## Abstract

We consider the problem of searching for an object on a line at an unknown distance OPT from the original position of the searcher, in the presence of a cost of d for each time the searcher changes direction. This is a generalization of the well-studied linear-search problem. We describe a strategy that is guaranteed to find the object at a cost of at most 9 · OPT + 2 d, which has the optimal competitive ratio 9 with respect to OPT plus the minimum corresponding additive term. Our argument for upper and lower bound uses an infinite linear program, which we solve by experimental solution of an infinite series of approximating finite linear programs, estimating the limits, and solving the resulting recurrences for an explicit proof of optimality. We feel that this technique is interesting in its own right and should help solve other searching problems. In particular, we consider the star search or cow-path problem with turn cost, where the hidden object is placed on one of m rays emanating from the original position of the searcher. For this problem we give a tight bound of (1 + 2 m^{m} / (m - 1)^{m - 1}) OPT + m ((m / (m - 1))^{m - 1} - 1) d. We also discuss tradeoffs between the corresponding coefficients and we consider randomized strategies on the line.

Original language | English |
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Pages (from-to) | 342-355 |

Number of pages | 14 |

Journal | Theoretical Computer Science |

Volume | 361 |

Issue number | 2-3 |

DOIs | |

State | Published - 1 Sep 2006 |

### Bibliographical note

Funding Information:We thank two anonymous referees for helpful suggestions that helped to extend the scope of this paper and improve overall presentation. Parts of this research were supported by NATO Grants PST.CLG976391 and CRG 972991. Other parts were done while Sándor Fekete was visiting MIT, with partial funding by DFG travel Grant FE 407/7-1.

## Keywords

- Competitive ratio
- Cow-path problem
- Infinite linear programs
- Linear-search problem
- Online problems
- Randomized strategies
- Search games
- Star search
- Turn cost

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science