Given two permutations σ and π, the Permutation Pattern problem asks if σ is a subpattern of π. We show that the problem can be solved in time 2 O( 2 log ) ·n, where = |σ| and n = |π|. In other words, the problem is fixed-parameter tractable parameterized by the size of the subpattern to be found. We introduce a novel type of decompositions for permutations and a corresponding width measure. We present a linear-time algorithm that either finds σ as a subpattern of π, or finds a decomposition of π whose width is bounded by a function of |σ|. Then we show how to solve the Permutation Pattern problem in linear time if a bounded-width decomposition is given in the input.
For the exact case, Guillemot and Marx REF showed that an order pattern π of length k can be found in a sequence f of length n in time 2 O(k 2 log k) n; in particular, the problem of finding order patterns is fixed-parameter tractable (in the parameter k).
Finding small patterns in permutations in linear time
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Computer Science Mathematics