On the Editing Distance Between Undirected Acyclic Graphs

Abstract

We consider the problem of comparing CUAL graphs (Connected, Undirected, Acyclic graphs with nodes being Labeled). This problem is motivated by the study of information retrieval for bio-chemical and molecular databases. Suppose we define the distance between two CUAL graphs G1 and G2 to be the weighted number of edit operations (insert node, delete node and relabel node) to transform G1 to G2. By reduction from exact cover by 3-sets, one can show that finding the distance between two CUAL graphs is NP-complete. In view of the hardness of the problem, we propose a constrained distance metric, called the degree-2 distance, by requiring that any node to be inserted (deleted) have no more than 2 neighbors. With this metric, we present an efficient algorithm to solve the problem. The algorithm runs in time O(N1N2D2) for general weighting edit operations and in time [Formula: see text] for integral weighting edit operations, where Ni, i=1, 2, is the number of nodes in Gi, D=min{d1, d2} and di is the maximum degree of Gi.

Matching Free Trees, Maximal Cliques, and Monotone Game Dynamics

“…This idea is attractive especially when the structures being matched are subject to significant structural distortions. Unfortunately, it turns out that computing the edit-distance on free trees is NP-hard [25], although it is solvable in polynomial time by restricting ourselves to ordered trees, where each node is assigned a cyclic ordering of its incidente edges [15]. Moreover, determining the set of elementary edit operations and the associated costs depends heavily on the application domain and can be problematic (see [16], [17] for some examples of edit operations motivated by shape matching problems).…”

A survey of content based 3D shape retrieval methods

“…Efficient computation of existing graph metrics for general graphs is not possible: computing the edit distance is NP-hard [90] and computing the maximal common subgraph [32] is even NP-complete. Polynomial solutions can be obtained for directed acyclic graphs such as shock graphs.…”

Efficient Similarity Search for Hierarchical Data in Large Databases

“…A degree-2 edit sequence consists only of insertions or deletions of nodes n with degree(n) ≤ 2, or of relabelings: ED 2 (t 1 , t 2 ) = min{c(S)|S is a degree-2 edit sequence transf orming t 1 into t 2 } One should note that the degree-2 edit distance is well defined in the sense that two trees can always be transformed into each other by using only degree-2 edit operations. In [14] an algorithm is presented to compute the degree-2 edit distance in O(|t 1 ||t 2 |D) time, where D is the maximum of the degrees of t 1 and t 2 and |t i | denotes the number of nodes in t i . Whereas this measure has a polynomial time complexity, it is still too complex for the use in large databases.…”

“…clustering, k-nn-classification). As similarity measure for trees, we implemented the degree-2 edit distance algorithm as presented in [14]. The filter histograms were organized in an X-tree [17].…”