WO2022082860A1

WO2022082860A1 - Lightweight and efficient graph vertex rearrangement method

Info

Publication number: WO2022082860A1
Application number: PCT/CN2020/125962
Authority: WO
Inventors: 刘志丹; 黄保福; 伍楷舜
Original assignee: 深圳大学
Priority date: 2020-10-21
Filing date: 2020-11-02
Publication date: 2022-04-28
Also published as: CN112380397B; CN112380397A

Abstract

A lightweight and efficient graph vertex rearrangement method. The method comprises: loading a graph to be subjected to ID rearrangement and performing preprocessing to determine a seed point (S1); selecting the seed point and generating a super vertex by means of the seed point (S2); and assigning new IDs to vertices of said graph according to the super vertex to achieve graph vertex rearrangement (S3). According to the method, vertices having low in-degrees are merged before ID reassignment, such that common neighbors of the merged vertices have similar IDs, and because the ID rearrangement is always performed according to information of the vertices directly connected to a current vertex, which is local information, a rearrangement operation can be completed quickly while achieving local optimization. The method can improve graph vertex rearrangement efficiency.

Description

A Lightweight and Efficient Graph Vertex Rearrangement Method

technical field

The invention relates to the field of graph computing, and more particularly, to a lightweight and efficient graph vertex rearrangement method.

Background technique

Since graphs can well represent various relationships in reality, and various graph algorithms can be used to discover potential facts in graph structures, graph computing has many applications. However, the calculation of large-scale graphs requires a very large amount of data to operate. In order to complete graph computations more efficiently, scholars have tried to optimize from many aspects, including the improvement of graph computing environment, optimization of algorithms, and preprocessing of graph data. aspect. The preprocessing of the graph data does not involve the specific graph algorithm before the execution of the graph algorithm, and the original algorithm does not need to be modified. Therefore, after a certain graph is preprocessed once, a single graph algorithm can be performed on the new graph. or multiple calculations.

The above graph calculation preprocessing method specifically refers to a method for reassigning graph vertex IDs. When using the graph structure to represent the real relationship, it is necessary to assign a unique ID to each graph vertex, which is convenient for graph storage and graph computation. Graph calculation mainly includes two stages: neighbor information acquisition and local calculation. A vertex needs to acquire the information of its neighbors. Specifically, it obtains the ID of its neighbor through the edge table, and then accesses the data corresponding to the ID. In specific applications, arrays are often used to store the data of all vertices in order. The ID of the vertex corresponds to the subscript of the array, and the data of the vertex can be directly accessed through <array+ID>. The access to the elements of the array is divided into sequential access and irregular access, both of which are quite different in terms of data access efficiency. The sequential access method can make good use of the principle of spatial locality to improve the hit rate of the cache, thereby speeding up the calculation speed. Therefore, the access to the array should be implemented in order as much as possible. In addition, frequently accessed elements in the array should be resident in the cache to obtain more time locality benefits of data access and speed up computation.

There has been extensive and in-depth research on the ID rearrangement of graph vertices. According to the time consumed by different methods to complete the rearrangement, it can be divided into time-consuming and lightweight ID rearrangement methods. The new ID of the graph vertex is obtained by a certain ID rearrangement method. If the time required to complete the calculation of the same graph on the new graph is less than that of the original graph, the rearrangement result of this method is said to be efficient, otherwise it is inefficient; If the time required for the method to complete the rearrangement is much greater than the calculation time of the graph algorithm, the rearrangement operation of the method is said to be inefficient, which is called a time-consuming ID rearrangement method, otherwise it is efficient, called lightweight Class ID rearrangement method.

The time-consuming ID rearrangement method is represented by Gorder (H.Wei, J.X.Yu, C.Lu, and X.Lin.Speedup graph processing by graph ordering.In ACM SIGMOD, 2016.). Gorder gives through analysis that structurally adjacent vertices should have similar IDs, that is, vertices with many common neighbors, their IDs should be similar. The formal description is as Equation 1. The first part represents the number of common neighbors of vertex u and vertex v, and the value is the total number of common neighbors. The second part represents whether u and v are neighbor vertices, and the value is 1 or 0. Gorder pursues the maximization of F(Φ) when assigning new IDs to vertices, and a more efficient rearrangement result can be obtained, where F(Φ) is calculated by Equation ②. But Gorder also pointed out that minimizing F(Φ) is an NP-hard problem. Using this method directly for rearrangement on large-scale graph data requires a long time for rearrangement, and the rearrangement operation is inefficient. It is suitable for practical applications, so Gorder proposes an approximate calculation method for rearrangement. Its rearrangement idea is consistent with maximizing F(Φ), so its rearrangement effect is also efficient, but its rearrangement operation is still low. effective.

