Recently, great efforts have been dedicated to researches on the management of large scale graph based data such as WWW, social networks, biological networks. In the study of graph based data management, node disjoint subgraph homeomorphism relation between graphs is more suitable than (sub)graph isomorphism in many cases, especially in those cases that node skipping and node mismatching are allowed. However, no efficient node disjoint subgraph homeomorphism determination (ndSHD) algorithms have been available. In this paper, we propose two computationally efficient ndSHD algorithms based on state spaces searching with backtracking, which employ many heuristics to prune the search spaces. Experimental results on synthetic data sets show that the proposed algorithms are efficient, require relative little time in most of the testing cases, can scale to large or dense graphs, and can accommodate to more complex fuzzy matching cases.
Recently, large scale graph based data management has received more and more research attentions, due to the broad application of graph based data. In the study of graph based data management, graph based pattern matching, i.e., to determine whether the structure of a pattern graph can match to that of a data graph, is the key of many problems about graph data management.
Existing graph pattern matchings can be classified into two preliminary categories: exact matching and inexact matching. Exact matching requires that the matched two graphs are isomorphic to each other; i.e., exact graph pattern matching is based on graph isomorphism relations between graphs. While the inexact graph matching is often considered as subgraph isomorphism between graphs, which means that pattern graph P matches to data graph G if and only if P is subgraph isomorphic to G.
However, in real applications, inexact graph pattern matching based on subgraph isomorphism cannot represent the fuzzy matching in some cases that node skipping or node mismatching is allowed. For example, as shown in Figure 1, although G2 is not a subgraph of G1, G2 still can be regarded as matched to G1 if node skipping or node mismatching is allowed. In other words, G2 is matched to G1 from the abstract topological structure perspective, because G2 retains the abstract topological structure of G1 if paths in G1 can be contracted into the corresponding edges in G2.
However, this kind of fuzzy matching is more desired in many real applications than subgraph isomorphism based inexact matching. For instance, the discovery of frequent
Fig. 1. Inexact Matching Fig. 2. Topological Minor conserved subgraph patterns from protein interaction networks [1,2] is an important and challenging work in evolutionary and comparative biology, where ‘conserved’ just means the inexact graph pattern matching allowing node mismatch and node skipping.
Similarly, in social network analysis, the direct connection between nodes usually is not the focus; instead, the high-level topological structure with independent paths contracted is of great interest. Using Graph Minor theory [4], the abstract topological structure in many real applications can be described as topological minor, and the relation between abstract topological structure and its detailed original graph can be described as node/vertex disjoint subgraph homeomorphism. However, to determine whether a pattern graph P is a topological minor of data graph G is not a trivial thing, and this problem has been proved to be NP-complete when P and G are not fixed [3]. Although Robertson and Seymour [4] have proposed a framework to solve minor containment problem that is a generalization of topology containment problem and [5] has implemented the framework, no efficient algorithms have been dedicated to solve ndSHD (in other contexts, also known as topological minor containment, homeomorphic embedding or topological embedding), to the best of our knowledge.
To efficiently determine the node disjoint homeomorphism relation between two graphs, we propose two algorithms based on state space searching with backtrack, which integrate many heuristics into the searching procedure to prune the search spaces. The work in the paper is inspired by Ullmann’s [6] s ¯ubgraph i ¯somorphism d ¯etermination (SID) algorithm. However, for ndSHD, we need to do some more specific things. First, for ndSHD, not only node mapping space but also edge-path mapping space needs to be searched, whereas for SID only the former needs to be searched. Second, for ndSHD, according to the definition of topological minor, we need to perform pairwise independence determination of the paths to ensure the paths are disjoint. Third, for SID, only edge information is explored, while in ndSHD path information is explored too, which will be a great challenge to the efficiency of the algorithm since the amount of paths is exponential to the size of the graph.
In a summary, we make the following contributions in this paper:
- We propose two efficient algorithms for node disjoint subgraph homeomorphism determination. To the best of our knowledge, it’s the first paper dedicated to design practical efficient algorithms for node disjoint subgraph homeomorphism determination or topological minor containment determination problem. 2. We investigate the properties of topological minors, and employ these properties as the heuristics to prune the search space. 3. We present a systematic performance study of proposed algorithms. The experimental results show that the algorithms are efficient and scalable on synthetic data sets.
We begin with some basic notations that are used in [7]. Let G = (V, E, l) be a vertex labeled graph, where V is the set of vertices, E is the set of edges and E ⊆ V × V , and l is a label function l : V → L , giving every vertex a label.(In this paper, we only focus on vertex labeled graphs. Unlabeled graph can be considered as a labeled graph with al
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