the nth position of the vertex label p n AG of n-dimensional alternating group graph n Each vertex is assigned to an alternating group graph AG of n. n-1 In the diagram, there are n alternating group graphs AG. n-1 Each by It means, and The last digit of each vertex label is i, and l≤i≤n;

[0110] Then, according to the (n-1)th vertex label p n-1 AG of each n-1 dimensional alternating group graph n-1 Each vertex is assigned to an n-1 alternating group graph AG n-2 In, and each AG n-2 The (n-1)th position of the vertex label is the same;

[0111] Repeat the above recursive steps until each vertex of each alternating group graph AG5 is assigned to one of the five alternating group graphs AG4, and the fifth bit of the vertex label of each AG4 is the same, thus achieving partitioning.

[0112] Take the 5-dimensional alternating group graph AG5 as an example. Divide the 5-dimensional alternating group graph AG5 into 5 AG4 groups (i.e.: ),like Figure 5 As shown in (a), (b), (c), (d), and (e).

[0113] This invention places all vertices of the obtained alternating group graph AG4 into a set T. Then, the following algorithm is used to obtain the alternating group graph AG4. n The independent set S.

[0114] Step A: Determine if set T is not empty;

[0115] If not empty, proceed to step B;

[0116] Otherwise, the algorithm terminates;

[0117] Step B: Take all vertices of an alternating group graph AG4 from set T and put them into set X;

[0118] Step C: Determine if set X is not empty;

[0119] If not empty, proceed to step D;

[0120] Otherwise, proceed to step A;

[0121] Step D: Take a vertex x from set X and add it to set S, where x is not adjacent to any vertex in S; through and (3≤i≤n) Obtain all neighbor vertices of x in the current AG4 and other AG4s, then delete the neighbor vertices in set X and set T, and then proceed to step C.

[0122] By following the steps above, we can obtain the independent set S.

[0123] This embodiment also provides code for generateDominatingSet() to obtain the independent set S, as shown below:

[0124]

[0125]

[0126]

[0127]

[0128] In some embodiments, for a 4-dimensional bubble sorting star diagram alternating group AG4, calling generateDominatingSet(4) yields the following result:

[0129] The independent set of AG4 is: [1234, 1342, 1423].

[0130] In other embodiments, for a 5-dimensional bubble sorting star diagram alternating group AG5, calling generateDominatingSet(5) yields the following result:

[0131] The independent set of AG5 is: [23451, 24531, 25341, 13542, 14352, 15432, 14523, 15243, 24153, 25413, 13254, 15324, 23514, 25134, 13425, 14235, 23145, 24315].

[0132] In summary, the method for determining independent sets on alternating group graphs in this invention mainly comprises:

[0133] First, consider an n-dimensional alternating group graph AG. n Decompose into n AGs n-1 ;

[0134] Then, for each n-1 dimensional alternating group graph AG n-1 Decompose into n-1 AGs n-2 ;

[0135] Continue in this manner until each 5-dimensional alternating group graph AG5 is decomposed into 5 AG4s.

[0136] Secondly, for each 4-dimensional alternating group graph AG4, select a vertex x that is not adjacent to any vertex in set S and add it to set S. and (3≤i≤n) Obtain all neighbor vertices of x in the current AG4 and other AG4s, and then delete these neighbor vertices; repeat the above operation until all AG4s have been processed.

[0137] Finally, we obtain a set S (which is an independent set), in which any two points u and v are not adjacent to each other.

[0138] Furthermore, in some other embodiments, such as Figure 6 As shown, this invention discloses an apparatus for determining independent sets on an alternating group graph, comprising:

[0139] Vertex determination module 201 is used to determine an n-dimensional alternating group graph AG. n All vertices of n, n≥4;

[0140] Alternating group graph partitioning module 202 is used to partition an n-dimensional alternating group graph AG according to the vertex labeling rules. n The recursive partitioning is performed step by step until it is divided into several alternating group graphs AG4;

[0141] The first set determination module 203 is used to put all vertices of all alternating group graphs AG4 obtained by the alternating group graph partitioning module 202 into set T;

[0142] The second set determination module 204 is used to extract all vertices of an alternating group graph AG4 from set T and put them into set X;

[0143] Independent set determination module 205 is used to take a vertex x from set X and put it into set S, and vertex x is not adjacent to any vertex in set S;

[0144] Get all neighboring vertices of vertex x in the current alternating group graph AG4 and other alternating group graphs AG4 in set T, and delete all corresponding neighboring vertices in set X and set T;

[0145] The first repeat execution module 206 is used to repeatedly execute the method in the independent set determination module 205 until the set X is empty, and then continue to execute the method ω in the second repeat execution module 207.

