A method and device for constructing a folded hypercube edge-disjoint hamiltonian cycle
By combining recursive construction with vertex mapping, the problem of constructing non-intersecting Hamiltonian cycles of folded hypercubes was solved, achieving efficient and reliable multiprocessor node connections and improving network throughput and fault tolerance.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- SUZHOU IND PARK SERVICE OUTSOURCING VOCATIONAL COLLEGE (SUZHOU SERVICE OUTSOURCING TALENT TRAINING & TRAINING CENT)
- Filing Date
- 2026-03-10
- Publication Date
- 2026-06-09
AI Technical Summary
The lack of existing methods for constructing edge-disjoint Hamiltonian cycles for folded hypercubes makes it impossible to efficiently and reliably connect multiprocessor nodes, affecting network throughput and fault tolerance.
By combining recursive construction with vertex mapping, the first Hamiltonian cycle is transformed into the second Hamiltonian cycle through the automorphism f. Parity and edge deletion operations are used to ensure that the two cycles have no common edges, thus constructing a Hamiltonian cycle with completely disjoint edges.
It implements a recursive construction with linear time complexity, provides determinism and reliability, supports uniform traffic distribution, improves network throughput, and has instantaneous fault switching capability. It is applicable to any n≥3 dimensional folded hypercube.
Smart Images
Figure CN122173440A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of interconnection network topology technology, and more specifically to a method and apparatus for constructing a Hamiltonian cycle with non-intersecting edges of a folded hypercube. Background Technology
[0002] High-performance parallel computers are an important indicator of a nation's comprehensive scientific and technological strength, and are widely used in fields with extremely high computing power requirements, such as scientific computing, weather forecasting, oil exploration, and artificial intelligence. With the exponential growth in the number of processors, how to efficiently and reliably connect thousands or even tens of thousands of processor nodes has become a core issue in the design of interconnect network topologies. In the study of interconnect networks, processor nodes are typically abstracted as vertices V of a graph G=(V, E), and communication links between nodes are abstracted as edges E.
[0003] Hypercube (notation) A hypercube is a classic interconnected network topology. An n-dimensional hypercube contains 2n vertices, each corresponding to an n-bit binary string. Two vertices are adjacent if and only if their binary representations differ by exactly one bit. A folded hypercube (noted...) is another type of hypercube. The is a variant of the hypercube obtained by adding complementary edges: each vertex is connected to its bitwise inverse vertex. This structure has a shorter diameter, better fault tolerance, and higher connectivity, and has therefore attracted widespread attention.
[0004] In graph theory and interconnection network design, a Hamiltonian cycle is a closed loop that traverses every vertex in a graph exactly once and returns to the starting point. If two Hamiltonian cycles share no common edges, they are called edge-disjoint Hamiltonian cycles. In large-scale multiprocessor systems, constructing multiple edge-disjoint Hamiltonian cycles has significant engineering value: load balancing: communication traffic can be evenly distributed across multiple non-overlapping loops, avoiding localized link congestion and improving network throughput; fault-tolerant routing: when an edge fails, data can be quickly switched to other edge-disjoint loops to continue transmission, ensuring system availability; parallel communication: supporting simultaneous fully switched communication between multiple processor pairs, greatly improving parallel computing efficiency.
[0005] In existing technologies, for hypercubes Extensive research has been conducted on the construction of Hamiltonian rings and the non-overlapping ring problem of other hypercube variants (such as intersecting cubes and Möbius cubes). However, folded hypercubes... Due to the introduction of complementary edges, the structural symmetry of folded hypercubes differs fundamentally from that of hypercubes, making it impossible to directly transfer existing methods to folded hypercubes. To date, no specific method or publicly available technical solution for constructing edge-disjoint Hamiltonian cycles for folded hypercubes has been documented. How to efficiently and deterministically construct two completely edge-disjoint Hamiltonian cycles using the unique complementary edge properties of folded hypercubes is a pressing problem in this field. Summary of the Invention
[0006] The purpose of this invention is to provide a method and apparatus for constructing edge-disjoint Hamiltonian cycles of a folded hypercube. This method employs a strategy combining recursive construction and vertex mapping, offering the following significant advantages: In terms of efficiency, the time complexity of recursive construction is linearly related to the number of vertices, the mapping step is a single traversal, the algorithm is deterministic and reproducible, and it is suitable for hardware logic implementation and rapid online reconstruction. In terms of reliability, by defining an automorphic mapping f conditioned on the least significant bit of a vertex, the first Hamiltonian cycle is transformed into a second Hamiltonian cycle. Furthermore, by utilizing the correspondence between parity and edge deletion operations, it is theoretically proven that the two cycles have no common edges, providing a completely independent primary and backup communication path for the network. In terms of application value, the resulting edge-disjoint Hamiltonian cycles can be directly deployed in multiprocessor interconnect networks, supporting even traffic distribution to avoid link congestion, improving system throughput, and providing instantaneous failover capability. Single-edge failure does not affect overall connectivity, and it is applicable to any n≥3D folded hypercube, exhibiting good scalability. It provides key technical support for topology design and routing protocols of high-performance parallel computing systems.
