Improved algorithm for permutation testing

Abstract

For a permutation π:[K]→[K], a sequence f:{1,2,⋯,n}→R contains a π-pattern of size K, if there is a sequence of indices (i1,i2,⋯,iK) (i1<i2<⋯<iK), satisfying that f(ia)<f(ib) if π(a)<π(b), for a,b∈[K]. Otherwise, f is referred to as π-free. For the special case where π=(1,2,⋯,K), it is referred to as the monotone pattern. [1] initiated the study of testing π-freeness with one-sided error. They focused on two specific problems, testing the monotone permutations and the (1,3,2) permutation. For the problem of testing monotone permutation (1,2,⋯,K), [2] improved the (log⁡n)O(K2) non-adaptive query complexity of [1] to O((log⁡n)⌊log2⁡K⌋). Further, [3] proposed an adaptive algorithm with O(log⁡n) query complexity. However, no progress has yet been made on the problem of testing (1,3,2)-freeness. In this work, we present an adaptive algorithm for testing (1,3,2)-freeness. The query complexity of our algorithm is O(ϵ−2log4⁡n), which significantly improves over the O(ϵ−7log26⁡n)-query adaptive algorithm of [1]. This improvement is mainly achieved by the proposal of a new structure embedded in the patterns.