Single and multiple consecutive permutation motif search

Abstract

Let t be a permutation (that shall play the role of the text) on [n] and a motif p be a sequence of m distinct integer(s) of [n], m ≤ n. The motif p occurs in t in position i if and only if p1...pm is order-isomorphic to ti...ti + m - 1, that is, for all 1 ≤ k < ℓ ≤ m, pk > pℓ if and only if ti+k-1 > ti+ℓ-1. Searching for a motif p in a text t consists in identifying all occurrences of p in t. We first present a forward automaton which allows us to search for p in t in O(m 2loglogm+n) time. We then introduce a Morris-Pratt automaton representation of the forward automaton which allows us to reduce this complexity to O(mloglogm+n) at the price of an additional amortized constant term. The latter automaton occupies O(m) space. We then extend the problem to search for a set of motifs and exhibit a specific Aho-Corasick like algorithm. Next we present a sub-linear average case search algorithm running in O(m log m/log log m + n log m/m log log m) time, that we eventually prove to be optimal on average. © 2013 Springer-Verlag.