[0007] To achieve the above objectives, the present invention provides the following technical solution: In a first aspect, the present invention provides a method for constructing a non-intersecting Hamiltonian cycle of a folded hypercube, the method comprising: Recursively construct an n-dimensional folded hypercube The first Hamiltonian cycle ; For the first Hamiltonian cycle Apply a mapping f to each vertex u in the equation, and connect the mapped vertices in the original order to obtain the second Hamiltonian cycle. ; Wherein, the mapping f is defined as: ; This represents the vertex obtained by inverting the first bit of vertex u. This represents the complementary vertex obtained by inverting all bits of vertex u; First Hamiltonian Cycle With the second Hamiltonian cycle folded hypercube There are no public edges in it.
[0008] In some embodiments, an n-dimensional folded hypercube is recursively constructed. The first Hamiltonian cycle ,include: When n=3, the preset Hamiltonian cycle vertex sequence is returned directly. The preset Hamiltonian cycle vertex sequence is: 000, 001, 011, 010, 110, 111, 101, 100. When n>3, first recursively construct an n-1 dimensional folded hypercube. Hamiltonian Circle Then based on structure .
[0009] In some embodiments, when n>3, based on structure ,include: Will Add a "0" bit to the left of the binary string of each vertex, keeping the vertex order unchanged, to obtain the sub-cycle. ; Will Add a "1" bit to the left of the binary string of each vertex and reverse the order of the vertices to obtain the sub-cycle. ; make ,exist Deleting edges and in Deleting edges ; Add edge ( , )and( , ),Will and Connect them to form a complete Hamiltonian cycle, and we get .
[0010] In some embodiments, the binary representation of vertex u is: ,in The least significant bit is the lowest bit; vertices whose least significant bit in binary representation is 0 are defined as even vertices, and vertices whose least significant bit in binary representation is 1 are defined as odd vertices. Corresponding to the vertex ,in ; Corresponding to the vertex .
[0011] In some embodiments, the mapping f is an n-dimensional folded hypercube. An automorphism on f, i.e., f is a vertex set V( A bijection from E to itself, and for any edge (a, b) ∈ E( ), (f(a), f(b))∈E( The second Hamiltonian cycle is guaranteed by the automorphism property. yes A Hamiltonian cycle.
[0012] In some embodiments, and The proof that the edges do not intersect includes: Mapping f will The edge mapping in is The edges in; If there exists a common edge (x, y) that belongs to both and Then it exists The edge (a, b) in the equation makes (f(a), f(b)) = (x, y); By analyzing the value of the least significant bit of the vertex and the mapping rules, it is deduced that (a, b) in The edge has been deleted during the construction process, leading to a contradiction and proving that no common edge exists.
[0013] Secondly, the present invention also provides a construction apparatus for a folded hypercube edge-non-intersecting Hamiltonian ring, the apparatus comprising: The first construction module is used to recursively construct an n-dimensional folded hypercube. The first Hamiltonian cycle ; The second construction module is used for the first Hamiltonian cycle. Apply a mapping f to each vertex u in the equation, and connect the mapped vertices in the original order to obtain the second Hamiltonian cycle. ; Wherein, the mapping f is defined as: ; This represents the vertex obtained by inverting the first bit of vertex u. This represents the complementary vertex obtained by inverting all bits of vertex u.
[0014] Thirdly, the present invention also provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the method for constructing the non-intersecting Hamiltonian loop of the folded hypercube provided in the first aspect.
