12 research outputs found
Parallel searching on m raysââThis research is supported by the DFG-Project âDiskrete Problemeâ, No. Ot 64/8-3.
AbstractWe investigate parallel searching on m concurrent rays. We assume that a target t is located somewhere on one of the rays; we are given a group of m point robots each of which has to reach t. Furthermore, we assume that the robots have no way of communicating over distance. Given a strategy S we are interested in the competitive ratio defined as the ratio of the time needed by the robots to reach t using S and the time needed to reach t if the location of t is known in advance.If a lower bound on the distance to the target is known, then there is a simple strategy which achieves a competitive ratio of 9âindependent of m. We show that 9 is a lower bound on the competitive ratio for two large classes of strategies if mâ©Ÿ2.If the minimum distance to the target is not known in advance, we show a lower bound on the competitive ratio of 1+2(k+1)k+1/kk where k=âlogmâ where log is used to denote the base-2Â logarithm. We also give a strategy that obtains this ratio
Revisiting the Problem of Searching on a Line
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, parameterized in terms of the given
bounds on D, along with an optimal search strategy that achieves this
competitive ratio. We prove that this optimal strategy is unique. 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, including
the maximal reach problem, and we discuss how to generalize these results to m
rays, for any m >= 2
Searching with increasing speeds
© Springer Nature Switzerland AG 2018. In the classical search problem on the line or in higher dimension one is asked to find the shortest (and often the fastest) route to be adopted by a robot R from the starting point s towards the target point t located at unknown location and distance D. It is usually assumed that robot R moves with a fixed unit speed 1. It is well known that one can adopt a âzig-zagâ strategy based on the exponential expansion, which allows to reach the target located on the line in time â€9D and this bound is tight. The problem was also studied in two dimensions where the competitive factor is known to be O(D). In this paper we study an alteration of the search problem in which robot R starts moving with the initial speed 1. However, during search it can encounter a point or a sequence of points enabling faster and faster movement. The main goal is to adopt the route which allows R to reach the target t as quickly as possible. We study two variants of the considered search problem: (1) with the global knowledge and (2) with the local knowledge. In variant (1) robot R knows a priori the location of all intermediate points as well as their expulsion speeds. In this variant we study the complexity of computing optimal search trajectories. In variant (2) the relevant information about points in P is acquired by R gradually, i.e., while moving along the adopted trajectory. Here the focus is on the competitive factor of the solution, i.e., the ratio between the solutions computed in variants (2) and (1). We also consider two types of search spaces with points distributed on the line and subsequently with points distributed in two-dimensional space
Parallel Searching on m Rays
. We investigate parallel searching on m concurrent rays. We assume that a target t is located somewhere on one of the rays; we are given a group of m point robots each of which has to reach t. Furthermore, we assume that the robots have no way of communicating over distance. Given a strategy S we are interested in the competitive ratio dened as the ratio of the time needed by the robots to reach t using S and the time needed to reach t if the location of t is known in advance. If a lower bound on the distance to the target is known, then there is a simple strategy which achieves a competitive ratio of 9 | independent of m. We show that 9 is a lower bound on the competitive ratio for two large classes of strategies if m 2. If the minimum distance to the target is not known in advance, we show a lower bound on the competitive ratio of 1 + 2(k + 1) k+1 =k k where k = dlog me. We also give a strategy that obtains this ratio. 1 Introduction Searching for a target is an important a..
Parallel Searching on m Rays
We investigate parallel searching on concurrent rays. We assume that a target is located somewhere on one of the rays; we are given a group of point robots each of which has to reach . Furthermore, we assume that the robots have no way of communicating over distance. Given a strategy we are interested in the competitive ratio defined as the ratio of the time needed by the robots to reach using and the time needed to reach if the location of is known in advance. If a lower bound on the distance to the target is known, then there is a simple strategy which achieves a competitive ratio of~9 --- independent of . We show that 9 is a lower bound on the competitive ratio for two large classes of strategies if . If the minimum distance to the target is not known in advance, we show a lower bound on the competitive ratio of where where is used to denote the base 2 logarithm. We also give a strategy that obtains this ratio
Parallel Searching on m Rays
The search of the robot can be viewed as an on-line problem since the robot's decisions about the search are based only on the part of its environment that it has seen so far. We use the framework of competitive analysis to measure the performance of an on-line search strategy S [19]. The competitive ratio of S is defined as the maximum of the ratio of the distance traveled by a robot using S to the optimal distance from its starting point to the target, over all possible locations in the environment of the target