CN111597404A

CN111597404A - Based on k2Maximum common subgraph matching method of labeled graph of MDD (minimization drive distribution)

Info

Publication number: CN111597404A
Application number: CN202010441592.6A
Authority: CN
Inventors: 朱奕杰; 冷贝贝; 关善文; 沈薪童; 罗力耕
Original assignee: Guilin Intelligent Industrial Park Co ltd
Current assignee: Guilin Intelligent Industrial Park Co ltd
Priority date: 2020-05-22
Filing date: 2020-05-22
Publication date: 2020-08-28

Abstract

The invention discloses a method based onk ²-a method for maximum common subgraph matching of labeled graphs of MDD, comprising the steps of: screening the vertex labels of the matching images, selecting the vertexes of the matching images which are the same as the target image labels, and changing the labels of the vertexes into the labels of the vertexes corresponding to the vertexes in the target image to obtain a new matching image; according tok ²The rules of the tree respectively encode the vertexes of the target graph and the new matching graph; encoding the edges of the target graph and the new matching graph according to the vertex codes of the target graph and the new matching graph; constructing a multi-value decision diagram structure according to the encoding of the edges of the target diagram and the new matching diagram to obtain the target diagram and the new matching diagramk ²-MDD structural diagram; in the constructed target graph and matching graphk ²In the MDD structure diagram, the purpose is obtained by using logical operation of symbol decision diagramThe largest common subgraph of the plots and the matching plots. The method can make the storage structure more compact, and greatly reduce the number of generated vertexes, thereby reducing the search space and improving the search efficiency.

Description

Based on k2Maximum common subgraph matching method of labeled graph of MDD (minimization drive distribution)

Technical Field

The invention relates to the technical field of maximum public subgraph matching of labeled graphs, in particular to a graph based on a label graphk ²Maximum common subgraph matching method of labeled graphs of MDD.

Background

The graph is an important mathematical model and data structure for describing specific relations between things, and is widely applied to various fields in life, such as the world wide web, social networks, protein interaction networks, chemical molecular structures, and the like. With the continuous increase of the scale of Graph data, effective and rapid analysis and processing of the Graph data still face a serious challenge, and a Graph Pattern Matching technology (Graph Pattern Matching) as an important method for realizing efficient query on the Graph data has become a research hotspot of scholars at home and abroad. From the perspective of whether the matching result is completely consistent with the pattern graph, the graph matching problem can be divided into exact matching and non-exact matching, wherein the exact matching generally analyzes the relationship between the data graph and the pattern graph by defining graph isomorphism and sub-graph isomorphism; non-exact graph matching generally measures the degree of similarity between two graphs by defining edit distance, minimum common hypergraph, and maximum common subgraph.

The largest common subgraph has wide application in biology, chemistry, computer vision, source code analysis, binary programming, circuit design, character recognition problems, and many other fields, and in some fields it is directed to a method of measuring similarity or difference between two graphs.

In order to compactly represent graph data, Brisaboa et al, in 2009, proposed a graph based on a traditional adjacency matrix representationk ²Tree (k ²-tree), each level in the tree corresponding to a partitioned sub-matrix of the adjacency matrix or of the partitioned sub-matrix, the nodes corresponding to the partitioned sub-matrices of the adjacency matrix, generatedk ²The tree is stored by using two bit vectors T and L, and the method not only can compactly represent the adjacency matrix, but also can realize forward or reverse efficient query operation of the adjacent nodes. To address this challenge, Li et al propose using Brisaboak ²Tree structures to solve this problem. Although usingk ²The tree structure is used for storing the large-scale graph, so that the structure is more compact, the number of nodes is greatly reduced, and the tree structure still has certain limitation when large-scale graph data are represented. Sch\ 20346k ²Two optimization techniques for tree representation: heuristic depth-first node reordering and adaptive correction k enable the expressed structure to be more compact and nodes to be obviously reduced.

Whether it bek ²Whether trees are optimized by k ²The tree still has certain limitation when representing large-scale graph data storage, and is particularly represented as follows:

1. when the scale of the graph becomes larger, a large number of isomorphic subgraphs exist in the graph, and similarly, when the graph is according tok ²The tree concept divides the adjacent matrix into a large number of identical sub-matrices. This results ink ²There are also a large number of isomorphic subtrees within the tree.