[0015] Fourthly, the present invention also provides a computer-readable storage medium storing a computer program, which, when executed by a processor, implements the method for constructing a non-intersecting Hamiltonian cycle of a folded hypercube provided in the first aspect.
[0016] Fifthly, the present invention also provides a computer program product, including a computer program that, when executed by a processor, implements the method for constructing a non-intersecting Hamiltonian cycle of a folded hypercube provided in the first aspect.
[0017] The beneficial effects of this invention are as follows: Firstly, this invention proposes a method for folding hypercubes. This method for constructing edge-disjoint Hamiltonian cycles fills a technological gap in the parallel communication and fault-tolerant routing fields for this topology. The method employs a strategy combining recursive construction and vertex mapping, offering the following significant advantages: In terms of efficiency, the time complexity of recursive construction is linearly related to the number of vertices, the mapping step is a single traversal, the algorithm is deterministic and reproducible, and it is suitable for hardware logic implementation and rapid online reconfiguration. In terms of reliability, by defining an automorphic mapping f conditioned on the least significant bit of a vertex, the first Hamiltonian cycle is transformed into a second Hamiltonian cycle. Furthermore, by utilizing the correspondence between parity and edge deletion operations, it is theoretically proven that the two cycles have no common edges, providing the network with completely independent primary and backup communication paths. In terms of application value, the resulting edge-disjoint Hamiltonian cycles can be directly deployed in multiprocessor interconnect networks, supporting even traffic distribution to avoid link congestion, improving system throughput, and providing instantaneous failover capability. Single-edge failure does not affect overall connectivity, and it is applicable to any n≥3D folded hypercube, exhibiting good scalability. This provides key technical support for topology design and routing protocols in high-performance parallel computing systems.
[0018] The above description is merely an overview of the technical solution of the present invention. In order to better understand the technical means of the present invention and to implement it in accordance with the contents of the specification, the preferred embodiments of the present invention are described in detail below with reference to the accompanying drawings. Attached Figure Description
[0019] Figure 1 This is a flowchart illustrating a method for constructing a non-intersecting Hamiltonian loop of a folded hypercube according to an embodiment of the present invention. Figure 2 The present invention includes 3D and 4D hypercubes and 3D and 4D folded hypercubes, as shown in one embodiment of the present invention. Figure 3 As shown in one embodiment of the present invention A Hamiltonian cycle whose two edges do not intersect; Figure 4 This is a schematic diagram of a construction device for a folded hypercube with non-intersecting Hamiltonian rings, according to an embodiment of the present invention. Figure 5 This is a schematic diagram of an electronic device structure provided in an embodiment of this application. Detailed Implementation
[0020] The technical solution of the present invention will now be clearly and completely described with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. 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.
[0021] It should be noted that references to "an embodiment," "embodiment," "example embodiment," etc., in this specification refer to the described embodiment including specific features, structures, or characteristics; however, not every embodiment must include these specific features, structures, or characteristics. Furthermore, such expressions do not refer to the same embodiment. Moreover, when describing specific features, structures, or characteristics in conjunction with embodiments, whether or not explicitly described, it is indicated that incorporating such features, structures, or characteristics into other embodiments is within the knowledge of those skilled in the art.
[0022] Furthermore, the technical features involved in the different embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.
[0023] In some embodiments, such as Figure 1 The diagram shows a flowchart illustrating a method for constructing a non-intersecting Hamiltonian cycle of a folded hypercube. The specific method includes: S101, Recursively construct an n-dimensional folded hypercube The first Hamiltonian cycle .
[0024] It should be noted that one 3D hypercube have There are n vertices, each of which can be represented as an n-bit binary string, i.e. .in, .if ,say Even point; if ,say It is a singularity. For Another vertex If and only if and Only one digit is different, that is When u and v are connected, they are said to form an edge, which is represented as . This is recorded as... .
[0025] exist Based on this, if and When connected, they form an n-dimensional folded hypercube. This is recorded as... In particular, remember . Figure 2 It showcases 3D and 4D hypercubes and folded hypercubes. Figure 2 In (d), , .
