Longest increasing subsequences in windows based on canonical antichain partition

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Given a sequence π 1 π 2 … π n , a longest increasing subsequence (LIS) in a window π 〈 l , r 〉 = π l π l + 1 … π r is a longest subsequence σ = π i 1 π i 2 … π i T such that l ≤ i 1 < i 2 < ⋯ < i T ≤ r and π i 1 < π i 2 < ⋯ < π i T . 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 S FIX = { 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 S FIX . In addition, we propose a more generalized problem called Lisset problem, which is to find a LIS for every window in a set S VAR 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.