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Linear Search with Probabilistic Detection and Variable Speeds

Jared Coleman, Oscar Morales-Ponce

TL;DR

This work extends the classic linear search (cow-path) problem by introducing two movement speeds and probabilistic detection, adding a speed/detection trade-off to search efficiency. The authors derive tight results for the extreme regimes (p = 0 and v = 0) and propose strategies for the general case where both p and v lie in (0,1), including a hybrid approach that blends slow exploration with occasional fast scouting. The fast-only and slow-only strategies yield competitive ratios CR_Fast = 8/p + p/(2-p) and CR_Slow = 3 + 2 sqrt(2 + 2/v) + 2/v respectively, with the hybrid method improving performance in many parameter regions, and numerical optimization guiding its parameters. The paper also situates the work within prior literature on non-uniform speed and uncertain detection, and outlines future directions such as tighter lower bounds for the v = 0 case, learning-augmented variants, and higher-dimensional extensions.

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.

Linear Search with Probabilistic Detection and Variable Speeds

TL;DR

This work extends the classic linear search (cow-path) problem by introducing two movement speeds and probabilistic detection, adding a speed/detection trade-off to search efficiency. The authors derive tight results for the extreme regimes (p = 0 and v = 0) and propose strategies for the general case where both p and v lie in (0,1), including a hybrid approach that blends slow exploration with occasional fast scouting. The fast-only and slow-only strategies yield competitive ratios CR_Fast = 8/p + p/(2-p) and CR_Slow = 3 + 2 sqrt(2 + 2/v) + 2/v respectively, with the hybrid method improving performance in many parameter regions, and numerical optimization guiding its parameters. The paper also situates the work within prior literature on non-uniform speed and uncertain detection, and outlines future directions such as tighter lower bounds for the v = 0 case, learning-augmented variants, and higher-dimensional extensions.

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 or at a slower speed . When traveling at the slower speed , the agent is guaranteed to detect the target upon passing through its location. When traveling at speed , however, the agent, upon passing through the target's location, detects it with probability . We present algorithms and provide upper bounds for the competitive ratios for three cases separately: when , , and when . We also prove that the provided algorithm for the case is optimal.
Paper Structure (8 sections, 3 theorems, 29 equations, 4 figures, 3 algorithms)

This paper contains 8 sections, 3 theorems, 29 equations, 4 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. The Fast-Only Case
  4. The Slow-Only Case
  5. The General Case
  6. Moving Fast or Slow
  7. A Hybrid Approach
  8. Conclusion

Key Result

theorem thmcountertheorem

Algorithm alg:slow has a competitive ratio of $3 + 2\sqrt{2+\frac{2}{v}}+\frac{2}{v}$.

Figures (4)

  • Figure 1: Cone of exploration for the agent.
  • Figure 2: The numerically computed best expansion ratio $a$ (left) and scout-ahead ratio $b$ (right) values for different values of $p$ and $v$.
  • Figure 3: Competitive ratio (left) and difference in expected competitive ratios (right) of Algorithm \ref{['alg:slow']} and Algorithm \ref{['alg:hybrid']}, showing how much the hybrid approach improves on the slow approach.
  • Figure 4: Values for $p$ and $v$ where the Slow and Hybrid Approaches are better/worse than the Fast Approach.

Theorems & Definitions (6)

  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • proof