[0026] A Hamiltonian cycle in an interconnected network is a closed path that passes through every vertex in the network exactly once. Two Hamiltonian cycles are called disjoint Hamiltonian cycles if they have no common edges. Figure 3 The red border in the middle shows Two edge-disjoint Hamiltonian cycles are used in large-scale multiprocessor systems. Multiple edge-disjoint Hamiltonian cycles allow processor nodes to exchange data simultaneously without interference, thus accelerating computational tasks. By evenly distributing traffic across different edge-disjoint Hamiltonian cycles, network load can be effectively distributed, local congestion avoided, and overall network throughput and transmission efficiency improved. Multiple edge-disjoint Hamiltonian cycles provide multiple independent backup paths. When an edge in the network fails, data can be quickly switched to other non-overlapping cycles for transmission, ensuring uninterrupted system operation.
[0027] Where, the binary representation of vertex u is: ,in The least significant bit is the lowest bit; vertices whose least significant bit in binary representation is 0 are defined as even vertices, and vertices whose least significant bit in binary representation is 1 are defined as odd vertices. Corresponding to the vertex ,in ; Corresponding to the vertex .
[0028] Specifically, when n=3, the preset Hamiltonian cycle vertex sequence is directly returned. The preset Hamiltonian cycle vertex sequence is: 000, 001, 011, 010, 110, 111, 101, 100. When n>3, an n-1 dimensional folded hypercube is recursively constructed first. Hamiltonian Circle Then based on structure .
[0029] Where, when n>3, based on structure Including: Add a "0" bit to the left of the binary string of each vertex, keeping the vertex order unchanged, to obtain the sub-cycle. ;Will Add a "1" bit to the left of the binary string of each vertex and reverse the order of the vertices to obtain the sub-cycle. ;make ,exist Deleting edges and in Deleting edges Add edge ( , )and( , ),Will and Connect them to form a complete Hamiltonian cycle, and we get .
[0030] S102, for the first Hamiltonian cycle Apply a mapping f to each vertex u in the equation, and connect the mapped vertices in the original order to obtain the second Hamiltonian cycle. ; Wherein, the mapping f is defined as: ; This represents the vertex obtained by inverting the first bit of vertex u. This represents the complementary vertex obtained by inverting all bits of vertex u; the first Hamiltonian cycle. With the second Hamiltonian cycle folded hypercube There are no common edges. Where the mapping f is an n-dimensional folded hypercube An automorphism on f, i.e., f is a vertex set V( A bijection from E to itself, and for any edge (a, b) ∈ E( ), (f(a), f(b))∈E( The second Hamiltonian cycle is guaranteed by the automorphism property. yes A Hamiltonian cycle.
[0031] Optional, proof is also required. and The edges do not intersect: Mapping f will The edge mapping in is If there exists a common edge (x, y) that belongs to the same region; and Then it exists In the equation, there exists an edge (a, b) such that (f(a), f(b)) = (x, y); by analyzing the lowest bit value of the vertex and the mapping rule, it is derived that (a, b) in the equation... The edge has been deleted during the construction process, leading to a contradiction and proving that no common edge exists.
[0032] The method described in the above embodiments fills a technological gap in the field of parallel communication and fault-tolerant routing for this topology. This method employs a strategy combining recursive construction and vertex mapping, offering the following significant advantages: In terms of efficiency, the time complexity of recursive construction is linearly related to the number of vertices, the mapping step is a single traversal, the algorithm is deterministic and reproducible, and it is suitable for hardware logic implementation and rapid online reconfiguration; In terms of reliability, by defining an automorphic mapping f based on the least significant bit of a vertex, the first Hamiltonian cycle is transformed into a second Hamiltonian cycle, and by utilizing the correspondence between parity and edge deletion operations, it is theoretically proven that the two cycles have no common edges, providing the network with completely independent primary and backup communication paths; In terms of application value, the resulting non-intersecting Hamiltonian cycles can be directly deployed in multiprocessor interconnect networks, supporting even traffic distribution to avoid link congestion, improving system throughput, and providing instantaneous failover capability. Single-edge failure does not affect overall connectivity, and it is applicable to any n≥3D folded hypercube, exhibiting good scalability, and providing key technical support for topology design and routing protocols of high-performance parallel computing systems.
[0033] In another embodiment, a 4D folded hypercube is used. For example, the method in this application will be explained: Contains 2 4 There are 16 vertices, each represented by a 4-bit binary string. , where u i ∈{0,1}, u1 is the least significant bit. Vertex u is adjacent to vertex v if and only if one of the following conditions is satisfied: v=u i i∈{1,2,3,4}, that is, u and v differ by exactly one binary bit; v=u 5 ,Right now , is the bitwise inverted vertex (complementary vertex) of u.
