We consider the problem of fault-tolerant parallel search on an infinite line by @math robots. Starting from the origin, the robots are required to find a target at an unknown location. The robots can move with maximum speed @math and can communicate in wireless mode among themselves. However, among the @math robots, there are @math robots that exhibit byzantine faults . A faulty robot can fail to report the target even after reaching it, or it can make malicious claims about having found the target when in fact it has not. Given the presence of such faulty robots, the search for the target can only be concluded when the non-faulty robots have sufficient verification that the target has been found. We aim to design algorithms that minimize the value of @math , the time to find a target at a distance @math from the origin by @math robots among which @math are faulty. We give several different algorithms whose running time depends on the ratio @math , the density of faulty robots, and also prove lower bounds. Our algorithms are optimal for some densities of faulty robots.
In several studies, when the environment is not known in advance, search implies exploration, often involving mapping and localizing searchers within the environment @cite_17 @cite_5 @cite_30 @cite_15 @cite_20 @cite_18 . However, even for the case of a known, simple environment like a line, there were several interesting studies attempting to optimize the search time. They started with the independent works of Bellman @cite_29 and Beck @cite_6 , in which the authors attempted to minimize the competitive ratio in a stochastic setting. More exactly, they proved that time @math is needed to guarantee finding the target situated at a ( unknown) distance @math from the origin. Several other works on linear search followed (e.g. see @cite_9 @cite_6 @cite_23 @cite_16 @cite_24 @cite_26 @cite_29 ). More recently the search by a single searcher was studied for different models, e.g., when the turn cost was considered @cite_19 , when the bounds on the distance to the target are known in advance @cite_14 , and when the target was moving or for more general linear cost functions @cite_22 .
@cite_30 @cite_18 @cite_26 @cite_14 @cite_22 @cite_29 @cite_9 @cite_6 @cite_24 @cite_19 @cite_23 @cite_5 @cite_15 @cite_16 @cite_20 @cite_17
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The authors consider the problem faced by a newborn that must explore and learn an unknown room with obstacles in it. They seek algorithms that achieve a bounded ratio of the worst-case distance traversed in order to see all visible points of the environment (thus creating a map), divided by the optimum distance needed to verify the map. The situation is complicated by the fact that the latter offline problem (optimally verifying a map) is NP-hard and thus must be solved approximately. Although the authors show that there is no such competitive algorithm for general obstacle courses, they give a competitive algorithm for the case of a polygonal room with a bounded number of obstacles in it. > We study several versions of the shortest-path problem when the map is not known in advanced, but is specified dynamically. We are seeking dynamic decision rules that optimize the worst-case ratio of the distance covered to the length of the (statically) optimal path. We describe optimal decision rules for two cases: Layered graphs of bounded width, and two-dimensional scenes with unit square obstacles. The optimal rules turn out to be intuitive, common-sense heuristics. For slightly more general graphs and scenes, we show that no bounded ratio is possible. We also show that the computational problem of devising a strategy that achieves a given worst-case ratio to the optimum path in a graph is a universal two-person game, and thus PSPACE-complete, whereas optimizing the expected ratio is #P-hard. We revisit the problem of searching for a target at an unknown location on a line when given upper and lower bounds on the distance D that separates the initial position of the searcher from the target. Prior to this work, only asymptotic bounds were known for the optimal competitive ratio achievable by any search strategy in the worst case. We present the first tight bounds on the exact optimal competitive ratio achievable, parametrized in terms of the given range for D, along with an optimal search strategy that achieves this competitive ratio. We prove that this optimal strategy is unique and that it cannot be computed exactly in general. We characterize the conditions under which an optimal strategy can be computed exactly and, when it cannot, we explain how numerical methods can be used efficiently. In addition, we answer several related open questions and we discuss how to generalize these results to m rays, for any m ≥ 2. Search Theory is one of the original disciplines within the field of Operations Research. It deals with the problem faced by a Searcher who wishes to minimize the time required to find a hidden object, or “target. ” The Searcher chooses a path in the “search space” and finds the target when he is sufficiently close to it. Traditionally, the target is assumed to have no motives of its own regarding when it is found; it is simply stationary and hidden according to a known distribution (e. g. , oil), or its motion is determined stochastically by known rules (e. g. , a fox in a forest). The problems dealt with in this book assume, on the contrary, that the “target” is an independent player of equal status to the Searcher, who cares about when he is found. We consider two possible motives of the target, and divide the book accordingly. Book I considers the zero-sum game that results when the target (here called the Hider) does not want to be found. Such problems have been called Search Games (with the “ze- sum” qualifier understood). Book II considers the opposite motive of the target, namely, that he wants to be found. In this case the Searcher and the Hider can be thought of as a team of agents (simply called Player I and Player II) with identical aims, and the coordination problem they jointly face is called the Rendezvous Search Problem. A man in an automobile searches for another man who is located at some point of a certain road. He starts at a given point and knows in advance the probability that the second man is at any given point of the road. Since the man being sought might be in either direction from the starting point, the searcher will, in general, have to turn around many times before finding his target. How does he search so as to minimize the expected distance travelled? When can this minimum expectation actually be achieved? This paper answers the second of these questions. The linear search problem has been discussed previously by one of the present authors. In this paper, the probability distribution of the point sought in the real line is not known to the searcher. Since there is noa priori choice of distribution which recommends itself above all others, we treat the situation as a game and obtain minimax type solutions. Different minimaxima apply depending on the factors which one wishes to minimize (resp. maximize). Certain criteria are developed which help the reader judge whether the results obtained can be considered “good advice” in the solution of real problems analogous to this one. 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. 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 study exploration problems where a robot has to construct a complete map of an unknown environment using a path that is as short as possible. We present an on-line strategy that enables a mobile robot with vision to explore an unknown simple polygon. We prove that the resulting tour is less than 26.5 times as long as the shortest watchman tour that could be computed off-line. Our analysis is doubly founded on a novel geometric structure called angle hull. Let D be a connected region inside a simple polygon, P. We define the angle hull of D, @math , to be the set of all points in P that can see two points of D at a right angle. We show that the perimeter of @math cannot exceed in length the perimeter of D by more than a factor of 2. This upper bound is tight. I wish to find something which is located on a certain road. I start at a point on the road, but I do not know in which direction the object sought is to be found. Somehow, I must incorporate in my way of searching the possibility that it is either to the right or to the left. Thus, I must search first to the right, and then to the left, and then to the right again until it is found. What is a good way of conducting this search, and what is a bad way?