We also addressed how to support distance-aware queries such as to find the shortest distance between two nodes in a large directed graph using the 2-hop cover, and how to support graph
Trang 1Cheng et al in [11, 12] consider𝐴,→𝐷 as a R-join (like 𝜃-join), and process
a graph pattern matching as a sequence of R-joins The issue is how to select
join order They propose a dynamic programming algorithm to determine the
R-join order in [11] They also propose an R-join/R-semijoin approach in [12].
The basic idea is to divide the join-index based approach into two steps namely filter and fetch The filter steps shares the similarity with semijoin, and the
fetch step is to join Cheng et al study how to select R-join/R-semijoin order
by interleaving R-joins with R-semijoins, using dynamic programming in [12].
Wang et al in [35] propose a query graph 𝐺𝑞 based on the hash join approach, and consider how to share the processing cost when it needs to process several 𝐴𝑙𝑖𝑠𝑡 and 𝐷𝑙𝑖𝑠𝑡 simultaneously Wang et al propose three basic join operators, namely, IT-HGJoin, T-HGJoin, and Bi-HGJoin The IT-HGJoin processes a subgraph of a query with one descendant and multi-ple ancestors, for exammulti-ple, 𝐴,→𝐷 ∧ 𝐵,→𝐷 The T-HGJoin process a sub-graph of a query with one ancestor and multiple descendants, for example,
𝐴,→𝐶 ∧ 𝐴,→𝐷 The Bi-HGJoin processes a complete bipartite subgraph
of a query with multiple ancestors and multiple descendants, for example
𝐴,→𝐶 ∧𝐴,→𝐷∧𝐵,→𝐶 ∧𝐵,→𝐷 A general query graph 𝐺𝑞will be processed
by a set of subgraph queries using IT-HGJoin, T-HGJoin, and Bi-HGJoin
11 Conclusions and Summary
In this chapter, we presented a survey on reachability queries We dis-cussed several coding-based approaches using traversal, dual-labeling, tree cover, chain cover, path-tree cover, 2-hop cover, and 3-hop cover approaches
We also addressed how to support distance-aware queries such as to find the shortest distance between two nodes in a large directed graph using the 2-hop cover, and how to support graph pattern matching using the existing graph-based coding schema As future work, it becomes important how to use the graph-based coding schema to support more real large graph-based applica-tions
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Trang 5EXACT AND INEXACT GRAPH MATCHING: METHODOLOGY AND APPLICATIONS
Kaspar Riesen
Institute of Computer Science and Applied Mathematics, University of Bern
Neubr-uckstrasse 10, CH-3012 Bern, Switzerland
riesen@iam.unibe.ch
Xiaoyi Jiang
Department of Mathematics and Computer Science, University of M-unster
Einsteinstrasse 62, D-48149 M-unster, Germany
xjiang@math.uni-muenster.de
Horst Bunke
Institute of Computer Science and Applied Mathematics, University of Bern
Neubr-uckstrasse 10, CH-3012 Bern, Switzerland
bunke@iam.unibe.ch
Abstract Graphs provide us with a powerful and flexible representation formalism which
can be employed in various fields of intelligent information processing The process of evaluating the similarity of graphs is referred to as graph matching Two approaches to this task exist, viz exact and inexact graph matching The former approach aims at finding a strict correspondence between two graphs
to be matched, while the latter is able to cope with errors and measures the difference of two graphs in a broader sense The present chapter reviews some fundamental concepts of both paradigms and shows two recent applications of graph matching in the fields of information retrieval and pattern recognition.
