Linear search problem
{{Short description|Computational search problem}}
In computational complexity theory, the linear search problem is an optimal search problem introduced by Richard E. Bellman{{citation
| last=Bellman | first=Richard
| title=Problem 63-9, An Optimal Search
| journal=SIAM Review
| volume=5
| issue=3
| date=July 1963
| pages=274
| doi=10.1137/1005070
| jstor=2027629}} and independently considered by Anatole Beck.{{citation
| last=Beck | first=Anatole
| title=On the linear search Problem
| journal=Israel Journal of Mathematics
| volume=2
| pages=221–228
| date=December 1964
| issue=4
| doi=10.1007/BF02759737 | doi-access=free}}{{citation
| last=Beck | first=Anatole
| title=More on the linear search problem
| journal=Israel Journal of Mathematics
| volume=3
| pages=61–70
| date=June 1965
| issue=2
| doi=10.1007/BF02760028 | doi-access=free}}{{citation
| last1=Beck | first1=Anatole
| last2=Beck | first2=Micah
| title=The linear search problem rides again
| journal=Israel Journal of Mathematics
| volume=53
| pages=365–372
| date=December 1986
| issue=3
| doi=10.1007/BF02786568 | doi-access=free}}
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."
The problem is to find the hider in the shortest time possible. Generally, since the hider could be on either side of the searcher and an arbitrary distance away, the searcher has to oscillate back and forth, i.e., 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). (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.
The linear search problem for a general probability distribution is unsolved.{{citation|contribution=Chapter 8. Search on the Infinite Line|title=The Theory of Search Games and Rendezvous, Part 2|series=International Series in Operations Research & Management Science|volume=55|year=2003|pages=123–144|doi=10.1007/0-306-48212-6_8|doi-access=free|first1=Steve|last1=Alpern|first2=Shmuel|last2=Gal|isbn=0-7923-7468-1 }}. 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." However, there exists a dynamic programming algorithm that produces a solution for any discrete distribution{{citation
| last1=Bruss | first1=F. Thomas
| last2=Robertson | first2=James B.
| title=A survey of the linear-search problem
| journal=The Mathematical Scientist
| volume=13
| pages=75–89
| date=December 1988
| url=http://www.appliedprobability.org/data/files/TMS%20articles/13_2_1.pdf
| archive-url=https://web.archive.org/web/20210818153158/http://www.appliedprobability.org/data/files/TMS%20articles/13_2_1.pdf
| archive-date=2021-08-18|url-status=dead}} and also an approximate solution, for any probability distribution, with any desired accuracy.{{citation|contribution=Section 8.7. A Dynamic Programming Algorithm for the LSP|title=The Theory of Search Games and Rendezvous, Part 2|series=International Series in Operations Research & Management Science|volume=55|year=2003|pages=139–144|doi=10.1007/0-306-48212-6_8|doi-access=free|first1=Steve|last1=Alpern|first2=Shmuel|last2=Gal|isbn=0-7923-7468-1 }}
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.{{citation
| last1=Beck | first1=Anatole
| last2=Newman | first2=Donald J.
| title=Yet More on the linear search problem
| journal=Israel Journal of Mathematics
| volume=8
| pages=419–429
| date=December 1970
| issue=4
| doi=10.1007/BF02798690 | doi-access=free}} 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.{{citation
| last=Gal | first=Shmuel
| title=Search games
| publisher=Academic Press
| date=1980}} 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. Demaine et al. gave an online solution with a turn cost.{{citation
| last1=Demaine | first1=Erik D.
| last2=Fekete | first2=Sandor
| last3=Gal | first3=Shmuel
| title=Online searching with turn cost
| journal=Theoretical Computer Science
| volume=361
| issue=2–3
| date=September 2006
| pages=342–355
| doi=10.1016/j.tcs.2006.05.018 | doi-access=free| arxiv=cs/0406045
}}
These results were rediscovered in the 1990s by computer scientists as the cow path problem.