2、k ²The tree is only valid for sparse graphs, when the graph becomes dense, since there are fewer 0 nodes that can be compressed within the adjacency matrix, sok ²The tree compactness also becomes low.

3、k ²The tree does not involve the representation and operation of dynamic graphs (graphs that require the addition or deletion of vertices, edges, subgraphs, etc.).

At present, the use of the compositionk ²The method for representing the tree graph data compactly lacks necessary consideration for the structural characteristics of the graph, and still has great improvement space in compactness. For based onk ²There is a need for further optimization and improvement of the tree, so as to obtain a more compact structure representation method, which further reduces the storage space of the nodes.

Disclosure of Invention

The invention aims to overcome the defects of the prior art and provides a device based onk ²Maximum common subgraph matching method of labeled graphs of MDD.

The technical scheme for realizing the purpose of the invention is as follows:

based onk ²-a method for maximal common subgraph matching of labeled graphs of MDDs, comprising the steps of:

1) screening the vertex labels of the matching images, selecting the vertexes of the matching images which are the same as the target image labels, and changing the labels of the vertexes into the labels of the vertexes corresponding to the vertexes in the target image to obtain a new matching image;

2) according tok ²The rules of the tree respectively encode the vertexes of the target graph and the new matching graph;

3) encoding the edges of the target graph and the new matching graph according to the vertex codes of the target graph and the new matching graph;

4) constructing a multi-value decision diagram structure according to the encoding of the edges of the target diagram and the new matching diagram to obtain the target diagram and the new matching diagramk ²-MDD structural diagram;

5) in the constructed target graph and matching graphk ²In the MDD structure diagram, the maximum common subgraph of the target graph and the matching graph is respectively obtained by using the logical traffic operation of the symbol decision graph.

In step 2), the number of coded bits of each vertex of the target graph and the new matching graph is n bits, whereinkIs an integer of 2 or moreVAnd l is the number of the vertex points, and the vertex coding steps of the target graph and the new matching graph are as follows:

2-1) according tok ²Rules for tree-to-target graph and new matching graph adjacency matrix partitioning, i.e.k ²Dividing rule, determining coding length n of vertex of graph data as vertex numberkBase logarithmic ceiling, i.e.

WhereinkIs an integer of 2 or more;

2-2) encoding vertices using a binary approach, the lower bound L of the binary_T=1, upper bound H_T=2ⁿFor number isNThe vertex of (1) is less than or equal toN≤|V(|, to the total number |)VCoding the vertex of the I according to a recursive halving mode, wherein each bit in the n-bit coding of the vertex is one of 2 states, namely 0 or 1;

2-3) if L_T＜H_TThe median of the dichotomy is equal to half of the sum of the upper bound and the lower bound; if it isNLess than or equal to the median value, given a number ofNOne bit of the vertex of (1) is encoded as "0", and the median value minus 1 is taken as the upper bound H_T(ii) a Otherwise, get the serial number ofNOne bit of the vertex of (1) is encoded as "1", while the median value plus 1 is taken as the lower bound L_T；

2-4) step 2-3) is repeated until L_T≥H_TGet the number asNN-bit encoding of the vertices of (1), each bit of the n-bit encoding beingkOne of the states, namely (0, 1, …,k-1) of the above.

In step 3), the edges of the target graph and the new matching graph are encoded, the directed edges of the target graph and the new matching graph are relationships between vertices, and are described by using a feature function between the vertices, which is specifically as follows:

3-1) X = (X)₁，…，x_n)、Y=(y₁，…，y_n) For the vertex coded vector in the figure, the characteristic function of the vertex X to vertex Y edge is represented as:E(X，Y)：{0，1，…k-1}ⁿ×{0，1，…k-1}ⁿ→ {1，2，…k ²}ⁿi.e. on each bit of the two vertex codeskCombining the species states to obtaink ²In the seed state, the code length of the edge is still n bits, and each bit of the code isk ²One of the seed states;

3-2) coding the edge according to the vertex codes obtained in the step 2), and combining the coding states of certain corresponding bits of the two vertexes of the edge to obtain one-bit codes of the edge; and the n-bit coding states are correspondingly combined in sequence to obtain the n-bit coding of the edge.

