Finding small patterns in permutations in linear time

Given two permutations σ and π, the textscPermutation Pattern problem asks if σ is a subpattern of π. We show that the problem can be solved in time 2O(ℓ2log ℓ)· n, where ℓ=|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 textscPermutation Pattern problem in linear time if a bounded-width decomposition is given in the input.