Longest Increasing Subsequences in Windows Based on Canonical Antichain Partition

Given a sequence π1π2 . . . πn, a longest increasing subsequence (LIS) in a window π〈l, r〉= πlπl+1 . . . πr is a longest subsequence σ = πi1πi2 . . . πiT such that l ≤ i1 < i2 < · · · < iT ≤ r and πi1 < πi2 < · · · < πiT . We consider the Lisw problem, which is to find the longest increasing subsequences in a sliding window of fixed-size w over a sequence. Formally, it is to find a LIS for every window in a set SFIX = { π〈i + 1, i + w〉 ∣∣ 0 ≤ i ≤ n− w}∪{π〈1, i〉, π〈n− i, n〉 ∣∣ i < w}. By maintaining a canonical antichain partition in windows, we present an optimal output-sensitive algorithm to solve this problem in O(output) time, where output is the sum of the lengths of the n+w−1 LISs in those windows of SFIX. In addition, we propose a more generalized problem called Lisset problem, which is to find a LIS for every window in a set SVAR containing variable-size windows. By applying our algorithm, we provide an efficient solution for the Lisset problem to output a LIS (or all the LISs) in every window which is better than the straightforward generalization of classical LIS algorithms. An upper bound of our algorithm on the Lisset problem is discussed.