_S (u,v)＝Ss(u,v)+ _Sn (u,v)①

Most of the lightweight ID rearrangement methods only consider the influence of the degree of the vertex (the number of vertices directly connected to the vertex, the same below) on the rearrangement effect. The rearrangement operation is efficient, but the rearrangement effect is low. Effective, such as DBG (P.Faldu, J.Diamond, and B.Grot.A closer look at lightweight graph reordering.In IEEE IISWC, 2019.). DBG considers the degree of vertices when assigning new IDs to vertices, and assigns new IDs to vertices through the clustering method, so that frequently visited vertices have similar IDs, and theoretically higher locality benefits can be obtained. However, since this type of method only considers the degree of vertices and does not combine more graph structure characteristics, this type of method cannot guarantee its rearrangement effect. Because the scale of the actual application graph is large, there are relatively many vertices accessed frequently, and the cache capacity of the computer is limited, which cannot accommodate all the vertex information of high frequency access at the same time. If the frequently accessed vertices are far apart in the graph structure, and the adjacent IDs are incorrectly assigned to such frequently accessed vertices, the limitation of cache capacity will cause many high-frequency vertices to be accessed to be replaced. cache, which reduces the cache hit rate and affects computing efficiency. Norder (E.Lee,J.Kim,K.Lim,S.H.Noh,and J.Seo.Pre-select static caching and neighborhood ordering for BFS-like algorithms on disk-based graph engines.In USENIX ATC,2019.) in On the basis of considering the vertex degree, combined with the neighbor information of the vertex, the effect of the rearrangement operation is relatively efficient, and the operation effect is relatively efficient; but since the neighbor information considered by Noder is local, its rearrangement effect is relative to Gorder. is much lower.

SUMMARY OF THE INVENTION

The present invention provides a lightweight and high-efficiency graph vertex rearrangement method in order to overcome the defect of low efficiency of graph vertex rearrangement described in the prior art.

The method includes the following steps:

S1: Load the graph that needs to be rearranged by ID, and perform preprocessing to determine the seed point;

S2: Select a seed point, and generate a super vertex from the seed point;

S3: Allocate new IDs to the vertices of the graph that need to be reordered according to the super-vertices, so as to realize the reordering of graph vertices.

Preferably, S1 includes the following steps:

S1.1: Load the graph G=(V, E) that needs to be rearranged by ID, where V represents the vertex set of the graph, E represents the edge set of the graph, and |V| and |E| represent the number of vertices and the number of edges respectively;

Then store the whole image information, set the flag bit is_trivial of the vertex according to the in-degree of the vertex, set is_trivial=true if the in-degree is greater than the given threshold λ, otherwise set is_trivial=false;

S1.2: Set rearrangement rules and select seed points.

Preferably, the information of the whole image is stored in a compressed sparse row (Compressed Sparse Row, CSR) manner.

Preferably, S1.2 is specifically as follows: in order to reproduce the rearrangement result, set the ID rearrangement from the vertex with the smallest original ID of G to start the ID rearrangement, and select the seed to lead you to seed, where the vertex with the smallest original ID is the initial seed point.

Among them, the selection method of seed points is as follows:

(1) when the connected component of the graph that needs to carry out ID rearrangement also has a vertex that does not have a new ID assigned, select an unassigned vertex of the vertex that has been assigned a new ID as a seed point;

(2) when all vertices of the connected components of the graph that need to carry out ID rearrangement have all obtained new IDs, select a vertex as a seed point from the remaining connected components that do not carry out ID allocation;

(3) There may be multiple vertices that meet the requirements of (1) and (2) above, so the vertex with the relatively smallest original ID is selected as the seed point.

(4) The selection of the initial seed point is carried out according to the above (2).

Preferably, S2 is specifically:

Taking the initial seed point seed as the center, merge the vertices with low in-degree within k hops that have not yet been assigned a new ID into a super vertex H, and obtain its hypergraph structure;

The vertices of the hypergraph structure are merged hypervertices. A super vertex is a logical concept and is actually a collection of vertices.