Keywords: Exact and Inexact Graph Matching, Graph Edit Distance, Information Retrieval
by means of Graph Matching, Graph Embedding via Graph Matching
© Springer Science+Business Media, LLC 2010
C.C Aggarwal and H Wang (eds.), Managing and Mining Graph Data,
Advances in Database Systems 40, DOI 10.1007/978-1-4419-6045-0_7, 217
Trang 6218 MANAGING AND MINING GRAPH DATA
1 Introduction
After many years of research, the fields of pattern recognition, machine learning and data mining have reached a high level of maturity [4] Power-ful methods for classification, clustering, information retrieval, and other tasks have become available However, the vast majority of these approaches rely
on object representations given in terms of feature vectors Such object repre-sentations have a number of useful properties For instance, the dissimilarity,
or distance, of two objects can be easily computed by means of the Euclidean distance Moreover, a large number of well-established methods for data min-ing, information retrieval, and related tasks in intelligent information process-ing are available Recently, however, a growprocess-ing interest in graph-based object representation can be observed [16] Graphs are powerful and universal data structures able to explicitly model networks of relationships between substruc-tures of a given object Thereby, the size as well as the complexity of a graph can be adopted to the size and complexity of a particular object (in contrast to vectorial approaches where the number of features has to be fixed beforehand) Yet, after the initial enthusiasm induced by the “smartness” and flexibility of graph representations in the late seventies, a number of problems became evi-dent First, working with graphs is unequally more challenging than working with feature vectors, as even basic mathematic operations cannot be defined
in a standard way, but must be provided depending on the specific applica-tion Hence, almost none of the common methods for data mining, machine learning, or pattern recognition can be applied to graphs without significant modifications
Second, graphs suffer from of their own flexibility For instance, computing the distances of a pair of objects, which is an important task in many areas,
is linear in the number of data items in the case where vectors are employed The same task for graphs, however, is much more complex, since one cannot simply compare the sets of nodes and edges, which are generally unordered and of different size More formally, when computing graph dissimilarity or similarity one has to identify common parts of the graphs by considering all of their subgraphs Regarding that there are𝑂(2𝑛) subgraphs of a graph with 𝑛 nodes, the inherent difficulty of graph comparisons becomes obvious
Despite adverse mathematical and computational conditions in the graph domain, various procedures for evaluating proximity, i.e similarity or dissimi-larity, of graphs have been proposed in the literature [15] The process of
evalu-ating the similarity of two graphs is commonly referred to as graph matching.
The overall aim of graph matching is to find a correspondence between the nodes and edges of two graphs that satisfies some, more or less, stringent con-straints That is, by means of the graph matching process similar substructures
in one graph are mapped to similar substructures in the other graph Based on
Trang 7this matching, a dissimilarity or similarity score can eventually be computed indicating the proximity of two graphs
Graph matching has been the topic of numerous studies in computer sci-ence over the last decades Roughly speaking, there are two categories of tasks
in graph matching, viz exact matching and inexact matching In the former
case, for a matching to be successful, it is required that a strict correspondence
is found between the two graphs being matched, or at least among their sub-parts In the latter approach this requirement is substantially relaxed, since also matchings between completely non-identical graphs are possible That is, in-exact matching algorithms are endowed with a certain tolerance to errors and noise, enabling them to detect similarities in a more general way than the exact matching approach Therefore, inexact graph matching is also referred to as
error-tolerant graph matching.
For an extensive review of graph matching methods and applications, the reader is referred to [15] In this chapter, basic notations and definitions are in-troduced (Sect 2) and an overview of standard techniques for exact as well as error-tolerant graph matching is given (Sect 3 and 4) In Sect 3, dissimilarity models derived from graph isomorphism, subgraph isomorphism, and maxi-mum common subgraph are discussed for exact graph matching In Sect 4, inexact graph matching and in particular the paradigm of edit distance applied
to graphs is discussed Finally, two recent applications of graph matching are reviewed First, in Sect 5 an algorithmic framework for information retrieval based on graph matching is described This approach is based on both exact and inexact graph matching procedures and aims at querying large database graphs Secondly, a graph embedding procedure based on graph matching is reviewed in Sect 6 This framework aims at an explicit embedding of graphs in real vector spaces, which establishes access to the rich repository of algorith-mic tools for classification, clustering, regression, and other tasks, originally developed for vectorial representations
2 Basic Notations
Various definitions for graphs can be found in the literature, depending upon the considered application It turns out that the definition given below is suffi-ciently flexible for a large variety of tasks
Definition 7.1 (Graph) Let 𝐿𝑉 and 𝐿𝐸 be a finite or infinite label alphabet for nodes and edges, respectively A graph 𝑔 is a four-tuple 𝑔 = (𝑉, 𝐸, 𝜇, 𝜈), where
𝑉 is the finite set of nodes,
𝐸 ⊆ 𝑉 × 𝑉 is the set of edges,
𝜇 : 𝑉 → 𝐿𝑉 is the node labeling function, and
Trang 8220 MANAGING AND MINING GRAPH DATA
a b
c d
e f g
(d)
Figure 7.1 Different kinds of graphs: (a) undirected and unlabeled, (b) directed and unlabeled,
(c) undirected with labeled nodes (different shades of gray refer to different labels), (d) directed with labeled nodes and edges.
