In
computational complexity theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved ...
, the linear search problem is an optimal search problem introduced by
Richard E. Bellman and independently considered by
Anatole Beck.
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." In order to find the hider the searcher has to go a distance x
1 in one direction, return to the origin and go distance x
2 in the other direction etc., (the length of the n-th step being denoted by x
n), 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. However, there exists a
dynamic programming
Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics.
I ...
algorithm that produces a solution for any discrete distribution and also an approximate solution, for any probability distribution, with any desired accuracy.
The linear search problem was solved by Anatole Beck and
Donald J. Newman (1970) as a two-person zero-sum game. Their
minimax
Minimax (sometimes MinMax, MM or saddle point) is a decision rule used in artificial intelligence, decision theory, game theory, statistics, and philosophy for ''mini''mizing the possible loss for a worst case (''max''imum loss) scenario. Whe ...
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. 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
In computer science, an online algorithm is one that can process its input piece-by-piece in a serial fashion, i.e., in the order that the input is fed to the algorithm, without having the entire input available from the start.
In contrast, an o ...
by
Shmuel Gal, who also generalized this result to a set of concurrent rays. 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.
These results were rediscovered in the 1990s by computer scientists as the cow path problem.
See also
*
Linear search
In computer science, a linear search or sequential search is a method for finding an element within a list. It sequentially checks each element of the list until a match is found or the whole list has been searched.
A linear search runs in at ...
*
Search games
A search game is a two-person zero-sum game which takes place in a set called the search space. The searcher can choose any continuous trajectory subject to a maximal velocity constraint. It is always assumed that neither the searcher nor the hide ...
References
{{DEFAULTSORT:Linear Search Problem
Computational problems
Mathematical optimization