On strictly output sensitive color frequency reporting

TL;DR

This paper introduces a new data structure for color frequency reporting in multi-dimensional spaces that is strictly output sensitive, near-optimal, and adaptable to higher dimensions, improving query efficiency for colored point sets.

Contribution

The paper presents a simple, space-efficient data structure with output-sensitive query times, proves a lower bound for the weighted case, and extends results to higher dimensions.

Findings

Achieves $O( ext{ns} ext{log}_s n)$ size and $O( ext{log} n + k ext{log}_s n)$ query time in 2D.

Proves a near-matching lower bound for the weighted version, indicating near-optimality.

Provides a space reduction transformation and an efficient algorithm for dominance queries with linear space.

Abstract

Given a set of $n$ colored points $P \subset \mathbb{R}^d$ we wish to store $P$ such that, given some query region $Q$, we can efficiently report the colors of the points appearing in the query region, along with their frequencies. This is the \emph{color frequency reporting} problem. We study the case where query regions $Q$ are axis-aligned boxes or dominance ranges. If $Q$ contains $k$ colors, the main goal is to achieve ``strictly output sensitive'' query time $O(f(n) + k)$. Firstly, we show that, for every $s \in \{ 2, \dots, n \}$, there exists a simple $O(ns\log_s n)$ size data structure for points in $\mathbb{R}^2$ that allows frequency reporting queries in $O(\log n + k\log_s n)$ time. Secondly, we give a lower bound for the weighted version of the problem in the arithmetic model of computation, proving that with $O(m)$ space one can not achieve query times better than $\Omega\left(\phi \frac{\log (n / \phi)}{\log (m / n)}\right)$, where $\phi$ is the number of possible colors. This means that our data structure is near-optimal. We extend these results to higher dimensions as well. Thirdly, we present a transformation that allows us to reduce the space usage of the aforementioned datastructure to $O(n(s \phi)^\varepsilon \log_s n)$. Finally, we give an $O(n^{1+\varepsilon} + m \log n + K)$-time algorithm that can answer $m$ dominance queries $\mathbb{R}^2$ with total output complexity $K$, while using only linear working space.