[0146] The second repeated execution module 207 is used to repeatedly execute the methods in the second set determination module 204, the independent set determination module 205, and the first repeated execution module 206 in sequence until all alternating group graphs AG4 in set T have been judged and set T is empty. The final set S is the n-dimensional alternating group graph AG. n An independent set.

[0147] Furthermore, the alternating group graph partitioning module 202 is used to perform the following methods:

[0148] First, the n-dimensional alternating group graph AG n Vertex set V(AG) n ) = A n A n for <n>The set of all even permutations of the vertex, where each vertex is labeled with a number of length n: p1p2...p n For example, n≥4;

[0149] Secondly, according to the nth position of the vertex label p n AG of n-dimensional alternating group graph n Each vertex is assigned to an alternating group graph AG of n. n-1 In the diagram, there are n alternating group graphs AG. n-1 Each by It means, and The last digit of each vertex label is i, and 1 ≤ i ≤ n;

[0150] Then, according to the (n-1)th vertex label p n-1 AG of each n-1 dimensional alternating group graph n-1 Each vertex is assigned to an n-1 alternating group graph AG n-2 In, and each AG n-2 The (n-1)th position of the vertex label is the same;

[0151] Repeat the above recursive steps until each vertex of each alternating group graph AG5 is assigned to one of the five alternating group graphs AG4, and the fifth bit of the vertex label of each AG4 is the same, thus achieving partitioning.

[0152] Furthermore, the first set determination module 206 is used to put all vertices of all alternating group graphs AG4 obtained by the alternating group graph partitioning module 202 into set T with vertex labels.

[0153] Furthermore, the independent set determination module 205 is used to determine the independent set through... and The operation obtains all neighboring vertices of vertex x in the current alternating group graph AG4 and other alternating group graphs AG4 in set T;

[0154] in: This represents the permutation obtained by rotating the first, second, and i-th elements of vertex x by one position clockwise.

[0155] This represents the permutation obtained by rotating the first, second, and i-th elements of vertex x counterclockwise by one position.

[0156] 3≤i≤n.

[0157] Furthermore, it should be noted that the device for determining independent sets on alternating group graphs provided in the above embodiments is only illustrated by the division of the above functional modules when determining independent sets. In actual applications, the above functions can be assigned to different functional modules as needed, that is, the internal structure of the device for determining independent sets on alternating group graphs can be divided into different functional modules to complete all or part of the functions described above.

[0158] Furthermore, the embodiments of the apparatus for determining independent sets on alternating group graphs and the method for determining independent sets on alternating group graphs provided in the above embodiments belong to the same concept, and their specific implementation process can be found in the method embodiments, which will not be repeated here.

[0159] Furthermore, in some other embodiments, such as Figure 7 As shown, the present invention also discloses a computing device, comprising:

[0160] One or more processors 301;

[0161] Memory 302;

[0162] And one or more programs, wherein the one or more programs are stored in memory 302 and configured to be executed by one or more processors 301, the one or more programs including instructions for the method of determining independent sets on alternating group graphs disclosed in the above embodiments.

[0163] Processor 301 may include one or more processing cores, such as a quad-core processor or an octa-core processor. Processor 301 may be implemented using at least one hardware form selected from DSP (Digital Signal Processing), FPGA (Field-Programmable Gate Array), and PLA (Programmable Logic Array). Processor 301 may also include a main processor and a coprocessor. The main processor, also known as a CPU (Central Processing Unit), is used to process data in the wake-up state; the coprocessor is a low-power processor used to process data in the standby state. In some embodiments, processor 301 may integrate a GPU (Graphics Processing Unit), which is responsible for rendering and drawing the content to be displayed on the screen. In some embodiments, processor 301 may also include an AI (Artificial Intelligence) processor, which is used to handle computational operations related to machine learning.

