Streaming and Query Once Space Complexity of Longest Increasing Subsequence

TL;DR

This paper investigates the space complexity of computing and approximating the Longest Increasing Subsequence (LIS) in streaming and query models, establishing new lower bounds and highlighting open problems in the field.

Contribution

It provides the first non-trivial space lower bounds for LIS in adaptive query-once and streaming models with reordered input.

Findings

Established space lower bounds in query-once models

Proved space lower bounds in streaming models with reordered input

Identified open problems for future research

Abstract

Longest Increasing Subsequence (LIS) is a fundamental problem in combinatorics and computer science. Previously, there have been numerous works on both upper bounds and lower bounds of the time complexity of computing and approximating LIS, yet only a few on the equally important space complexity. In this paper, we further study the space complexity of computing and approximating LIS in various models. Specifically, we prove non-trivial space lower bounds in the following two models: (1) the adaptive query-once model or read-once branching programs, and (2) the streaming model where the order of streaming is different from the natural order. As far as we know, there are no previous works on the space complexity of LIS in these models. Besides the bounds, our work also leaves many intriguing open problems.