In step 4), the target graph and the new matching graphk ²-MDD structural diagram, obtained in particular by the following steps:

4-1) initializing an MDD comprising n variables, i.e. the code length n of a vertex or edge, the value range of n being 1 tok ²；

4-2) assuming that the target graph or the matching graph has m edges, coding one edge obtained in the step 3), and generating an initial MDD by using a createEdge () function in the MEDDLY library, wherein the initial MDD is marked as R; taking one edge from the rest edges, generating MDD of the other edge by the same method, and marking as T;

4-3) using UNION operator provided in MEDDLY library to perform UNION operation on MDDs (namely R and T) generated by the strip edges, and storing the combined result in the initial MDD, namely covering the combined result with the original R and still recording the combined result as R;

4-4) continuously taking one edge from the rest edges, generating MDD by the same method, marking as T, carrying out UNION operation on R and T, and marking the combined result as R;

4-5) repeating the steps until all edges generate MDD and combining the MDD into initial MDD, and obtaining the final initial MDD R which is the graphk ²-MDDAnd (5) structure.

In step 5), the step of obtaining the maximum common subgraph of the target graph and the matching graph by using the logic traffic operation of the symbolic decision graph is to find the maximum common subgraph of the target graph and the matching graph in the edge queryk ²Any 2 vertices v on the MDD structure diagram₁And v₂Performing an edge query operation, if the query return result is T, representing a vertex v₁And v₂If the query returns a result of F, the vertex v is represented₁And v₂The edge in between does not exist.

The invention provides a method based onk ²The maximum common subgraph matching method of MDD labeled graph includes screening the top point labels of the matched graph, selecting the top points in the matched graph identical to the target graph labels, and matching the top pointsThe labels are changed into the labels of the vertexes corresponding to the labels in the target graph, and then a new matching graph can be obtained. Then, the vertex of the target graph and the new matching graph are coded, the edge is coded according to the coding of the vertex, and then the coding set of the edge is constructedk ²-an MDD; thereafter using the symbolk ²-logical traffic operation of the MDDs finding a common subgraph. For the inventionk ²The idea of the tree is to divide the adjacent matrix and then use the multi-valued decision diagram for storage, so thatk ²Redundant vertexes caused by a large number of isomorphic subtrees in the tree are combined, the purpose of more compact storage structure is achieved, the number of generated vertexes is greatly reduced, the search space is reduced, and the search efficiency is also improved.

Drawings

FIG. 1 is an original target graph G_TAnd the original matching graph G_P；

FIG. 2 is an original target graph G_TAnd a new matching graph G_Q；

FIG. 3 is a target graph G_TAnd matching graph G_QIs/are as followsk ²-MDD_TAndk ²-MDD_Q；

FIG. 4 is a drawing showingk ²-MDD results graph.

Detailed Description

The invention will be further described with reference to the following drawings and examples, but the invention is not limited thereto.

A Multi-valued Decision Diagram (MDD) is a directed acyclic graph with multiple terminal nodes, describing a node with a node numbernA discrete multi-valued function of the individual variables,f：D₁×D₂×…×Di×…×Dn→ S, wherein:

1、Di={1，2，…，n _iis a multivalued variablex _iThe value ranges of the different variables may be different; s is a multi-valued functionfThe finite value range of (1), i.e. the value set of the MDD terminal node, which may be boolean values (true and false, or 0 and 1), a finite set of integersA sum or a finite set of real numbers.

2. The nodes of the MDD include terminal nodes and non-terminal nodes.

3. For non-terminal nodesx _iIs shown as comprisingn _iPointers to other nodes, the pointers and functionsfCorrespondingly, the formalization is described as formula (1):

f _{xi c=}=f（x ₁，x ₂，…，x _i-1，c，x _i+1，…，x _n）（1）

multivalued variable x₁To x_nAnd giving a group of values to obtain a unique terminal node value.

