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Competitive Search in the Line and the Star with Predictions

Spyros Angelopoulos

TL;DR

The paper investigates competitive search for a hidden target on a line and in an $m$-ray star under predictions, introducing three models of prediction: untrusted $k$-bit advice, directional, and positional. It develops consistency–robustness tradeoffs and extends them with a tolerance parameter $H$ for weak predictions, providing both tight upper and matching lower bounds. A key methodological bridge is the reduction from single-searcher with advice to a $2^k$-searcher parallel framework and the use of Gal's theorem (via $ ho_m^*$ and $oldsymbol{eta}$ functionals) to derive fundamental limits, complemented by Rivest et al.'s lying-responder results for weak predictions. The results deliver Pareto-optimal tradeoffs across all three prediction models, reveal the impact of prediction quality on robustness, and offer explicit constructive strategies (including geometric and biased-search variants) with clear implications for resource allocation and rescue/navigation tasks in unbounded environments.

Abstract

We study the classic problem of searching for a hidden target in the line and the $m$-ray star, in a setting in which the searcher has some prediction on the hider's position. We first focus on the main metric for comparing search strategies under predictions; namely, we give positive and negative results on the consistency-robustness tradeoff, where the performance of the strategy is evaluated at extreme situations in which the prediction is either error-free, or adversarially generated, respectively. For the line, we show tight bounds concerning this tradeoff, under the untrusted advice model, in which the prediction is in the form of a $k$-bit string which encodes the responses to $k$ binary queries. For the star, we give tight, and near-tight tradeoffs in the positional and the directional models, in which the prediction is related to the position of the target within the star, and to the ray on which the target hides, respectively. Last, for all three prediction models, we show how to generalize our study to a setting in which the performance of the strategy is evaluated as a function of the searcher's desired tolerance to prediction errors, both in terms of positive and inapproximability results.

Competitive Search in the Line and the Star with Predictions

TL;DR

The paper investigates competitive search for a hidden target on a line and in an -ray star under predictions, introducing three models of prediction: untrusted -bit advice, directional, and positional. It develops consistency–robustness tradeoffs and extends them with a tolerance parameter for weak predictions, providing both tight upper and matching lower bounds. A key methodological bridge is the reduction from single-searcher with advice to a -searcher parallel framework and the use of Gal's theorem (via and functionals) to derive fundamental limits, complemented by Rivest et al.'s lying-responder results for weak predictions. The results deliver Pareto-optimal tradeoffs across all three prediction models, reveal the impact of prediction quality on robustness, and offer explicit constructive strategies (including geometric and biased-search variants) with clear implications for resource allocation and rescue/navigation tasks in unbounded environments.

Abstract

We study the classic problem of searching for a hidden target in the line and the -ray star, in a setting in which the searcher has some prediction on the hider's position. We first focus on the main metric for comparing search strategies under predictions; namely, we give positive and negative results on the consistency-robustness tradeoff, where the performance of the strategy is evaluated at extreme situations in which the prediction is either error-free, or adversarially generated, respectively. For the line, we show tight bounds concerning this tradeoff, under the untrusted advice model, in which the prediction is in the form of a -bit string which encodes the responses to binary queries. For the star, we give tight, and near-tight tradeoffs in the positional and the directional models, in which the prediction is related to the position of the target within the star, and to the ray on which the target hides, respectively. Last, for all three prediction models, we show how to generalize our study to a setting in which the performance of the strategy is evaluated as a function of the searcher's desired tolerance to prediction errors, both in terms of positive and inapproximability results.
Paper Structure (10 sections, 13 theorems, 46 equations, 1 figure)

This paper contains 10 sections, 13 theorems, 46 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Searching with predictions
  3. Contribution
  4. Preliminaries
  5. Linear search with untrusted advice
  6. Extension to weak predictions
  7. Ray search with directional prediction
  8. Extension to weak predictions
  9. Ray search with positional prediction
  10. Extension to weak predictions

Key Result

Theorem 1

Let $X = (x_0,x_1,\ldots)$ be a sequence of positive numbers, $r$ an integer, and $\alpha_X = \limsup_{n\rightarrow\infty} (x_n)^{1/n}$, for $\alpha\in \mathbb{R} \cup \{+\infty\}$. Let $F_i$, $i \geq 0$ be a sequence of functionals which satisfy the following properties: then

Figures (1)

  • Figure 1: A snapshot of the first iteration in a $p$-parallel strategy.

Theorems & Definitions (23)

  • Theorem 1: Gal80
  • Theorem 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Theorem 6
  • ...and 13 more