We revisit the linear search problem where a robot, initially placed at the origin on an infinite line, tries to locate a stationary target placed at an unknown position on the line. Unlike previous studies, in which the robot travels along the line at a constant speed, we consider settings where the robot's speed can depend on the direction of travel along the line, or on the profile of the terrain, e.g. when the line is inclined, and the robot can accelerate. Our objective is to design search algorithms that achieve good competitive ratios for the time spent by the robot to complete its search versus the time spent by an omniscient robot that knows the location of the target. We consider several new robot mobility models in which the speed of the robot depends on the terrain.These include 1) different constant speeds for different directions, 2) speed with constant acceleration and or variability depending on whether a certain segment has already been searched, 3) speed dependent on the incline of the terrain. We provide both upper and lower bounds on the competitive ratios of search algorithms for these models, and in many cases, we derive optimal algorithms for the search time.
The search problem has been extensively studied, e.g. see the survey by Benkoski @cite_2 ; deterministic algorithms for optimal linear search @cite_4 ; incorporating a when a robot changes direction during the search @cite_13 ; when bounds on the distance to the target are known in advance @cite_11 ; and for moving targets or more general linear cost functions @cite_15 . Other approaches include optimal randomized algorithms for the related @cite_0 , and stochastic and game theoretic investigations @cite_6 @cite_17 .
@cite_4 @cite_17 @cite_6 @cite_0 @cite_2 @cite_15 @cite_13 @cite_11
1995434882 2085497871 2005693620 2774807875 2014544956 2085216434 2068224788 2477849570
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 ċ OPT + 2d, 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 cowpath 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 + 2mm (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. Searching for a goal is a central and extensively studied problem in computer science. In classical searching problems, the cost of a search function is simply the number of queries made to an oracle that knows the position of the goal. In many robotics problems, as well as in problems from other areas, we want to charge a cost proportional to the distance between queries (e.g., the time required to travel between two query points). With this cost function in mind, the abstract problem known as the @math -lane cow-path problem was designed. There are known optimal deterministic algorithms for the cow-path problem, and we give the first randomized algorithms in this paper. We show that our algorithm is optimal for two paths ( @math ), and give evidence that it is indeed optimal for larger values of @math . The randomized algorithms give expected performance that is almost twice as good as is possible with a deterministic algorithm. The linear search problem concerns a search on the real line for a point selected at random according to a given probability distribution. The search begins at zero and is made by a continuous motion with constant speed, first in one direction and then the other. The problem is to determine when it is possible to devise a “best” search plan. In former papers the best plan has been selected according to the criterion of minimum expected path length. In this paper we consider a more general, nonlinear criterion for a “best” plan and show that the substantive requirements of the earlier results are not affected by these changes. Consider a setting where robots must visit sites represented as nodes in a graph, but each robot may fail when traversing an edge. The goal is to find a set of paths for a team of robots which maximizes the expected number of nodes collectively visited, while guaranteeing that the paths satisfy a notion of “independence” formalized by a matroid (e.g. limits on team size, number of visits to regions), and that the probabilities that each robot survives to its destination are above a given threshold. We call this problem the Matroid Team Surviving Orienteers (MTSO) problem, which has broad applications such as environmental monitoring in risky regions and search and rescue in dangerous conditions. We present the MTSO formally and detail numerous examples of matroids in a path planning context. We then propose an approximate greedy algorithm for selecting a feasible set of paths and prove that the value of the output is within a factor p s p s + λ of the optimum, where p s is the per-robot survival probability threshold and 1 λ ≤ 1 is the approximation factor of an oracle routine for the well known orienteering problem. We demonstrate the efficiency of our approach by applying it to a scenario where a team of robots must gather information while avoiding pirates in the Coral Triangle. Let us consider an ordered vector A[1 : n]. If the cost of testing each position is similar, then the standard binary search is the best strategy to search the vector. This is true in both average and worst case. However, if the costs are non-uniform, then the best strategy is not necessarily the standard binary search. The best algorithm to construct a strategy that minimizes the expected search cost runs in O(n3) time and requires O(n2) space. The same complexities hold for the best algorithm to construct a strategy that minimizes the worst case search cost. Here, we show how to efficiently construct search strategies that are at most at a constant factor from the optimal one. These constructions take linear time and use only linear space. For the problem of minimizing the expected search cost, we present an algorithm that requires O(n) space and gives a (2 + ∈ + O(1))-approximated solution in O(n) time, for any fixed value of ∈ > 0. On the other hand, for the problem of minimizing the worst case search cost, we describe an algorithm that requires O(n) space and gives a (2 + ∈ + O(1))- approximated solution in O(n) time, for any fixed value of ∈ > 0. These two problems arise when processing a query in a distributed text database indexed by a suffix array. We consider the problem of searching on m current rays for a target of unknown location. If no upper bound on the distance to the target is known in advance, then the optimal competitive ratio is 1+2mm (m−1)m−1. We show that even if an upper bound of D on the distance to the target is known in advance, then the competitive ratio of any search strategy is at least 1+2mm (m−1)m−1−O(1 log2D). This is again optimal – but in a stricter sense. In particular, this result implies the same lower bound for a robot searching for a target on infinite rays and finding it at a distance of D. To show that our lower bound is, indeed, optimal we construct a search strategy that achieves this ratio. Our strategy does not need to know an upper bound on the distance to the target in advance; it achieves a competitive ratio of 1+2mm (m−1)m−1−O(1 log2D) if the target is found at distance D. Finally, we also present a linear time algorithm to compute the strategy that allows the robot to search the farthest for a given competitive ratio C. The linear search problem concerns a search made in the real line for a point selected according to a given probability distribution. The search begins at zero and is made by continuous motion with constant speed along the line, first in one direction and then the other. The problem is to search in such a manner that the expected time required for finding the point according to the chosen plan of search is a minimum. This plan of search is usually conceived of as having a first step, a second,etc., and in that case, this author has previously shown a necessary and sufficient condition on the probability distribution for the existence of a search plan which minimizes the expected searching time. In this paper, we define a notion of search in which there is no first step, but the steps are instead numbered from negative to positive infinity. These new rules change the problem, and under them, there is always a minimizing search procedure. In those cases which satisfy the earlier criterion, the solutions obtained are essentially the same as those obtained previously. We consider the problem of searching on a line using n mobile robots, of which at most f are faulty, and the remaining are reliable. The robots start at the same location and move in parallel along the line with the same speed. There is a target placed on the line at a location unknown to the robots. Reliable robots can find the target when they reach its location, but faulty robots cannot detect the target. Our goal is to design a parallel algorithm minimizing the competitive ratio, represented by the worst case ratio between the time of arrival of the first reliable robot at the target, and the distance from the source to the target. If n ≥ 2f+2, there is a simple algorithm with competitive ratio 1. For f Our search algorithm is easily seen to be optimal for the case n=f+1. We also show that as @math tends to ∞ the competitive ratio of our algorithm for the case $n = 2f+1 approaches 3 and this is optimal. More precisely, we show that asymptotically, the competitive ratio of our proportional schedule algorithm A(2f+1,f) is at most 3 + 4ln n n , while any search algorithm has a lower bound 3 + 2ln n n on its competitive ratio.