𝜈 : 𝐸 → 𝐿𝐸 is the edge labeling function.
The number of nodes of a graph𝑔 is denoted by∣𝑔∣, while 𝒢 represents the set of all graphs over the label alphabets𝐿𝑉 and𝐿𝐸
Definition 7.1 allows us to handle arbitrarily structured graphs with uncon-strained labeling functions For example, the labels for both nodes and edges can be given by the set of integers𝐿 ={1, 2, 3, }, the vector space 𝐿 = ℝ𝑛,
or a set of symbolic labels 𝐿 = {𝛼, 𝛽, 𝛾, } Given that the nodes and/or
the edges are labeled, the graphs are referred to as labeled graphs Unlabeled graphs are obtained as a special case by assigning the same label 𝜀 to all nodes
and edges, i.e 𝐿𝑉 = 𝐿𝐸 ={𝜀}
Edges are given by pairs of nodes (𝑢, 𝑣), where 𝑢∈ 𝑉 denotes the source node and𝑣 ∈ 𝑉 the target node of a directed edge Commonly, the two nodes
𝑢 and 𝑣 connected by an edge (𝑢, 𝑣) are referred to as adjacent A graph is termed complete if all pairs of nodes are adjacent Directed graphs directly cor-respond to the definition above In addition, the class of undirected graphs can
be modeled by inserting a reverse edge (𝑣, 𝑢)∈ 𝐸 for each edge (𝑢, 𝑣) ∈ 𝐸 with identical labels, i.e 𝜈(𝑢, 𝑣) = 𝜈(𝑣, 𝑢) In Fig 7.1 some graphs (di-rected/undirected, labeled/unlabeled) are shown
Definition 7.2 (Subgraph) Let 𝑔1 = (𝑉1, 𝐸1, 𝜇1, 𝜈1) and 𝑔2 = (𝑉2, 𝐸2, 𝜇2, 𝜈2) be graphs Graph 𝑔1 is a subgraph of 𝑔2, denoted by
𝑔1 ⊆ 𝑔2, if
(1) 𝑉1⊆ 𝑉2,
(2) 𝐸1 ⊆ 𝐸2,
(3) 𝜇1(𝑢) = 𝜇2(𝑢) for all 𝑢∈ 𝑉1, and
(4) 𝜈1(𝑒) = 𝜈2(𝑒) for all 𝑒∈ 𝐸1.
By replacing condition (2) in Definition 7.2 by the more stringent condition (2’) 𝐸1 = 𝐸2∩ 𝑉1× 𝑉1,
𝑔1 becomes an induced subgraph of𝑔2 If𝑔2is a subgraph of𝑔1, graph𝑔1 is
called a supergraph of𝑔2
Trang 9(a) (b) (c)
Figure 7.2 Graph (b) is an induced subgraph of (a), and graph (c) is a non-induced subgraph of
(a).