[0164] Memory 302 may include one or more computer-readable storage media, which may be non-transitory. Memory 302 may also include high-speed random access memory and non-volatile memory, such as one or more disk storage devices or flash memory devices. In some embodiments, the non-transitory computer-readable storage media in memory 302 is used to store at least one instruction, which is executed by processor 301 to implement the method for determining independent sets on alternating group graphs provided in the method embodiments of the present invention.

[0165] In addition, the computing device may optionally include: a peripheral device interface and at least one peripheral device. The processor 301, memory 302, and peripheral device interface can be connected via a bus or signal lines. Each peripheral device can be connected to the peripheral device interface via a bus, signal lines, or a circuit board. Illustratively, peripheral devices include, but are not limited to: radio frequency circuitry, a touchscreen display, audio circuitry, and a power supply.

[0166] Of course, the computing device may also include fewer or more components, and this embodiment does not limit this.

[0167] Furthermore, in other embodiments, the present invention also discloses a storage medium storing one or more computer-readable programs, the one or more programs including instructions adapted to be loaded by memory and executed by the method for determining independent sets on alternating group graphs disclosed in the above embodiments.

[0168] This invention discloses a method, apparatus, device, and storage medium for determining independent sets on an alternating group graph, which has the following beneficial effects:

[0169] First, this invention utilizes the structural properties of alternating group graphs to construct an alternating group graph AG for any integer n ≥ 4. n Let S be an independent set of vertices, in which all vertices in S are not adjacent to each other.

[0170] Secondly, this invention can be applied to interconnected networks to help optimize data transmission and resource allocation strategies, thereby improving network performance and efficiency. When independent cluster nodes are identified, independent network channels and server resources can be allocated to these nodes.

[0171] For example, a dedicated network bandwidth segment can be allocated to each independent node set to ensure they can transmit data in parallel and efficiently. Simultaneously, resource allocation strategies can be dynamically adjusted based on node load and resource requirements to balance network load and improve resource utilization.

[0172] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the present invention. Various changes and modifications can be made to the present invention without departing from its spirit and scope. All such changes and modifications fall within the scope of the present invention as claimed, which is defined by the appended claims and their equivalents.< / n> < / n> < / n> < / n> < / n> < / n> < / n> < / n>

Claims

1. A method for determining independent sets on an alternating group graph, characterized in that, include: Step S1: Determine the n-dimensional alternating group graph AG n All vertices of n, n≥4; Step S2: According to the vertex labeling rules, transform the n-dimensional alternating group graph AG n The recursive partitioning is performed step by step until it is divided into several alternating group graphs AG4; Step S3: Put all vertices of all alternating group graphs AG4 obtained in step S2 into set T; Step S4: Take all vertices of an alternating group graph AG4 from set T and put them into set X; Step S5: Take a vertex x from set X and put it into set S, and vertex x is not adjacent to any vertex in set S; Get all neighboring vertices of vertex x in the current alternating group graph AG4 and other alternating group graphs AG4 in set T, and delete all corresponding neighboring vertices in set X and set T; Step S6: Repeat step S5 until set X is empty, then continue to execute step S7; Step S7: Repeat steps S4-S6 until all alternating group graphs AG4 in set T have been evaluated and set T is empty. The final set S is the n-dimensional alternating group graph AG. n An independent set.

2. The determination method according to claim 1, characterized in that, Step S2 includes the following: First, the n-dimensional alternating group graph AG n Vertex set V(AG) n ) = A n A n for <n>The set of all even permutations of the vertex, where each vertex is labeled with a number of length n: p1p2...p n For example, n≥4;< / n> Secondly, according to the nth position of the vertex label p n AG of n-dimensional alternating group graph n Each vertex is assigned to an alternating group graph AG of n. n-1 In the diagram, there are n alternating group graphs AG. n-1 Each by It means, and The last digit of each vertex label is i, and 1 ≤ i ≤ n; Then, according to the (n-1)th vertex label p n-1 AG of each n-1 dimensional alternating group graph n-1 Each vertex is assigned to an n-1 alternating group graph AG n-2 In, and each AG n-2 The (n-1)th position of the vertex label is the same; Repeat the above recursive steps until each vertex of each alternating group graph AG5 is assigned to one of the five alternating group graphs AG4, and the fifth bit of the vertex label of each AG4 is the same, thus achieving partitioning.

