Space-Efficient Algorithms for Longest Increasing Subsequence

TL;DR

This paper introduces space-efficient algorithms for finding the longest increasing subsequence that significantly reduce memory usage while maintaining near-optimal time complexity.

Contribution

It presents novel algorithms that lower space consumption for LIS problems with minimal impact on time complexity, extending the feasible space-time trade-offs.

Findings

Algorithms use $O(s \,\log n)$ bits of space

Time complexity is near-optimal within polylogarithmic factors

Space reduction is effective for large sequences

Abstract

Given a sequence of integers, we want to find a longest increasing subsequence of the sequence. It is known that this problem can be solved in $O(n \log n)$ time and space. Our goal in this paper is to reduce the space consumption while keeping the time complexity small. For $\sqrt{n} \le s \le n$, we present algorithms that use $O(s \log n)$ bits and $O(\frac{1}{s} \cdot n^{2} \cdot \log n)$ time for computing the length of a longest increasing subsequence, and $O(\frac{1}{s} \cdot n^{2} \cdot \log^{2} n)$ time for finding an actual subsequence. We also show that the time complexity of our algorithms is optimal up to polylogarithmic factors in the framework of sequential access algorithms with the prescribed amount of space.