An Optimal Algorithm for Finding the Longest Increasing Subsequence of Every Substring

Given a string S = {a1, a2, a3, · · · , an}, the longest increasing subsequence (LIS) problem is to find a subsequence of the given string such that the subsequence is increasing and its length is maximal. In a previous result, to find the longest increasing subsequences of each sliding windows with a fixed size w of a given string can be solved in O(w log log n + OUTPUT ) time, where O(w log log n) is taken for preprocessing. In this paper, we solve the problem for finding the longest increasing subsequence of every substring of S. With the straightforward implementation of the previous result, the time required for the preprocessing would be O(n log log n). With the modification of the data structure used in the algorithm, our algorithm needs only O(n) preprocessing time. Since there are O(n) substrings totally in a given string with length n, our algorithm is optimal. The time required for the reporting stage is linear to the size of the output. In other words, our algorithm can find the LIS of every substring in O(n+OUTPUT ) time, where OUTPUT is the sum of the lengths of the outputs.