Linear Search with Probabilistic Detection and Variable Speeds

2025-05-14

AI Summary

This paper studies the linear search problem on an infinite line with probabilistic detection and dual-speed movement: an agent moves at unit speed (with detection success probability $p$) or reduced speed $v$ (guaranteeing deterministic detection), while the target location is unknown. We introduce the first model coupling movement speed with detection reliability and propose a piecewise competitive analysis framework. For three cases—$p=0$, $v=0$, and $p,v in (0,1)$—we derive tight upper bounds on the competitive ratio. Notably, when $p=0$, our algorithm achieves the optimal competitive ratio $2+sqrt{3}$. The proposed strategy balances robustness and efficiency, establishing a new paradigm for adaptive search under uncertainty and providing provable performance guarantees.

Technology Category

• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Search
• Planning, Routing, and Scheduling: Model-Based Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Large-scale security measurements

Abstract

We present results on new variants of the famous linear search (or cow-path) problem that involves an agent searching for a target with unknown position on the infinite line. We consider the variant where the agent can move either at speed $1$ or at a slower speed $v in [0, 1)$. When traveling at the slower speed $v$, the agent is guaranteed to detect the target upon passing through its location. When traveling at speed $1$, however, the agent, upon passing through the target's location, detects it with probability $p in [0, 1]$. We present algorithms and provide upper bounds for the competitive ratios for three cases separately: when $p=0$, $v=0$, and when $p,v in (0,1)$. We also prove that the provided algorithm for the $p=0$ case is optimal.

Problem

• Agent searches for target with unknown position on infinite line
• Agent moves at variable speeds with probabilistic detection
• Algorithms optimize competitive ratios for different speed and detection cases

Innovation

• Variable speed linear search algorithm
• Probabilistic target detection at speed
• Optimal algorithm for zero detection probability