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 pattern p be a sequence of m distinct integer(s) of [n], m ≤ n. The pattern 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 > t1+-1 . Searching for a pattern 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(m2 log log m + n) time. We then introduce a Morris-Pratt automaton representation of the forward automaton which allows us to reduce this complexity to O(m log log m + n) at the price of an additional amortized constant term by integer of the text. Both automata occupy O(m) space. We then extend the problem to search for a set of patterns 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.
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Djamal Belazzougui, Adeline Pierrot, Mathieu Raffinot, Stéphane Vialette. Single and multiple consecutive permutation motif search. ISAAC 2013, Dec 2013, Hong-Kong, Hong Kong SAR China. pp.66-77, ⟨10.1007/978-3-642-45030-3_7⟩. ⟨hal-00859363⟩

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