Preferably, the generation of hypervertices includes:

S2.1: Initialize the ID allocation flag of all vertices of the graph that needs to be rearranged by ID assigned=false, set the new ID of all vertices to Φ(*)=-1, seed point seed=-1, set auxiliary variable move_id=1 (representing the new ID that can be allocated currently), auxiliary variable v=-1 (representing the original ID of the vertex currently participating in the allocation).

S2.2: The super vertex H is obtained by the super vertex generation function Fusion(seed):

Preferably, S2.2 includes the following steps:

S2.2.1: Input the seed point seed, and set the super vertex H={seed};

S2.2.2:

(N _Hv represents the set of neighbor vertices of H), check the flag bit is_trivial of vertex u, if the flag bit is_trivial is false, skip further operations on the vertex (that is, do not operate on the vertex), if the flag bit is_trivial is true, execute S2 .2.3;

S2.2.3: Check the assigned flag bit assigned of vertex u, if the assigned flag bit assigned is true, then skip the further operation of the vertex (that is, do not operate the vertex), if the assigned flag bit assigned is false, add the point to H middle.

S2.2.4: Repeat the above operations (S2.2.1-S2.2.3) for other connected outgoing neighbors one by one. After merging all unassigned neighbor vertices, add one to the number of merged hops;

S2.2.5: Repeat the above steps (S2.2.1-S2.2.4) for the newly added vertices in H, until the number of hops hop reaches the given number of merged hops k, one merge operation is completed, and the super vertex H is output; The low-in-degree vertices within k-hops are merged into H and output.

Preferably, S3 includes the following steps:

S3.1: First assign a continuous ID to the super vertex H generated by S2, and then assign a new ID to the neighbor vertices of the super vertex H; update the assignment flag bit assigned=true of the vertices that have obtained the new ID above.

S3.2: Select a vertex from the neighbor vertices of the super vertex H as the seed point seed. When all the vertices of a connected component of the seed point seed have obtained new IDs, select the vertex with the smallest original ID among the remaining connected components as the seed point seed. , perform S2 until all vertices get unique new IDs.

Preferably, the allocation rule for allocating new IDs to the neighbor vertices of the super vertex H is as follows: firstly, the high-in-degree neighbor vertices are allocated consecutive IDs, and then the low-in-degree neighbor vertices are allocated consecutive IDs.

Compared with the prior art, the beneficial effect of the technical solution of the present invention is that the method of reassigning IDs by merging low-in-degree vertices enables the common neighbors of the merging vertices to have similar IDs. The method of the present invention can effectively improve the graph vertex rearrangement efficiency.

Description of drawings

FIG. 1 shows the lightweight and efficient graph vertex rearrangement method described in Embodiment 1. FIG.

FIG. 2 is a schematic diagram of reassignment of graph vertex IDs.

FIG. 3 is a flowchart of graph vertex ID reassignment.

Figure 4 is a flow chart of generating hypervertices from seed points.

FIG. 5 is a process diagram of rearranging the numbers of the pictures that need to be rearranged by ID.

Detailed ways

The accompanying drawings are for illustrative purposes only, and should not be construed as limitations on this patent;

In order to better illustrate this embodiment, some parts of the drawings are omitted, enlarged or reduced, which do not represent the size of the actual product;

It will be understood by those skilled in the art that some well-known structures and their descriptions may be omitted from the drawings.

The technical solutions of the present invention will be further described below with reference to the accompanying drawings and embodiments.

Example 1

This embodiment provides a lightweight and efficient graph vertex rearrangement method. As shown in FIG. 1 , the method includes the following steps:

S1: Load the graph that needs to be rearranged by ID, and perform preprocessing to determine the seed point.

S1.1: Load the graph G=(V, E) that needs to be rearranged by ID, where V represents the vertex set of the graph, E represents the edge set of the graph, and |V| and |E| represent the number of vertices and the number of edges, respectively. The whole image information is stored in the form of Compressed Sparse Row (CSR). The flag bit is_trivial of the vertex is set according to the in-degree of the vertex. If the in-degree is greater than the given threshold λ, is_trivial=true, otherwise, is_trivial=false.

S1.2: In order to reproduce the rearrangement result, this method sets the ID rearrangement from the vertex with the smallest original ID, and selects the seed point seed, where the vertex with the smallest original ID is selected as the initial seed point.

