Searching in Euclidean Spaces with Predictions

TL;DR

This paper investigates search strategies in Euclidean spaces utilizing approximate distance predictions, achieving a competitive ratio of (10c)^{d+1} without knowing c, and establishing a lower bound of roughly (c/4)^{d-1}.

Contribution

It introduces a search strategy that is competitive in high-dimensional Euclidean spaces using approximate distance predictions, even when the accuracy constant c is unknown.

Findings

Achieves (10c)^{d+1}-competitive search strategy without knowing c.

Establishes a lower bound of roughly (c/4)^{d-1} on the competitive ratio.

Provides bounds for search efficiency in Euclidean spaces with predictions.

Abstract

We study the problem of searching for a target at some unknown location in $\mathbb{R}^d$ when additional information regarding the position of the target is available in the form of predictions. In our setting, predictions come as approximate distances to the target: for each point $p\in \mathbb{R}^d$ that the searcher visits, we obtain a value $\lambda(p)$ such that $|p\bm{t}|\le \lambda(p) \le c\cdot |p\bm{t}|$, where $c\ge 1$ is a fixed constant, $\bm{t}$ is the position of the target, and $|p\bm{t}|$ is the Euclidean distance of $p$ to $\bm{t}$. The cost of the search is the length of the path followed by the searcher. Our main positive result is a strategy that achieves $(10c)^{d+1}$-competitive ratio, even when the constant $c$ is unknown. We also give a lower bound of roughly $(c/4)^{d-1}$ on the competitive ratio of any search strategy in $\RR^d$, assuming that $c\ge 4$.