Low-Latency Sliding Window Algorithms for Formal Languages

TL;DR

This paper investigates low-latency sliding window algorithms for various formal language classes, establishing constant or logarithmic latency solutions for regular, visibly pushdown, and deterministic 1-counter languages, while also providing lower bounds under complexity conjectures.

Contribution

It introduces the first constant-latency algorithms for regular and visibly pushdown languages in the two-way model and presents a logarithmic latency algorithm for deterministic 1-counter languages, along with conditional lower bounds.

Findings

Constant-latency solutions exist for regular and visibly pushdown languages.

An $ ext{O}( ext{log} n)$ latency algorithm is provided for deterministic 1-counter languages.

Conditional lower bounds show no sub-$n^{1/2- ext{epsilon}}$ latency algorithms for some context-free languages under OMV conjecture.

Abstract

Low-latency sliding window algorithms for regular and context-free languages are studied, where latency refers to the worst-case time spent for a single window update or query. For every regular language $L$ it is shown that there exists a constant-latency solution that supports adding and removing symbols independently on both ends of the window (the so-called two-way variable-size model). We prove that this result extends to all visibly pushdown languages. For deterministic 1-counter languages we present a $\mathcal{O}(\log n)$ latency sliding window algorithm for the two-way variable-size model where $n$ refers to the window size. We complement these results with a conditional lower bound: there exists a fixed real-time deterministic context-free language $L$ such that, assuming the OMV (online matrix vector multiplication) conjecture, there is no sliding window algorithm for $L$ with latency $n^{1/2-\epsilon}$ for any $\epsilon>0$, even in the most restricted sliding window model (one-way fixed-size model). The above mentioned results all refer to the unit-cost RAM model with logarithmic word size. For regular languages we also present a refined picture using word sizes $\mathcal{O}(1)$, $\mathcal{O}(\log\log n)$, and $\mathcal{O}(\log n)$.