A Fast Algorithm for Permutation Pattern Matching Based on Alternating Runs

Abstract

The NP-complete Permutation Pattern Matching problem asks whether a k-permutation P is contained in a n-permutation T as a pattern. This is the case if there exists an order-preserving embedding of P into T . In this paper, we present a fixed-parameter algorithm solving this problem with a worst-case runtime of O(1.79 run(T ) · n · k), where run(T ) denotes the number of alternating runs of T . This algorithm is particularly well-suited for instances where T has few runs, i.e., few ups and downs. Moreover, since run(T ) < n, this can be seen as a O(1.79 n · n · k) algorithm which is the first to beat the exponential 2 n runtime of brute-force search. Furthermore, we prove that under standard complexity theoretic assumptions such a fixedparameter tractability result is not possible for run(P ).

Faster Exponential Algorithm for Permutation Pattern Matching

“…The counting version of this problem asks about the number of occurrences. For instance, for σ = (3, 2, 5, 4, 1) and π = (1, 3, 2) there are two such occurrences, since both (3,5,4) and (2,5,4) are subsequences of σ with the same relative order as π, and there are no other such subsequences.…”

“…In this paper we are interested in algorithms with running time depending only on the value of n. From such point of view, none of the algorithms mentioned above improve upon the trivial solution working in time 2 n . However, Bruner and Lackner [5] were able to design an algorithm solving the counting version in time O(1.79 n ). Berendsohn, Kozma and Marx [3] improved on their result with an elegant polynomial-space algorithm working in time O(1.6181 n ).…”

Griddings of permutations and hardness of pattern matching

“…When k is large in terms of n, a brute-force approach solves PPM in time O (2 n+o(n) ). The first improvement upon this bound was obtained by Bruner and Lackner [8], whose algorithm achieves the running time O(1.79 n ), which was in turn improved by Berendsohn, Kozma and Marx [6] to O(1.6181 n ).…”

Pattern Matching for Separable Permutations

“…Improvements to this algorithm were presented in [2] and [1], the latter describing a nice O(|π| 0.47k+o(|σ|) ) time algorithm. Bruner and Lackner [7] gave a fixed-parameter algorithm solving the pattern containment problem with an exponential worstcase runtime of O (1.79 run(π) ), where run(π) denotes the number of alternating runs of π. Of particular importance, it has been proved in [10] that the pattern containment problem is fixed-parameter tractable for parameter |σ|.…”

“…Vincular patterns, also called generalized patterns, resemble (classical) patterns, with the constraint that some of the letters in an occurrence must be consecutive. Of particular importance in our context, Bruner and Lackner [7] proved that deciding whether a vincular pattern σ of length k occurs in a permutation π of length n is W [1]-complete for parameter k. Bivincular patterns generalize classical patterns even further than vincular patterns. Indeed, in bivincular patterns, not only positions but also values of elements involved in a occurrence may be forced to be adjacent The paper is organised as follows.…”