Minimal Height and Sequence Constrained Longest Increasing Subsequence

Abstract

Given a string S = a1a2a3 ¢ ¢ ¢ an, the longest increasing subsequence (LIS) problem is to ¯nd a subsequence of S such that the subsequence is increasing and its length is maximal. In this paper, we propose and solve two variants of the LIS problem. The ¯rst one is the minimal height LIS where the height means the di®erence between the largest and smallest elements. We propose an algorithm with O(n log n) time and O(n) space to solve it. The second one is the sequence constrained LIS that given a sequence S and a constraint C, we are to ¯nd the LIS of S containing C as its subsequence. We propose an algorithm with O(n log(n + jCj)) time to solve it.