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Register a new userA Quadratic Assignment Formulation of the Graph Edit Distance
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Added by
Sebastien Bougleux
Publication date
2015
fields
Informatics Engineering
and research's language is
English
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Computing efficiently a robust measure of similarity or dissimilarity between graphs is a major challenge in Pattern Recognition. The Graph Edit Distance (GED) is a flexible measure of dissimilarity between graphs which arises in error-tolerant graph matching. It is defined from an optimal sequence of edit operations (edit path) transforming one graph into an other. Unfortunately, the exact computation of this measure is NP-hard. In the last decade, several approaches have been proposed to approximate the GED in polynomial time, mainly by solving linear programming problems. Among them, the bipartite GED has received much attention. It is deduced from a linear sum assignment of the nodes of the two graphs, which can be efficiently computed by Hungarian-type algorithms. However, edit operations on nodes and edges are not handled simultaneously, which limits the accuracy of the approximation. To overcome this limitation, we propose to extend the linear assignment model to a quadratic one, for directed or undirected graphs having labelized nodes and edges. This is realized through the definition of a family of edit paths induced by assignments between nodes. We formally show that the GED, restricted to the paths in this family, is equivalent to a quadratic assignment problem. Since this problem is NP-hard, we propose to compute an approximate solution by an adaptation of the Integer Projected Fixed Point method. Experiments show that the proposed approach is generally able to reach a more accurate approximation of the optimal GED than the bipartite GED, with a computational cost that is still affordable for graphs of non trivial sizes.
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We study the problem of estimating the edit distance between two $n$-character strings. While exact computation in the worst case is believed to require near-quadratic time, previous work showed that in certain regimes it is possible to solve the following {em gap edit distance} problem in sub-linear time: distinguish between inputs of distance $le k$ and $>k^2$. Our main result is a very simple algorithm for this benchmark that runs in time $tilde O(n/sqrt{k})$, and in particular settles the open problem of obtaining a truly sublinear time for the entire range of relevant $k$. Building on the same framework, we also obtain a $k$-vs-$k^2$ algorithm for the one-sided preprocessing model with $tilde O(n)$ preprocessing time and $tilde O(n/k)$ query time (improving over a recent $tilde O(n/k+k^2)$-query time algorithm for the same problem [GRS20].
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