Towards a General Framework for Searching on a Line and Searching on $m$ Rays

TL;DR

This paper develops a unified framework for search problems on a line and multiple rays, optimizing strategies considering turn costs and movement costs, and provides provably optimal solutions under various conditions.

Contribution

It introduces a general cost framework for searching on a line with turn costs and extends it to multiple rays, offering optimal strategies and bounds.

Findings

Optimal search strategy for fixed lower bound D and turn costs.

New bounds on search costs surpass previous strategies.

Proposed strategies are optimal for small turn costs and conjectured to be always optimal.

Abstract

Consider the following classical search problem: given a target point $p\in \Re$, starting at the origin, find $p$ with minimum cost, where cost is defined as the distance travelled. Let $D$ be the distance of $p$ from the origin. When no lower bound on $D$ is given, no competitive search strategy exists. Demaine, Fekete and Gal (Online searching with turn cost, Theor. Comput. Sci., 361(2-3):342-355, 2006) considered the situation where no lower bound on $D$ is given but a fixed \emph{turn cost} $t>0$ is charged every time the searcher changes direction. When the total cost is expressed as $c D+\phi$, where $c$ and $\phi$ are positive constants, they showed that if $c$ is set to $9$, then the optimal search strategy has a cost of $9D+2t$. Although their strategy is optimal for $c=9$, we prove that the minimum cost in their framework is $5D+t+2\sqrt{2D(2D+t)} < 9D+2t$. Note that the minimum cost requires knowledge of $D$. However, given $D$, the optimal strategy has a smaller cost of $3D+t$. Therefore, this problem cannot be solved optimally and exactly when no lower bound on $D$ is given. To resolve this issue, we introduce a general framework where the cost of moving distance $x$ away from the origin is $\alpha_1 x+\beta_1$ and the cost of moving distance $y$ towards the origin is $\alpha_2 y+\beta_2$ for constants $\alpha_1,\alpha_2,\beta_1,\beta_2$. Given a lower bound $\lambda$ on $D$, we provide a provably optimal competitive search strategy when $\alpha_1,\alpha_2,\beta_1,\beta_2 \geq 0$ and $\alpha_1+\alpha_2 > 0$. Finally, we address the problem of searching for a target lying on one of $m$ rays extending from the origin where the cost is measured as the total distance travelled plus $t \geq 0$ times the number of turns. We provide a search strategy and compute its cost. We prove our strategy is optimal for small values of $t$ and conjecture it is always optimal.