Exploratory Equivalence in Graphs: Definition and Algorithms

Abstract

Abstract-Motivated by improving the efficiency of pattern matching on graphs, we define a new kind of equivalence on graph vertices. Since it can be used in various graph algorithms that explore graphs, we call it exploratory equivalence. The equivalence is based on graph automorphisms. Because many similar equivalences exist (some also based on automorphisms), we argue that this one is novel. For each graph, there are many possible exploratory equivalences, but for improving the efficiency of the exploration, some are better than others. To this end, we define a goal function that models the reduction of the search space in such algorithms. We describe two greedy algorithms for the underlying optimization problem. One is based directly on the definition using a straightforward greedy criterion, whereas the second one uses several practical speedups and a different greedy criterion. Finally, we demonstrate the huge impact of exploratory equivalence on a real application, i.e., graph grammar parsing.

Maximum Exploratory Equivalence in Trees

“…Motivated by this observation, we recently introduced the so-called exploratory equivalence [9], on the basis of which such constraints can be defined and safely imposed during …”

“…However, of particular interest is one that gives rise to the set of constraints that results in the largest speedup when searching for the occurrences of G. Such an EE partition is called a maximum EE partition ('a' instead of 'the' because there can be several of them), and the problem of finding such a partition for a given graph is denoted MAXEXPLOREQ. In our previous paper [9], we defined the problem and showed two algorithms, both of which are polynomial only in the number of automorphisms, rather than in the number of graph vertices. Besides that, the algorithms fail to find a maximum EE partition for all graphs, although counterexamples appear to be very rare; for example, the second algorithm finds a maximum EE partition for all but 2 graphs out of 261080 connected unlabeled undirected 9-vertex graphs.…”

“…To the best of our knowledge, our definition introduces a new form of equivalence. We call it exploratory equivalence (EE), since its primary intent is to be utilized in graph search algorithms; see [9] for the introductory paper. Nevertheless, the exploitation of symmetries of a problem to reduce the amount of time for exploring the solution search space is not new.…”

“…In our preceding paper [9], we already presented the definition of exploratory equivalences and the corresponding problem of finding maximum exploratory equivalent partition of graph vertices, i.e., the MAXEXPLOREQ problem. In this paper, we show that the MAXEXPLOREQ is GI -hard, which means that a polynomial-time algorithm (in terms of the number of graph vertices) is unlikely to exist.…”

Kaprekar's transformations. Part I—theoretical discussion

“…Another important fact (which we intend to discuss in this study as well) concerns not only the periods of a given map but also the so called orbit type. It was at first defined by S. Baldwin in [3] for maps of an interval (see also [24] and references therein) and next extended by others (for example in [4] for the maps of a circle and in [21] for the groups and the groups of graphs). We will use here the following definition [1].…”