Obviously, a subgraph 𝑔1 is obtained from a graph 𝑔2 by removing some nodes and their incident, as well as possibly some additional, edges from
𝑔2 For 𝑔1 to be an induced subgraph of 𝑔2, some nodes and only their in-cident edges are removed from𝑔2, i.e no additional edge removal is allowed Fig 7.2(b) and 7.2(c) show an induced and a non-induced subgraph of the graph in Fig 7.2(a), respectively
3 Exact Graph Matching
The aim in exact graph matching is to determine whether two graphs, or at least part of them, are identical in terms of structure and labels A common
approach to describe the structure of a graph is to define the adjacency matrix
A= (𝑎𝑖𝑗)𝑛×𝑛of graph𝑔 = (𝑉, 𝐸, 𝜇, 𝜈) (∣𝑔∣ = 𝑛) In this matrix the entry 𝑎𝑖𝑗
is equal to1 if there is an edge (𝑣𝑖, 𝑣𝑗) ∈ 𝐸 connecting the 𝑖-th node 𝑣𝑖 ∈ 𝑉 with the𝑗− 𝑡ℎ node 𝑣𝑗 ∈ 𝑉 , and 0 otherwise
Generally, for the nodes (and also the edges) of a graph there is no unique canonical order Thus, for a single graph with𝑛 nodes, 𝑛! different adjacency matrices exist, since there are 𝑛! possibilities to order the nodes of 𝑔 Con-sequently, for checking two graphs for structural identity, we cannot simply compare their adjacency matrices The identity of two graphs 𝑔1 and 𝑔2 is commonly established by defining a function, termed graph isomorphism, that maps𝑔1 to𝑔2
Definition 7.3 (Graph Isomorphism) Let us consider two graphs denoted by
𝑔1 = (𝑉1, 𝐸1, 𝜇1, 𝜈1) and 𝑔2 = (𝑉2, 𝐸2, 𝜇2, 𝜈2) respectively A graph isomor-phism is a bijective function 𝑓 : 𝑉1→ 𝑉2satisfying
(1) 𝜇1(𝑢) = 𝜇2(𝑓 (𝑢)) for all nodes 𝑢∈ 𝑉1
(2) for each edge 𝑒1 = (𝑢, 𝑣)∈ 𝐸1, there exists an edge
𝑒2= (𝑓 (𝑢), 𝑓 (𝑣))∈ 𝐸2
such that 𝜈1(𝑒1) = 𝜈2(𝑒2)
(3) for each edge 𝑒2 = (𝑢, 𝑣)∈ 𝐸2, there exists an edge
𝑒1 = (𝑓−1(𝑢), 𝑓−1(𝑣))∈ 𝐸1
Trang 10222 MANAGING AND MINING GRAPH DATA
Figure 7.3 Graph (b) is isomorphic to (a), and graph (c) is isomorphic to a subgraph of (a) Node
attributes are indicated by different shades of gray.
such that 𝜈1(𝑒1) = 𝜈2(𝑒2)
Two graphs are called isomorphic if there exists an isomorphism between them.
Obviously, isomorphic graphs are identical in both structure and labels That
is, a one-to-one correspondence between each node of the first graph and each node of the second graph has to be found such that the edge structure is pre-served and node and edge labels are consistent
Unfortunately, no polynomial runtime algorithm is known for the problem
of graph isomorphism [25] That is, in the worst case, the computational com-plexity of any of the available algorithms for graph isomorphism is exponential
in the number of nodes of the two graphs However, since most scenarios en-countered in practice are often different from the worst case, and furthermore, the labels of both nodes and edges very often help to substantially reduce the complexity of the search, the actual computation time can still be manageable Polynomial algorithms for graph isomorphism have been developed for spe-cial kinds of graphs, such as trees [1], ordered graphs [38], planar graphs [34], bounded-valence graphs [45], and graphs with unique node labels [18] Standard procedures for testing graphs for isomorphism are based on tree search techniques with backtracking The basic idea is that a partial node matching, which assigns nodes from the two graphs to each other, is itera-tively expanded by adding new node-to-node correspondences This expan-sion is repeated until either the edge structure constraint is violated or node
or edge labels are inconsistent In this case a backtracking procedure is ini-tiated, i.e the last node mappings are iteratively undone until a partial node mapping is found for which an alternative extension is possible Obviously, if there is no further possibility for expanding the partial node matching without violating the constraints, the algorithm terminates indicating that there is no isomorphism between the considered graphs Conversely, finding a complete node-to-node correspondence without violating any of the structure or label constraints proves that the investigated graphs are isomorphic In Fig 7.3 (a) and (b) two isomorphic graphs are shown
A well known, and despite its age still very popular, algorithm implementing the idea of a tree search with backtracking for graph isomorphism is described
in [89] A more recent algorithm for graph isomorphism, also based on the idea of tree search, is the VF algorithm and its successor VF2 [17] Here the