Searching in Euclidean Spaces with Predictions

TL;DR

This work introduces a predictions-based search problem in Euclidean spaces, where the searcher obtains at each visited point $p$ a $c$-prediction $\lambda(p)$ of the target distance with $|p t|\le \lambda(p)\le c\,|p t|$. The authors develop an epsilon-net based strategy that drives the predicted distance down geometrically, achieving a competitive ratio of $4\cdot 5^d\cdot (c^*)^{d+1}$ when $c$ is known and $6\cdot 10^d\cdot (c^*)^{d+1}$ when $c^*$ is unknown, both improving with dimension and the prediction factor. They also prove a lower bound of roughly $(c/4)^{d-1}\cdot \min\{\sqrt{\pi/d},1\}$ for $c\ge 4$, demonstrating near-optimality gaps in high dimensions. Overall, the paper establishes constant-competitive strategies in $d\ge 2$ under a continuous prediction model and outlines several promising directions for refinement and extension to other targets and metric spaces.

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 $λ(p)$ such that $|p\bm{t}|\le λ(p) \le c\cdot |p\bm{t}|$, 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 $c\ge 4$.

Figures (4)

• Figure 1: Example of $c$-prediction function $\lambda(p)$ for $c=3/2$. In this example the function is not monotone in $|p\bm{t}|$ and it is not continuous.
• Figure 3: Spherical shell $S(p_i,\lambda(p_i)/c,\lambda(p_i))$ where $\bm{t}$ must lie and (part of) a net for the shell.
• Figure 5: Parts in the domains for $\lambda_{\bm{t}}$ and $\lambda_{\bm{t}'}$ for two targets when $d=2$.
• Figure 6: Examples of functions $\lambda_{\bm{t}}$ for $d=1$. Note that the axes have different scales. Left: example for $c=3$; the target has to be at distance at most $1/6$ from the origin. Right: two functions for $c=6$; the target has to be at distance at most $1/3$ from the origin.

Theorems & Definitions (14)

• Lemma 1
• Lemma 2
• Lemma 3
• Proposition 4
• Lemma 5
• Proposition 6: Case $c=1$
• Lemma 7
• Theorem 8: Known $c$
• Theorem 9: Unknown $c$
• Lemma 10