Computing a Longest Increasing Subsequence of Length k in Time O(n log log k)

Abstract

We consider the complexity of computing a longest increasing subsequence parameterised by the length of the output. Namely, we show that the maximal length k of an increasing subsequence of a permutation of the set of integers -1, 2,..., n} can be computed in time O(n log log k) in the RAM model, improving the previous 30-year bound of O(n log log k). The optimality of the new bound is an open question.