The simplification rules for MDD are the following three:

rule 1, merging the same terminal nodes:

only one terminal node with the same attribute is reserved, other terminal nodes with the same attribute are deleted, and the pointer originally pointing to the deleted terminal nodes is redirected to the reserved terminal nodes.

Rule 2, merge the same internal nodes:

only one internal node with the same attribute, namely a non-terminal node, is reserved, other internal nodes with the same attribute are deleted, and the pointer originally pointing to the deleted nodes is redirected to the reserved internal nodes.

Rule 3, delete redundant nodes:

if all the pointers of a node point to the same node, then the node is a redundant node, it is deleted, and the pointer to the node points to the child node of the deleted node.

The invention provides a method based onk ²-maximum common subgraph matching algorithm of labeled graphs of MDD, given the process of constructing the algorithm: firstly, encoding the vertexes in the two graphs; encoding edges again according to the encoding of the vertex; encoding set construction by edgesk ²-an MDD; by means of symbolsk ²The logical operation of the MDD enables the solution of the maximum common subgraph match.

Example (b):

1) screening the vertex labels of the matching graph to select the vertexes in the matching graph, which are the same as the target graph labels, and obtaining the original target graph G_TAnd the original matching graph G_PAs shown in fig. 1, the new matching graph is obtained by changing the labels of the vertices to the labels of the vertices corresponding to the vertices in the target graph, i.e., the vertex P in fig. 1 (b)₀、P₁、P₂And the vertex T in FIG. 1 (a)₁、T₂、T₄So that fig. 1 (b) can be converted as shown in fig. 2, and a new matching graph is shown in fig. 2;

4) constructing a multi-value decision diagram structure according to the encoding of the edges of the target diagram and the new matching diagram to obtain the target diagram and the new matching diagramk ²MDD structure diagram, FIG. 3, it can be seen from FIG. 3 that of the new matching graphk ²The MDD structure, also called the process of FIG. 2, transformed into FIG. 3;

5) in the constructed target graph and matching graphk ²In the MDD structure diagram, the maximum common subgraph of the target diagram and the matching diagram is obtained by using the logical operation of the symbol decision diagram respectively, namely the maximum common subgraph of the target diagramk ²-MDD_TWith new matching figuresk ²-MDD_QA new one can be obtained by performing the logical intersection operationk ²MDD, as shown in fig. 4. FIG. 4 is obtained by the logical "intersection" operation of FIG. 3, from which FIG. 4 the vertices of the matching graph that match the target graph, and thus the matching result (P)₀，T₁），（P₁，T₂），（P₂，T₄) (ii) a This is the desired end result.

In step 2), the number of coded bits of each vertex of the target graph and the new matching graph is n bits, wherein k is an integer greater than or equal to 2, and the caltrops are not overlapped with each otherVAnd l is the number of the vertex points, and the vertex coding steps of the target graph and the new matching graph are as follows:

WhereinkIs an integer of 2 or more;

2-2) this examplek=2, encode vertex using dichotomy, lower bound of dichotomy L_T=1, upper bound H_T=2ⁿFor number isNThe vertex of (1) is less than or equal toN≤|V(|, to the total number |)VCoding the vertex of the I according to a recursive halving mode, wherein each bit in the n-bit coding of the vertex is one of 2 states, namely 0 or 1;

3-1) e.g. the edge between vertex X and vertex Y, described by a characteristic function E (X, Y), let X = (X)₁，…，x_n)、Y=(y₁，…，y_n) For the vertex coded vector in the figure, the characteristic function of the vertex X to vertex Y edge is represented as:E(X，Y)：{0，1，…k-1}ⁿ×{0，1，…k-1}ⁿ→ {1，2，…k ²}ⁿi.e. on each bit of the two vertex codeskCombining the species states to obtaink ²In the seed state, the code length of the edge is still n bits, and each bit of the code isk ²One of the seed states;

3-2) according to the vertex codes obtained in the step 2), then coding the edge, and combining the coding states of certain corresponding bits of the two vertexes of the edge to obtain a bit code of the edge; and the n-bit coding states are correspondingly combined in sequence to obtain the n-bit coding of the edge.