S2: Take the seed point seed as the center, merge the vertices with low in-degree within k hops and have not yet been assigned a new ID into a super-vertex H to obtain the hyper-graph structure, and the vertices of the hyper-graph are the merged hyper-vertices.

In the above step S2, the seed is selected and the super vertex H is generated by the seed, and then a new ID is allocated to the vertex. The reassignment of the graph vertex ID is shown in Figure 2. The specific allocation process is as follows (as shown in Figure 3):

S2.2: Obtain H (see Figure 4) through the hypervertex generating function Fusion(seed), which includes the following steps:

S2.2.1: Set the super vertex H={seed},

S2.2.2:

(N _Hv represents the set of neighbor vertices of H), check the flag bit is_trivial of vertex u, false skips further operations on the vertex, and true continues the next operation.

S2.2.3: Check the assigned flag bit assigned of vertex u, true skips further operations on the vertex, and false adds the point to H.

S2.2.4: Repeat the above steps for other connected outgoing neighbors one by one. After merging all unassigned neighbor vertices, increase the merge hop count by one.

S2.2.5: Repeat the above steps for the newly added vertices in H until the number of hops hop reaches the given number of merged hops k, one merge operation is completed, and the super vertex H is output.

S3.1: First assign continuous IDs to the H generated above, and then assign new IDs to the neighbor vertices of H. First, assign continuous IDs to high-in-degree neighbor vertices, and then assign continuous IDs to low-in-degree neighbor vertices. Update the assigned flag bit assigned=true of the vertex that has obtained the new ID above.

In other words, the specific operation of S3.1 is: assign a new ID to the H obtained by S2, and then assign a new ID to the vertices directly connected to H. Specifically, firstly, a new ID is assigned to a neighbor vertex whose flag bit is_trivial=false, and then a new ID is assigned to a vertex whose flag bit is_trivial=true. Sets the flag bit is_assigned to true for vertices that have been assigned.

S3.2: Select a vertex from the neighbor vertices of H above as the seed. When all the vertices of a connected component of the seed point seed have obtained new IDs, select the vertex with the smallest original ID among the remaining connected components as the seed point seed, and execute the steps S2 until all vertices get unique new IDs.

In other words, the specific operation of S3.2 is: select a vertex from H's neighbor vertices as a seed, and execute 2.2. If all vertices of the current connected component have obtained new IDs, select the vertex with the smallest original ID from the remaining connected components as seed, and execute 2.2 until all vertices have obtained unique new IDs.

As a specific embodiment, the present embodiment will be described below in conjunction with specific examples:

Taking Fig. 5(a) as an example, the process of performing the numbering rearrangement in this embodiment is as follows (see Fig. 5), the in-degree threshold λ=1 is set, the number of merged hops k=1, and the currently assignable new ID number is 0. ; Due to the small scale of the example graph, this example only selects seeds from the neighbors on the outgoing edge:

Step 1: Select the vertex with the smallest original ID as the seed from the vertices that have not obtained the new ID: the vertex that meets the conditions is No. 1 vertex, and select seed=1.

Step 2: Generate super vertices: merge the vertices within the seed vertex k hops, the in-degree is less than λ, and the new ID has not yet been assigned, and the vertices of vertex 1 within 1 hop, the in-degree is less than λ, and the vertices that have not yet been assigned a new ID are 3, 7 number vertex, merge (1, 3, 7) into a super vertex, see Figure 5(a).

Step 3: Allocate new IDs for the vertices that have not obtained new IDs in the super-vertices: the new ID number that can be assigned is 0, so the vertices with the original IDs of 1, 3, and 7 correspond to the new IDs of 1, 2, and 3. The new ID number currently assignable is 4.

Step 4: Assign a new ID to the neighbor of the super vertex that has not obtained a new ID: the

number

5, 9, and 6 among the neighbors of the super vertex are high-in-degree vertices (sequentially assign new IDs to the high-in-degree vertices of each vertex of the super vertex , that is, first assign a new ID to the high-in-degree neighbor of vertex 1, then to the high-in-degree neighbor of vertex 3, and finally to the high-in-degree neighbor of vertex 7, the same below), 10 is the low-in-degree vertex (low The ID allocation order of the in-degree neighbors is the same as that of the high-in-degree neighbors above, the same below). The current assignable new ID number is 4, so the new IDs corresponding to the vertices whose original IDs are 5, 9, 6, and 10 are 4, 5, 6, and 7. The currently assignable new ID number is 8, see Figure 5(b).

