Faster Linear-Space Data Structures for Path Frequency Queries

TL;DR

This paper introduces new linear-space data structures for efficient path frequency queries on trees, improving query times and supporting multiple query types with optimized preprocessing.

Contribution

It presents the first linear-space data structures for several path frequency queries with improved query times and new capabilities, including path maximum g-value color queries.

Findings

Achieves $O(rac{1}{ ooted{2}} rac{ ooted{n}}{ ooted{w}})$ query time for path mode and least frequent element.

Reduces path $ ooted{ ext{ extalpha}- ext{minority}}$ query time to $O(rac{1}{ extroot{}})$ using a randomized algorithm.

Supports path maximum g-value color queries in $O(rac{ ooted{n}}{ ooted{w}})$ time with linear space.

Abstract

We present linear-space data structures for several frequency queries on trees, namely: path mode, path least frequent element, and path $\alpha$-minority queries. We present the first linear-space data structures, requiring $O(n \sqrt{nw})$ preprocessing time, that can answer path mode and path least frequent element queries in $O(\sqrt{n/w})$ time. This improves upon the best previously known bound of $O(\log\log n \sqrt{n/w})$ achieved by Durocher et al. in 2016. For the path $\alpha$-minority problem, where $\alpha$ is specified at query time, we reduce the query time of the linear-space data structure of Durocher et al. from $O(\alpha^{-1}\log\log n)$ down to $O(\alpha^{-1})$ by employing a simple randomized algorithm with a success probability $\geq 1/2$. We also present the first linear-space data structure supporting "Path Maximum $g$-value Color" queries in $O(\sqrt{n/w})$ time, requiring $O(n \sqrt{nw})$ preprocessing time. This general framework encapsulates both path mode and path least frequent element queries. For our data structures, we consider the word-RAM model with $w\in \Omega(\log n)$, where $w$ is the word size in bits.