[0034] Vertices with the lowest bit u1=0 are defined as even vertices, and vertices with u1=1 are defined as odd vertices.
[0035] Since n=4>3, we first construct the recursive structure. Hamiltonian Circle According to the method of the present invention, when n=3, the preset sequence is directly returned: ={000, 001, 011, 010, 110, 111, 101, 100}; This sequence has been verified and is... A Hamiltonian cycle that traverses all 8 vertices without repeating edges.
[0036] Will Add a "0" bit to the left of the binary string of each vertex, keeping the vertex order unchanged, to obtain 8 4-bit vertices, which are then used to form a sub-cycle in their original order. : ={0000, 0001, 0011, 0010, 0110, 0111, 0101, 0100}; Will Add a "1" bit to the left of the binary string of each vertex, and reverse the order of the vertices to obtain 8 4-bit vertices. Form a sub-cycle by reversing the order of the vertices. : Reverse order = {100, 101, 111, 110, 010, 011, 001, 000}; Adding a "1" to the left side results in: ={1100, 1101, 1111, 1110, 1010, 1011, 1001, 1000}; Let u = 0000, which is a 4-bit vertex consisting entirely of zeros. middle, side (u, u 3 ) corresponds to (0000,0000) 3 u 3 This means inverting the 3rd bit, 0000 3 =0010, therefore delete edge (0000,0010).
[0037] exist in, u 4 =0000 4 =1000, u 4,3 =(1000) 3 =1010, therefore delete edge (1000,1010).
[0038] Add a new edge (u,u) 4 )=(0000,1000), and the new edge (u 3 ,u 4,3 = (0010, 1010).
[0039] After the above deletion and addition operations. and They are connected to form a complete Hamiltonian cycle, and the vertex sequence is: ={0000, 0001, 0011, 0010, 0110, 0111, 0101, 0100, 1100, 1101,1111, 1110, 1010, 1011, 1001, 1000}; It has been verified that this sequence traversal... All 16 vertices and exactly once, adjacent vertices are all in The first and last vertices, 1000 and 0000, are adjacent, forming a closed loop, which is indeed true. A Hamiltonian cycle.
[0040] As listed above, It contains 16 vertices, labeled v1, v2, ..., v 16 .
[0041] The mapping f is defined as: ; Among them, u 1 This means inverting the first bit of vertex u. 5 This means inverting all 4 bits of vertex u (i.e., the complementary vertex).
[0042] Process them one by one Each vertex in: The mapped vertices are arranged according to Arrange the original order sequentially to obtain Vertex sequence: ={0001, 1110, 1100, 0011, 0111, 1000, 1010, 0101, 1101, 0010,0000, 1111, 1011, 0100, 0110, 1001}; This embodiment is... and Perform a rigorous edge comparison to verify whether there is a common edge between the two.
[0043] Will The vertex sequence forms edges based on adjacent positions and the first and last positions: E( ={(0000,0001), (0001,0011), (0011,0010), (0010,0110), (0110,0111), (0111,0101), (0101,0100), (0100,1100), (1100,1101), (1101,1111), (1111,1110), (1110,1010), (1010,1011), (1011,1001), (1001,1000), (1000,0000)}.
[0044] Will The vertex sequence forms edges based on adjacent positions and the first and last positions: E( ={(0001,1110), (1110,1100), (1100,0011), (0011,0111), (0111,1000), (1000,1010), (1010,0101), (0101,1101), (1101,0010), (0010,0000),(0000,1111), (1111,1011), (1011,0100), (0100,0110), (0110,1001), (1001,0001)}.
[0045] Comparison E( ) and E( No identical edges were found. Further examination of the equivalence case of undirected edges (i.e., (a,b) and (b,a) are considered the same edge) still yielded no overlap. Proof: and exist The middle and sides do not intersect.
[0046] After thorough examination, and Each contains All 16 vertices are identical, with no duplicates, and each vertex appears exactly once; adjacent vertices are all within the range of vertices. The first and last vertices are directly connected; the first and last vertices are connected to form a closed loop. Both are... Hamiltonian cycle.
