Binary Jumbled Pattern Matching on Trees and Tree-Like Structures

TL;DR

This paper presents efficient algorithms for binary jumbled pattern matching on trees and compressed strings, and proves the problem is fixed-parameter tractable with respect to treewidth, advancing the state of the art in pattern matching.

Contribution

It introduces new algorithms with improved time bounds for trees and compressed strings, and establishes fixed-parameter tractability based on treewidth.

Findings

O(n^2 / log^2 n) time for trees

Improved bounds for grammar-compressed strings

Fixed-parameter tractability with respect to treewidth

Abstract

Binary jumbled pattern matching asks to preprocess a binary string $S$ in order to answer queries $(i,j)$ which ask for a substring of $S$ that is of length $i$ and has exactly $j$ 1-bits. This problem naturally generalizes to vertex-labeled trees and graphs by replacing "substring" with "connected subgraph". In this paper, we give an $O(n^2 / \log^2 n)$-time solution for trees, matching the currently best bound for (the simpler problem of) strings. We also give an $\Oh{g^{2 / 3} n^{4 / 3}/(\log n)^{4/3}}$-time solution for strings that are compressed by a grammar of size $g$. This solution improves the known bounds when the string is compressible under many popular compression schemes. Finally, we prove that the problem is fixed-parameter tractable with respect to the treewidth $w$ of the graph, thus improving the previous best $n^{O(w)}$ algorithm [ICALP'07].