#jsDisabledContent { display:none; } My Account |  Register |  Help

# Linear search problem

Article Id: WHEBN0022727353
Reproduction Date:

 Title: Linear search problem Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Linear search problem

In computational complexity theory, the linear search problem is an optimal search problem introduced by Richard E. Bellman.[1] (independently considered by Anatole Beck [2][3][4]).

## The problem

"An immobile hider is located on the real line according to a known probability distribution. A searcher, whose maximal velocity is one, starts from the origin and wishes to discover the hider in minimal expected time. It is assumed that the searcher can change the direction of his motion without any loss of time. It is also assumed that the searcher cannot see the hider until he actually reaches the point at which the hider is located and the time elapsed until this moment is the duration of the game." It is obvious that in order to find the hider the searcher has to go a distance x1 in one direction, return to the origin and go distance x2 in the other direction etc., (the length of the n-th step being denoted by xn), and to do it in an optimal way. (However, an optimal solution need not have a first step and could start with an infinite number of small 'oscillations'.) This problem is usually called the linear search problem and a search plan is called a trajectory. It has attracted much research, some of it quite recent.

The linear search problem for a general probability distribution is unsolved.[5] However, there exists a dynamic programming algorithm that produces a solution for any discrete distribution [6] and also an approximate solution, for any probability distribution, with any desired accuracy.[7]

The linear search problem was solved by Anatole Beck and Donald J. Newman (1970) as a two-person zero-sum game. Their minimax trajectory is to double the distance on each step and the optimal strategy is a mixture of trajectories that increase the distance by some fixed constant.[8] This solution gives search strategies that are not sensitive to assumptions concerning the distribution of the target. Thus, it also presents an upper bound for a worst-case scenario. This solution was obtained in the framework of an online algorithm by Shmuel Gal, who also generalized this result to a set of concurrent rays.[9] The best online competitive ratio for the search on the line is 9 but it can be reduced to 4.6 by using a randomized strategy. The online solution with a turn cost is given in.[10]

These results were rediscovered in the 1990s by computer scientists as the cow path problem.

## References

1. ^ R. Bellman. An optimal search problem, SIAM Rev. (1963).
2. ^ A. Beck. On the linear search Problem, Israel J. Mathematics (1964).
3. ^ A. Beck. More on the linear search problem, Israel J. Mathematics (1965).
4. ^ A. Beck and M. Beck. The linear search problem rides again, Israel J. Mathematics (1986).
5. ^ Alpern, Steve; Gal, Shmuel (2003), "Chapter 8. Search on the Infinite Line", The Theory of Search Games and Rendezvous, Part 2, International Series in Operations Research & Management Science 55, pp. 123–144, . On p. 124, Alpern and Gal write "no algorithm for solving the problem for a general probability distribution function has been found during about 37 years since the LSP was first presented."
6. ^ T.F. Bruce and J.B. Robertson. A survey of the linear-search problem. Math. Sci. 13 (1988).
7. ^ S. Alpern and S. Gal. The Theory of Search Games and Rendezvous. Springer (2003): 139--143.
8. ^ A. Beck and D.J. Newman. Yet More on the linear search problem. Israel J. Math. (1970).
9. ^ S. Gal. SEARCH GAMES, Academic Press (1980).
10. ^ E. Demaine, S. Fekete and S. Gal. Online searching with turn cost. Theoretical Computer Science (2006).
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.

Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.