# Visibly Pushdown Languages over Sliding Windows

## TL;DR

This paper characterizes the space complexity of visibly pushdown languages in sliding window models, showing it is always constant, logarithmic, or linear, thus extending known results from regular languages.

## Contribution

It provides a complete classification of the space complexity for visibly pushdown languages in sliding window models.

## Key findings

- Space complexity is either constant, logarithmic, or linear.
- Extends previous regular language results to visibly pushdown languages.
- Provides a theoretical framework for streaming algorithms on these languages.

## Abstract

We investigate the class of visibly pushdown languages in the sliding window model. A sliding window algorithm for a language $L$ receives a stream of symbols and has to decide at each time step whether the suffix of length $n$ belongs to $L$ or not. The window size $n$ is either a fixed number (in the fixed-size model) or can be controlled by an adversary in a limited way (in the variable-size model). The main result of this paper states that for every visibly pushdown language the space complexity in the variable-size sliding window model is either constant, logarithmic or linear in the window size. This extends previous results for regular languages.