Permutation patterns in streams

TL;DR

This paper investigates the complexity of permutation pattern matching in streaming data, revealing that the space requirements vary significantly with the pattern type, and introduces algorithms that leverage the permutation structure.

Contribution

It introduces the first streaming algorithms for permutation pattern matching that exploit the permutation structure to reduce space complexity.

Findings

Space complexity varies with pattern type, from logarithmic to near-linear.

Algorithms effectively utilize the permutation input to improve efficiency.

Lower bounds demonstrate the inherent difficulty for certain pattern types.

Abstract

Permutation patterns and pattern avoidance are central, well-studied concepts in combinatorics and computer science. Given two permutations $\tau$ and $\pi$, the pattern matching problem (PPM) asks whether $\tau$ contains $\pi$. This problem arises in various contexts in computer science and statistics and has been studied extensively in exact-, parameterized-, approximate-, property-testing- and other formulations. In this paper, we study pattern matching in a streaming setting, when the input $\tau$ is revealed sequentially, one element at a time. There is extensive work on the space complexity of various statistics in streams of integers. The novelty of our setting is that the input stream is a permutation, which allows inferring some information about future inputs. Our algorithms crucially take advantage of this fact, while existing lower bound techniques become difficult to apply. We show that the complexity of the problem changes dramatically depending on the pattern $\pi$. The space requirement is: $\Theta(k\log{n})$ for the monotone patterns $\pi = 12\dots k$, or $\pi = k\dots21$, $O(\sqrt{n\log{n}})$ for $\pi \in \{312,132\}$, $O(\sqrt{n} \log n)$ for $\pi \in \{231,213\}$, and $\widetilde{\Theta}_{\pi}(n)$ for all other $\pi$. If $\tau$ is an arbitrary sequence of integers (not necessary a permutation), we show that the complexity is $\widetilde{\Theta}_{\pi}(n)$ in all except the first (monotone) cases.