On the complexity of the (approximate) nearest colored node problem

TL;DR

This paper introduces a new color distance oracle with improved query time and proves its optimality, also exploring dynamic variants and establishing bounds for the approximate nearest colored node problem.

Contribution

The paper presents a faster color distance oracle with optimal query time and extends the analysis to dynamic graph settings with new bounds.

Findings

Achieved a color distance oracle with query time O(log k)

Proved the query time is optimal in the cell probe model

Explored dynamic settings with new upper and lower bounds

Abstract

Given a graph $G=(V,E)$ where each vertex is assigned a color from the set $C=\{c_1, c_2, .., c_\sigma\}$. In the (approximate) nearest colored node problem, we want to query, given $v \in V$ and $c \in C$, for the (approximate) distance $\widehat{\mathbf{dist}}(v, c)$ from $v$ to the nearest node of color $c$. For any integer $1 \leq k \leq \log n$, we present a Color Distance Oracle (also often referred to as Vertex-label Distance Oracle) of stretch $4k-5$ using space $O(kn\sigma^{1/k})$ and query time $O(\log{k})$. This improves the query time from $O(k)$ to $O(\log{k})$ over the best known Color Distance Oracle by Chechik \cite{DBLP:journals/corr/abs-1109-3114}. We then prove a lower bound in the cell probe model showing that our query time is optimal in regard to space up to constant factors. We also investigate dynamic settings of the problem and find new upper and lower bounds.