[0047] In another embodiment, to verify the effectiveness of the method of this application, the deployment method of the present invention in a real multiprocessor network system is demonstrated.
[0048] A certain high-performance computing system contains 32 processor nodes, with Topology interconnection. The system control plane integrates a routing control unit, which automatically performs the following operations: Read the current system dimension parameter n=5; Calling the recursive constructor module generates ; Generate by calling the mapping module ; Write the path information of the two Hamiltonian cycles into the routing table of each processor; Configure the primary communication loop as The backup communication loop is ; Enable dynamic load balancing strategy to distribute data packets to two independent loops for parallel transmission according to traffic ratio; The link failure monitoring process is initiated. When a failure of an edge of the main ring is detected, the affected data stream is switched to the backup ring within microseconds.
[0049] Experimental results show that the dual-ring communication mechanism deployed using the method of this invention increases network throughput by about 42% and reduces link congestion probability by 67% under continuous full-load transmission conditions, and achieves zero system communication interruption in single-sided failure scenarios.
[0050] Based on the same inventive concept, this application also provides a device for constructing a non-intersecting Hamiltonian ring of a folded hypercube, used to implement the above-described method for constructing such a ring. The solution provided by this device is similar to the solution described in the above method. Therefore, the specific limitations of the one or more embodiments of the device for constructing a non-intersecting Hamiltonian ring of a folded hypercube provided below can be found in the limitations of the method for constructing such a ring above, and will not be repeated here.
[0051] In one embodiment, such as Figure 4 As shown, a construction apparatus for folding a hypercube with non-intersecting Hamiltonian rings is provided, the apparatus comprising: The first construction module 30 is used to recursively construct an n-dimensional folded hypercube. The first Hamiltonian cycle ; The second construction module 31 is used for the first Hamiltonian cycle. Apply a mapping f to each vertex u in the equation, and connect the mapped vertices in the original order to obtain the second Hamiltonian cycle. ; Wherein, the mapping f is defined as: ; This represents the vertex obtained by inverting the first bit of vertex u. This represents the complementary vertex obtained by inverting all bits of vertex u; First Hamiltonian Cycle With the second Hamiltonian cycle folded hypercube There are no public edges in it.
[0052] This application also provides an electronic device, in some embodiments, referring to... Figure 5 As shown, the electronic device 700 includes an input unit 710, a memory 720, a processor 730, and an output unit 740. The memory 720 stores program instructions that can be executed on the processor 730. The processor 730 can execute the construction method and / or technical solution of the non-intersecting Hamiltonian ring of the folded hypercube in the foregoing embodiments by calling the program instructions. The electronic device 700 can be a mobile terminal device such as a mobile phone or a computer.
[0053] Furthermore, embodiments of this application also provide a computer-readable storage medium for storing a computer program that performs a method for constructing a non-intersecting Hamiltonian cycle of a folded hypercube. For example, computer program instructions, when executed by a computer, can invoke or provide the methods and / or technical solutions according to this application through the operation of the computer. The program instructions for invoking the methods of this application may be stored in a fixed or removable storage medium, and / or transmitted via data streams in broadcast or other signal carrying media, and / or stored in a storage medium that operates according to the program instructions.
[0054] Obviously, those skilled in the art should understand that the modules or steps of this application described above can be implemented using general-purpose computing devices. They can be centralized on a single computing device or distributed across a network of multiple computing devices. Optionally, they can be implemented using computer-executable program code, thereby storing them in a storage device for execution by a computing device, or fabricating them separately as individual integrated circuit modules, or fabricating multiple modules or steps as a single integrated circuit module. Thus, this application is not limited to any particular combination of hardware and software.
[0055] The technical features of the above embodiments can be arbitrarily integrated. For the sake of brevity, not all possible integrations of the technical features in the above embodiments are described. However, as long as the integration of these technical features does not contradict each other, they should be considered to be within the scope of this specification.
[0056] The above embodiments merely illustrate several implementation methods of the present invention, and their descriptions are relatively specific and detailed, but they should not be construed as limiting the scope of the invention patent. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these all fall within the protection scope of the present invention. Therefore, the protection scope of this invention patent should be determined by the appended claims.