Step 5: Select seeds from neighbors: Since it is set here that only seeds are selected from outgoing neighbors,

outgoing neighbors

5, 9, 6, and 10 all get new IDs.

Step 6: Select the vertex with the smallest original ID as the seed from the vertices that have not obtained the new ID: the vertex that meets the conditions is No. 2 vertex, and select seed=2.

Step 7: Generate super vertices: merge the vertices whose in-degree is less than λ in seed vertex k and have not yet been assigned a new ID, and vertex 2, whose in-degree is less than λ and has not yet been assigned a new ID, is vertex 8. Merge (2, 8) into a super vertex, see Figure 5(b).

Step 8: Allocate new IDs for the vertices that have not obtained new IDs in the super vertices: the current new ID number that can be allocated is 8, so the vertices whose original IDs are 2 and 8 correspond to new IDs of 8 and 9. The new ID number that is currently assignable is 10.

Step 9: Assign a new ID to the neighbor of the super vertex that has not obtained a new ID: the number 4 among the neighbors of the super vertex is a high-in-degree vertex. The current assignable new ID number is 10, so the new ID corresponding to the vertex whose original ID is 4 is 10. The currently assignable new ID number is 11, see Figure 5(c).

Step 10: Select seed from neighbors: select seed=4;

Step 11: Generate super vertices: merge the vertices with seed vertex k hops whose in-degree is less than λ and have not yet been assigned a new ID, and vertex 4, whose in-degree is less than λ and has not yet been assigned a new ID, is vertex 11. Merge (4, 11) into hypervertices, see Figure 5(c).

Step 12: Allocate a new ID to a vertex that has not obtained a new ID in the super-vertices: the new ID number that can be assigned is 11, so the new ID corresponding to the vertex whose original ID is 11 is 11. The new ID number currently assignable is 12. (Here, the vertex with the original ID of 4 has obtained a new ID and cannot be assigned repeatedly).

Step 13: Assign a new ID to the neighbors of the super-vertex that have not obtained the new ID: The neighbors of the super-vertex have all obtained new IDs and cannot be assigned repeatedly.

Step 14: The rearrangement is completed, and the result is shown in Figure 5(d).

In this embodiment, the reassignment mechanism of adjacent low-in-degree vertices is combined and the ID allocation method for distinguishing the high- and low-in-degree distribution order is adopted. Among them, the purpose of merging adjacent low-in-degree vertices is to determine the set of common neighbors between vertices in a fast method. Related studies have shown that vertices with more common neighbors should have adjacent IDs. The property should be consistent with the proximity of vertices in the graph structure. With this mapping relationship, higher locality benefits and computational efficiency can be obtained when data access is performed in specific graph computations. The order of ID allocation for distinguishing high and low in-degree vertices is to make the IDs of frequently accessed vertices similar. Such vertices are frequently accessed, and similar IDs can improve the possibility of such vertices resident in the cache. Because a certain vertex generally has high-in-degree neighbors and low-in-degree neighbors, due to the limitation of cache capacity, if the high-in-degree vertex IDs are far apart, the access to low-in-degree neighbors and high-in-degree neighbors may lead to certain Some high-in-degree vertices that are about to be accessed are replaced out of the cache, reducing the cache hit rate.

The terms describing the positional relationship in the accompanying drawings are only used for exemplary illustration, and should not be construed as a limitation on this patent;

Obviously, the above-mentioned embodiments of the present invention are only examples for clearly illustrating the present invention, rather than limiting the embodiments of the present invention. For those of ordinary skill in the art, changes or modifications in other different forms can also be made on the basis of the above description. There is no need and cannot be exhaustive of all implementations here. Any modifications, equivalent replacements and improvements made within the spirit and principle of the present invention shall be included within the protection scope of the claims of the present invention.

