Report on Nearest Dominating Point Queries

TL;DR

This work addresses nearest dominating point queries in a joint real- and feature-space setting: for each $p_i$, find the closest dominator $p_j$ with $q_i \succ q_j$ or report none. It presents an $O(n \log n)$ solution in the 2D case using a balanced priority search tree and quadrant dominance queries, and generalizes to $d$ dimensions with a $d$-dimensional range tree plus Voronoi diagrams, achieving $O(n \log^{d+1} n)$ construction and query time. The paper also analyzes speedups, arguing against merging Voronoi diagrams and approximate approaches, and proposes an offline dynamic nearest-neighbor strategy in 2D to leverage known queries. The results offer a practical, scalable framework for dominance-based nearest-neighbor queries in multi-criteria spaces, with implications for related domination and nearest-neighbor problems $($e.g., multi-criteria scoring and recommendation systems$)$.

Abstract

Given two points $p, q \in \mathbb R^d$, we say that $p$ dominates $q$ and write $p \succ q$ if each coordinate of $p$ is larger than the corresponding coordinate of $q$. That is, if $p = (p^{(1)}, p^{(2)}, \ldots, p^{(d)})$ and $q = (q^{(1)}, q^{(2)}, \ldots, q^{(d)})$, $p \succ q$ if and only if $p^{(i)} > q^{(i)}$ for all $1 \le i \le d$. For example, $p$ and $q$ could represent various ratings for $2$ restaurants, based on different metrics like taste, affordability, ratings on different platforms, et cetera. $p \succ q$ then means that the first restaurant outperformed the second on each metric. Given a list of restaurants and their rating, we solve the problem of determining, for each restaurant, the closest restaurant to it that dominates it. We improve upon the algorithm under some assumptions towards the end.

Figures (1)

• Figure 1: The algorithm for \ref{['prb:midterm']} in action.