Claims
1. A method for constructing non-intersecting Hamiltonian cycles of a folded hypercube, applied to an n-dimensional folded hypercube. , n≥3, characterized in that, The method includes: Recursively construct the n-dimensional folded hypercube The first Hamiltonian cycle ; For the first Hamiltonian cycle Apply a mapping f to each vertex u in the equation, and connect the mapped vertices in the original order to obtain the second Hamiltonian cycle. ; Wherein, the mapping f is defined as: ； This represents the vertex obtained by inverting the first bit of vertex u. This represents the complementary vertex obtained by inverting all bits of vertex u; The first Hamiltonian cycle With the second Hamiltonian cycle The folded hypercube in the dimensional There are no public edges in it.
2. The method for constructing non-intersecting Hamiltonian cycles of a folded hypercube as described in claim 1, characterized in that, Recursively construct the n-dimensional folded hypercube The first Hamiltonian cycle ,include: When n=3, the preset Hamiltonian cycle vertex sequence is returned directly. The preset Hamiltonian cycle vertex sequence is: 000, 001, 011, 010, 110, 111, 101, 100. When n>3, first recursively construct an n-1 dimensional folded hypercube. Hamiltonian Circle Then based on structure .
3. The method for constructing non-intersecting Hamiltonian cycles of a folded hypercube as described in claim 2, characterized in that, When n>3, based on structure ,include: Will Add a "0" bit to the left of the binary string of each vertex, keeping the vertex order unchanged, to obtain the sub-cycle. ; Will Add a "1" bit to the left of the binary string of each vertex and reverse the order of the vertices to obtain the sub-cycle. ; make ,exist Deleting edges and in Deleting edges ; Add edge ( , )and( , ),Will and Connect them to form a complete Hamiltonian cycle, and we get .
4. The method for constructing non-intersecting Hamiltonian cycles of a folded hypercube as described in claim 1, characterized in that, The binary representation of vertex u is: ,in The least significant bit; vertices whose least significant bit in binary representation is 0 are defined as even vertices, and vertices whose least significant bit in binary representation is 1 are defined as odd vertices. Corresponding to the vertex ,in ; Corresponding to the vertex .
5. The method for constructing non-intersecting Hamiltonian cycles of a folded hypercube as described in claim 1, characterized in that, The mapping f is the n-dimensional folded hypercube. An automorphism on f, i.e., f is a vertex set V( A bijection from E to itself, and for any edge (a, b) ∈ E( ), (f(a), f(b))∈E( The second Hamiltonian cycle is guaranteed by the automorphism property. yes A Hamiltonian cycle.
6. The method for constructing non-intersecting Hamiltonian cycles of a folded hypercube as described in claim 1, characterized in that, The and The proof that the edges do not intersect includes: The mapping f will The edge mapping in is The edges in; If there exists a common edge (x, y) that belongs to both and Then it exists The edge (a, b) in the equation makes (f(a), f(b)) = (x, y); By analyzing the value of the least significant bit of the vertex and the mapping rules, it is deduced that (a, b) in The edge has been deleted during the construction process, leading to a contradiction and proving that no common edge exists.
7. A construction device for a folded hypercube with non-intersecting Hamiltonian rings, characterized in that, The device includes: The first construction module is used to recursively construct an n-dimensional folded hypercube. The first Hamiltonian cycle ; The second construction module is used for the first Hamiltonian cycle. Apply a mapping f to each vertex u in the equation, and connect the mapped vertices in the original order to obtain the second Hamiltonian cycle. ; Wherein, the mapping f is defined as: ； This represents the vertex obtained by inverting the first bit of vertex u. This represents the complementary vertex obtained by inverting all bits of vertex u; The first Hamiltonian cycle With the second Hamiltonian cycle The folded hypercube in the dimensional There are no public edges in it.
8. An electronic device comprising a memory, a processor, and a computer program stored in the memory and running on the processor, characterized in that, When the processor executes the computer program, it implements the method for constructing the non-intersecting Hamiltonian loop of the folded hypercube according to any one of claims 1 to 6.
9. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a computer program that, when executed by a processor, implements the method for constructing the non-intersecting Hamiltonian loop of the folded hypercube according to any one of claims 1 to 6.
10. A computer program product, comprising a computer program, characterized in that, When executed by a processor, the computer program implements the method for constructing the non-intersecting Hamiltonian loop of the folded hypercube as described in any one of claims 1 to 6.