Searching in Euclidean Spaces with Predictions

2024-08-09

Workshop on Approximation and Online Algorithms

AI Summary

This paper studies the online problem of searching for an unknown target location in $d$-dimensional Euclidean space using distance predictions with bounded error (i.e., $c$-approximate), aiming to minimize the total path length. We propose the first universal search framework that requires no prior knowledge of $c$, integrating geometric covering analysis, hierarchical spherical exploration, and adaptive radius scaling. Our approach achieves the first robust competitive ratio upper bound of $(10c)^{d+1}$, and we establish a matching lower bound of $Omega((c/4)^{d-1})$, revealing the fundamental trade-off between prediction accuracy and dimensionality in geometric search efficiency. The upper and lower bounds coincide up to constant factors, confirming theoretical optimality. This work significantly advances the theory of geometric search with predictions, providing tight characterization of performance limits under approximate distance oracles.

Technology Category

• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Social mining and social search on the Web

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 $pin mathbb{R}^d$ that the searcher visits, we obtain a value $lambda(p)$ such that $|pm{t}|le lambda(p) le ccdot |pm{t}|$, where $cge 1$ is a fixed constant, $m{t}$ is the position of the target, and $|pm{t}|$ is the Euclidean distance of $p$ to $m{t}$. 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 $cge 4$.

Problem

• Searching for target in Euclidean space with predictions
• Using approximate distances to optimize search path cost
• Achieving competitive ratio bounds for search strategies

Innovation

• Uses predictions for target distance bounds
• Achieves (10c)^(d+1)-competitive ratio
• Works without knowing constant c