Graphs can be succinctly indexed for pattern matching in $ O(|E|^2 + |V|^{5 / 2}) $ time

Abstract

Pattern matching is a pervasive problem in computer science. In several applications (notably, bioinformatics), it is crucial to perform pattern matching on edge-labeled graphs: given a string, decide whether the string can be read on the graph. In this paper we consider the problem of indexing (preprocessing) a graph for pattern matching by building a succinct data structure, that is, a data structure that uses a number of bits close to the lower bounds from information theory, while allowing efficient queries. In [TCS 2017] Gagie et al. showed how to index a class of graphs -the so-called Wheeler graphs -by totally ordering the nodes of the graph (building a Wheeler order ) and using some techniques based on the Burrows-Wheeler transform. However, the class of Wheeler graphs is rather small: for example, a unary language is recognized by some Wheeler automaton (that is, an automaton whose underlying graph is Wheeler) if and only if the language is finite or cofinite. In [SODA 2021] these ideas where extended to arbitrary node-labeled graphs, and Wheeler orders were generalized to co-lex orders, which are partial orders: it was showed that a node-labeled graphs G = (V, E) can be succinctly indexed by means of a data structure of |E|(⌈log |Σ|⌉ + ⌈log p⌉ + 2)where p is the minimum width of a co-lex order on G, P is the pattern and Σ is the alphabet. However, determining p is NP-hard and building the data structure is also hard. Intuitively, a partial order imposes antisymmetry and transitivity constraints, which cause the hardness of all natural decision problems connected to Wheeler and co-lex orders. In this paper, we change perspective switching from (partial) orders to arbitrary relations, so defining co-lex relations. We work in the more general setting of edge-labeled graphs G = (V, E), and for the first time we provide a succinct index for arbitrary graphs that can be built in polynomial time, which requires less space and answers queries more efficiently than the one in [SODA 2021]. We show that, given an edge-labeled graph G = (V, E), there exists a data structure ofand q is the width of the maximum co-lex relation on G. The bounds achieved in this paper look similar to the ones in [SODA 2021], but, in fact, there are several sources of improvement:1. Most importantly, q can be determined in O(|E| 2 ) time, and our data structure can be built in) time (while determining p and building the data structure in [SODA 2021] are hard problems). 2. It always holds q ≤ p, and q can be arbitrarily smaller than p (that is, for every integer n there exists a graph such that p = n and q = 1). 3. Our bound only depends on the size of G/ ≤ G and it is independent of the size of G. In other words, the quotient graph G/ ≤ G eliminates the unnecessary redundancy in G from a pattern matching perspective. Our results have relevant applications in automata theory. First, we can build a succinct data structure to decide whether a string is accepted by a given automaton. Second, starting from an automaton A, one can define a ...