text: We study a new search problem on the plane involving a robot and an immobile treasure, initially placed at distance @math from each other. The length @math of an arc (a fence) within the perimeter of the corresponding circle, as well as the promise that the treasure is outside the fence, is given as part of the input. The goal is to device movement trajectories so that the robot locates the treasure in minimum time. Notably, although the presence of the fence limits searching uncertainty, the location of the fence is unknown, and in the worst case analysis is determined adversarially. Nevertheless, the robot has the ability to move in the interior of the circle. In particular the robot can attempt a number of chord-jump moves if it happens to be within the fence or if an endpoint of the fence is discovered. The optimal solution to our question can be obtained as a solution to a complicated optimization problem, which involves trigonometric functions, and trigonometric equations that do not admit closed form solutions. For the 1-Jump Algorithm, we fully describe the optimal trajectory, and provide an analysis of the associated cost as a function of @math . Our analysis indicates that the optimal k-Jump Algorithm requires that the robot has enough memory and computation power to compute the optimal chord-jumps. Motivated by this, we give an abstract performance analysis for every k-Jump Algorithm. Subsequently, we present a highly efficient Halving Heuristic k-Jump Algorithm that can effectively approximate the optimal k-Jump Algorithm, with very limited memory and computation requirements.
  @cite_4: Abstract In this paper we investigate parallel searching in the plane using robots as searchers. We show that a huge number of robots are not necessary for several problems and under different assumptions. This corresponds to real situations since, actually, the number of processors of parallel machines is fixed and independent of the dimension of the problem to be solved.
  @cite_7: Optimal search for targets with conditionally deterministic motion is investigated. The target motion takes place in @math , a copy of Euclidean n-space, and depends on a stochastic parameter @math which takes values in @math , another copy of Euclidean n-space. The target motion is deterministic given knowledge of @math . That is, there is a function @math , where T is a time interval, such that @math gives the target motion conditioned on @math . Search plans are specified by functions @math . A functional P is defined so that @math gives the probability of detecting the target by time t using plan @math .Let @math be the absolute value of the Jacobian of @math evaluated at x. If there exist functions @math and @math such that @math for @math , the target motion is called factorable Let $ _2 :T ...
  @cite_9: 
  @cite_6: 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.
  @cite_0: Call a path an escape path if it eventually leads out of the forest no matter what the initial starting point or the relative orientations of the path and forest. To solve the "lost in a forest" problem we must find the "best" escape path. Bellman proposed two different interpretations of "best," one in which the maximum time to escape is minimized, and one in which the expected time to escape is minimized. A third interpretation (see
  @cite_5: Consider k mobile robots inside a circular disk of unit radius. The robots are required to evacuate the disk through an unknown exit point situated on its boundary. We assume all robots having the same (unit) maximal speed and starting at the centre of the disk. The robots may communicate in order to inform themselves about the presence (and its position) or the absence of an exit. The goal is for all the robots to evacuate through the exit in minimum time.
  @cite_16: 
  @cite_12: In this paper we initiate a new area of study dealing with the best way to search a possibly unbounded region for an object. The model for our search algorithms is that we must pay costs proportional to the distance of the next probe position relative to our current position. This model is meant to give a realistic cost measure for a robot moving in the plane. We also examine the effect of decreasing the amount of a priori information given to search problems. Problems of this type are very simple analogues of non-trivial problems on searching an unbounded region, processing digitized images, and robot navigation. We show that for some simple search problems, knowing the general direction of the goal is much more informative than knowing the distance to the goal.
  @cite_11: 
summary: The type of search problem investigated in our work was first seen sixty years ago when Beck @cite_6 and Bellman @cite_16 asked an important, yet simplistic question tied to the minimization of distance. Motivated from this, several different natural search problems have been studied including the use of a fixed @cite_5 @cite_9 or mobile target @cite_7 , the tools searchers have access to, the number of searchers, the communication restrictions and many more. Often, the essential part of the robot activity is the recognition and or mapping of the terrain. In the case of a known structure, the main objective of the search is to minimize the time to find the treasure. Searching for a motionless target has been studied in the cow-path problem @cite_6 , lost in a forest problem @cite_0 @cite_11 and plane searching problem @cite_4 @cite_12 .