3. The determination method according to claim 2, characterized in that, Step S3 specifically involves: Step S3: Place all vertices of the alternating group graph AG4 obtained in step S2 into set T with vertex labels.

4. The determination method according to claim 3, characterized in that, In step S5 pass and The operation obtains all neighboring vertices of vertex x in the current alternating group graph AG4 and other alternating group graphs AG4 in set T; in: This represents the permutation obtained by rotating the first, second, and i-th elements of vertex x by one position clockwise. This represents the permutation obtained by rotating the first, second, and i-th elements of vertex x counterclockwise by one position. 3≤i≤n.

5. A device for determining independent sets on an alternating group graph, characterized in that, include: The vertex determination module is used to determine the n-dimensional alternating group graph AG. n All vertices of n, n≥4; The alternating group graph partitioning module is used to partition an n-dimensional alternating group graph AG according to the vertex labeling rules. n The recursive partitioning is performed step by step until it is divided into several alternating group graphs AG4; The first set determination module is used to put all vertices of all alternating group graphs AG4 obtained by the alternating group graph partitioning module into set T; The second set determination module is used to extract all vertices of an alternating group graph AG4 from set T and put them into set X; The independent set determination module is used to take a vertex x from set X and put it into set S, wherein vertex x is not adjacent to any vertex in set S; Get all neighboring vertices of vertex x in the current alternating group graph AG4 and other alternating group graphs AG4 in set T, and delete all corresponding neighboring vertices in set X and set T; The first repeated execution module is used to repeatedly execute the methods in the independent set determination module until the set X is empty, and then continue to execute the methods in the second repeated execution module. The second repeated execution module is used to repeatedly execute the methods in the second set determination module, the independent set determination module, and the first repeated execution module in sequence until all alternating group graphs AG4 in set T have been evaluated and set T is empty. The final set S is the n-dimensional alternating group graph AG4. n An independent set.

6. The determining device according to claim 5, characterized in that, The alternating group graph partitioning module is used to perform the following methods: First, the n-dimensional alternating group graph AG n Vertex set V(AG) n ) = A n A n for <n>The set of all even permutations of the vertex, where each vertex is labeled with a number of length n: p1p2...p n For example, n≥4;< / n> Secondly, according to the nth position of the vertex label p n AG of n-dimensional alternating group graph n Each vertex is assigned to an alternating group graph AG of n. n-1 In the diagram, there are n alternating group graphs AG. n-1 Each by It means, and The last digit of each vertex label is i, and 1 ≤ i ≤ n; Then, according to the (n-1)th vertex label p n-1 AG of each n-1 dimensional alternating group graph n-1 Each vertex is assigned to an n-1 alternating group graph AG n-2 In, and each AG n-2 The (n-1)th position of the vertex label is the same; Repeat the above recursive steps until each vertex of each alternating group graph AG5 is assigned to one of the five alternating group graphs AG4, and the fifth bit of the vertex label of each AG4 is the same, thus achieving partitioning.

7. The determining device according to claim 6, characterized in that, The first set determination module is used to put all vertices of all alternating group graphs AG4 obtained by the alternating group graph partitioning module into set T by vertex label.

8. The determining device according to claim 7, characterized in that, The independent set determination module is used to... and The operation obtains all neighboring vertices of vertex x in the current alternating group graph AG4 and other alternating group graphs AG4 in set T; in: This represents the permutation obtained by rotating the first, second, and i-th elements of vertex x by one position clockwise. This represents the permutation obtained by rotating the first, second, and i-th elements of vertex x counterclockwise by one position. 3≤i≤n.

9. A computing device, characterized in that, include: One or more processors; Memory; And one or more programs, wherein the one or more programs are stored in the memory and configured to be executed by one or more processors, and the one or more programs include instructions for the method of determining independent sets on an alternating group graph as described in any of claims 1-4.

10. A storage medium, characterized in that, The storage medium stores one or more computer-readable programs, the programs including instructions adapted to be loaded by memory and executed as described in any of claims 1-4 for determining independent sets on an alternating group graph.

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