4-1) initializing an MDD comprising n variables, i.e. the code length n of a vertex or edge, the value range of the variable n being 1 ℃; 1Ek ²Examples of the case are 1 to 4;

4-2) assuming that the target graph or the matched graph has m edges, coding one edge obtained in the step 3), and generating an initial MDD by using a createEdge () function in the MEDDLY library, wherein the initial MDD is marked as R; taking one edge from the rest edges, generating MDD of the other edge by the same method, and marking as T;

4-5) repeating the above stepsUntil all edges generate MDD and combine into initial MDD, the final initial MDD R is the graphk ²-MDDAnd (5) structure.

k ²-MDDThe structure is a special case of the MDD structure, and defines the number of variables and the value range of the variables, wherein the value range of each variable is {1,2, …,k ²}. Therefore, it isk ²-MDDHas the property of MDD, and applies the simplification rule of MDD.

k ²-MDDBy using a graph-structured adjacency matrix with a recursion of the original matrixk ²A multi-valued decision diagram structure constructed after equal division, wherein any unit in a adjacency matrix of the diagram corresponds tok ²-MDDAnd obtaining a unique function value, namely a terminal vertex value, according to the unique group of values of the n variables, wherein the value is equal to the element value of the corresponding cell in the original matrix. Constructed according to the invention and containing n variablesK ²-MDDMaking the values of n variables equal to the values in the edge coding set, the function value is T, otherwise, it is F, and the obtained valuek ²-MDDCorresponding to the original map.

Using a Multi-endpoint and Edge decision gallery, MEDDLY (Multi-terminal and Edge-value decision dictionary), a map is created with a range of values that are all {1,2, …,k ²n variables of. From these n variables, a boolean MDD is used whose end point is true (T) indicating that the conceptual parameter has a correlation with the service, and false (F) indicating that the conceptual parameter has no correlation with the service.

The MEDDLY library is a C/C + + open source project provided for the manipulation of MDDs, developed by Iowa State university under the LINUX platform, which provides rich functions of MDD construction and operation. For example: creating the number of variables and the value range of each variable for constructing the MDD by using a createVariablesBottomUp () function; generating an MDD from the values of the given one or more sets of variables using a createEdge () function; the two MDDs are merged using the apply () function and the UNION operator.

Claims

1. Based onk ²-method for maximum common subgraph matching of labeled graphs of MDDs, characterized in that it comprises the following steps:

2. A method according to claim 1k ²-method for maximum common subgraph matching of labeled graphs of MDD, characterized in that in step 2) the number of coded bits per vertex of the target graph and the new matching graph is n bits, wherekIs an integer of 2 or moreVAnd l is the number of the vertex points, and the vertex coding steps of the target graph and the new matching graph are as follows:

WhereinkIs an integer of 2 or more;

3. A method according to claim 1k ²The maximum common subgraph matching method of the labeled graphs of the MDD is characterized in that, in step 3), the edges of the target graph and the new matching graph are encoded, and the directed edges of the target graph and the new matching graph are relationships between vertices and are described by a feature function between the vertices, which is as follows:

3-1) X = (X)₁，…，x_n)、Y=(y₁，…，y_n) For the vertex code vector in the figure, the edge from vertex X to vertex YIs expressed as:E(X，Y)：{0，1，…k-1}ⁿ×{0，1，…k-1}ⁿ→ {1，2，…k ²}ⁿi.e. on each bit of the two vertex codeskCombining the species states to obtaink ²In the seed state, the code length of the edge is still n bits, and each bit of the code isk ²One of the seed states;

4. A method according to claim 1k ²-method for maximum common subgraph matching of labeled graphs of MDD, characterized in that in step 4) said target graph and new matching graph are obtainedk ²-MDD structural diagram, obtained in particular by the following steps:

5. A method according to claim 1k ²-MDD maximum common subgraph matching method of labeled graphs, characterized in that in step 5), the maximum common subgraph of the target graph and the matching graph is obtained by using the logic traffic operation of the symbolic decision graph in the edge queryk ²Any 2 vertices v on the MDD structure diagram₁And v₂Performing an edge query operation, if the query return result is T, representing a vertex v₁And v₂If the query returns a result of F, the vertex v is represented₁And v₂The edge in between does not exist.