Claims

A lightweight and efficient graph vertex rearrangement method, characterized in that the method comprises the following steps:

S1: Load the graph that needs to be rearranged by ID, and perform preprocessing to determine the seed point;

S2: Select a seed point, and generate a super vertex from the seed point;

S3: Allocate new IDs to the vertices of the graph whose IDs need to be rearranged according to the super vertices, so as to realize the rearrangement of the graph vertices.
The lightweight and efficient graph vertex rearrangement method according to claim 1, wherein S1 comprises the following steps:

S1.1: Load the graph G=(V, E) that needs to be rearranged by ID, where V represents the vertex set of the graph, E represents the edge set of the graph, and |V| and |E| represent the number of vertices and the number of edges respectively;

Then store the whole image information, set the flag bit is_trivial of the vertex according to the in-degree of the vertex, set is_trivial=true if the in-degree is greater than the given threshold λ, otherwise set is_trivial=false;

S1.2: Set rearrangement rules and select seed points.
The lightweight and high-efficiency graph vertex rearrangement method according to claim 2, characterized in that the storage of the whole graph information is performed by compressing sparse rows.
The lightweight high-efficiency graph vertex rearrangement method according to claim 3, wherein, S1.2 is specifically: setting the ID rearrangement from the vertex with the smallest original ID of the graph G that needs to be ID rearranged, Select the seed point seed, where the vertex with the smallest original ID is the initial seed point.
The lightweight high-efficiency graph vertex rearrangement method according to any one of claims 1-4, wherein S2 is specifically:

Taking the seed point seed as the center, merge the vertices with low in-degree within k hops that have not yet been assigned a new ID into a super vertex H.
The lightweight high-efficiency graph vertex rearrangement method according to claim 5, wherein the generation of the super-vertices comprises:

S2.1: Initialize the ID allocation flag of all vertices in the graph that needs to be rearranged by ID assigned=false, set the new ID of all vertices to Φ(*)=-1, seed point seed=-1, set auxiliary variable move_id=1 , indicating the new ID that can be assigned at present; the auxiliary variable v=-1, indicating the original ID of the vertices that can be assigned at present;

S2.2: Obtain the hypervertex H through the hypervertex generating function Fusion(seed).
The lightweight and efficient graph vertex rearrangement method according to claim 6, wherein S2.2 comprises the following steps:

S2.2.1: Input the seed point seed, and set the super vertex H={seed};

S2.2.2:
Among them, N Hv represents the set of neighbor vertices of H, check the flag bit is_trivial of vertex u, if the flag bit is_trivial is false, skip the further operation of the vertex, if the flag bit is_trivial is true, execute S2.2.3;

S2.2.3: Check the assigned flag bit assigned of vertex u. If the assigned flag bit assigned is true, the further operation of the vertex will be skipped. If the assigned flag bit assigned is false, the point will be added to H;

S2.2.4: Repeat S2.2.1-S2.2.3 for other connected outgoing neighbors one by one. After merging all unassigned neighbor vertices, add one to the number of merged hops;

S2.2.5: Repeat S2.2.1-S2.2.4 for the newly added vertices in H until the number of hops hop reaches the given number of merged hops k, and merge the low-in-degree vertices within the seed point k hops into H and then output .
The lightweight high-efficiency graph vertex rearrangement method according to claim 7, characterized in that, S3 comprises the following steps:

S3.1: First assign a continuous ID to the super vertex H generated by S2, and then assign a new ID to the neighbor vertices of the super vertex H; update the assignment flag bit assigned=true of the vertices that have obtained the new ID above;

S3.2: Select a vertex from the neighbor vertices of the super vertex H as the seed point seed. When all the vertices of a connected component of the seed point seed have obtained new IDs, select the vertex with the smallest original ID among the remaining connected components as the seed point seed. , perform S2 until all vertices have unique new IDs.
The lightweight high-efficiency graph vertex rearrangement method according to claim 8, characterized in that, the allocation rule for allocating new IDs for the neighbor vertices of the hypervertex H is: firstly, allocating continuous IDs for high-in-degree neighbor vertices, and then for low-in-degree neighbor vertices. In-degree neighbor vertices are assigned consecutive IDs.
The lightweight high-efficiency graph vertex rearrangement method according to any one of claims 1 or 9, wherein the method for selecting seed points is as follows:

(1) when the connected component of the graph that needs to carry out ID rearrangement also has the vertex that does not have the new ID assigned, select a vertex that does not assign the ID of the vertex that has been assigned the new ID as the seed point;

(2) When all the vertices of the connected components of the graph that need to carry out ID rearrangement have all obtained new IDs, select a vertex as a seed point from the remaining connected components that do not carry out ID allocation;

(3) If there are multiple vertices that meet the requirements of (1) and (2) above, select the vertex with the relatively smallest original ID as the seed point;

(4) The selection of the initial seed point is carried out according to the above (2).