R. KLEIN, Walking an unknown street with bounded detour, Comput. Geom., 1 (1992), pp. 325-- 351.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a competitive ratio of 2; we offer a simple randomized variant with competitive ratio 5=4. It is also interesting to consider special cases of the search problem in a polygon restricting the type of polygon so that better competitive ratios can be obtained. This was the approach taken by Klein [K], who considered the class of streets and gave a search algorithm with competitive ratio at most 1 2 ( 5:71) We give an algorithm for streets with competitive ratio at most 8 ( 2:61) improving on this bound by more than a factor of two. A number of other types of polygons may well be ....

....P . Another direction in which one could investigate such search problems is to restrict the class of polygons and target points in such a way that a constant competitive ratio can be achieved for the search problem. Such an approach has been adopted by Klein in his work on streets. The papers [IK, K] introduce the term street to define a class of general (not necessarily rectilinear) polygons with two distinguished points s and t, such that the two st boundary chains are mutually weakly visible (see below for an elaboration on this definition) In [K] Klein gives a 1 2 competitive ( ....

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R. Klein, "Walking an unknown street with bounded detour," Computational Geometry: Theory and Applications, 1(

.... wall O( n) competitive algorithm [21] s on wall of room, O(n2 lg n ) algorithm [21] t in center O(n lg n) algorithm [11] rectilinear polygon O(n) competitive algorithm [66] with m essential cuts streets = polygons with competitive algorithm 2 chains between s and t, ratio 5:71 [65] each point can see some ratio 2:83 [66] point on opposite chain arbitrarily shaped algorithm for path between [79] obstacles, goal s and t bounded by 1.5 coordinates given times the sum of perimeters of all obstacles not farther away from t than s. perfect sensing within O(k log k) ....

....t than s. perfect sensing within O(k log k) algorithm [73] disks around landmarks no sensing outside k landmarks Kleinberg [66] gives a competitive algorithm for exploring an unknown rectilinear polygon. He also addresses the problem of exploring streets which was introduced by Klein [65]. Streets are polygons defined by two chains between s and t where each point on one chain can see some point on the opposite chain. Kleinberg s algorithm has competitive ratio of 2:83 and improves on Klein s 5:71 competitive algorithm. For the case that the coordinates of the goal t are given, ....

Rolf Klein. Walking an unknown street with bounded detour. Computational Geometry: Theory and Applications, 1(6):325--351, June 1992.

....when the robot makes repeated trips between two points. The goal of the robot is to find better paths in each trip. In environments with axis parallel obstacles, they give an algorithm with the property that at the i th trip, the robot s path is O( n=i) times the shortest path length. Klein [51] considers the problem of a polygon with distinguished start and goal vertices. The robot s goal is to walk inside the polygon from the start location to the goal location. The goal location is recognized as soon as the robot sees it. For a special type of polygon known as a street, Klein gives an ....

Rolf Klein. Walking an unknown street with bounded detour. Computational geometry: theory and applications, 1(6):325--351, June 1992.

....Introduction Searching for a target is an important and well studied problem in robotics. In many realistic situations the robot does not possess complete knowledge about its environment, for instance, the robot may not have a map of its surroundings, or the location of the target may be unknown [3 6, 9, 11, 12, 14, 16, 17]. 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 ....

R. Klein. Walking an unknown street with bounded detour. Comput. Geom. Theory Appl., 1:325-351, 1992.

....and Gurvits [7] Kleinberg [16] and Romanik and Schuierer [22] address the problem of localizing a mobile robot in 4 its environment. Blum and Chalasani [9] consider the problem of finding a k trip shortest path in the environment. There are many other related papers in the literature (e.g. [15, 14, 20]) Our techniques are inspired by the work of Awerbuch and Gallager [2, 3] We observe that our learning model bears some similarity to the asynchronous distributed message passing model. This similarity is surprising and has not been explored in the past. Model and Definitions This section ....

Rolf Klein. Walking an unknown street with bounded detour. Computational Geometry: Theory and Applications, 1(6):325--351, June 1992.

.... Sutherland [79] Lumelsky et al. [50] Lumelsky and Skewis [51] discrete vision Rao [61] Choo et al. [15] Foux et al. 26] Class B searching in plane Baeza Yates [3] Kao et al. [38] figure of merit Papadimitriou and Yannakakis [58] Blum et al. [5] Bar Eli et al. [4] Deng et al. [20] Klein [40], Kleinberg [41] Kalyanasundaram and Pruhs [37, 35, 36] Class C restricted computation Budach [10, 11] Coy [19] Dopp [22] Shah [75] Blum and Kozen [7] Table 1: A taxonomy of non heuristic navigation algorithms. A) Touch Sensors: Typically a touch sensor detects when the robot touches an ....

....discuss algorithms to solve navigation problems that optimize figures of merit. Then we consider an algorithm to solve the terrain model acquisition problem due to Deng, Kameda Papadimitriou [20] Finally we consider the problem of crossing a street, which is a special case of a navigation problem [40]. 30 6.1 Searching in Plane Baeza Yates et al. [3] solve several problems dealing with an automaton capable of computation with real numbers in plane. The robot is searching for an object in plane such that for each new probe a cost proportional to the distance of the probe position relative to ....

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R. Klein. Walking an unknown street with bounded detour. In Proceedings of 32nd Annual Symposium on Foundations of Computer Science, pages 304--313, 1991.

....Introduction Searching for a target is an important and well studied problem in robotics. In many realistic situations the robot does not possess complete knowledge about its environment, for instance, the robot may not have a map of its surroundings, or the location of the target may be unknown [DI94, IK95, Kle92, LOS95, PY89]. Since the robot has to make decisions about the search based only on the part of its environment that it has explored before, the search of the robot can be viewed as an on line problem. One way to judge the performance of an on line search strategy is to compare the distance traveled by the ....

R. Klein. Walking an unknown street with bounded detour. Comput. Geom. Theory Appl., 1:325-351, 1992.

....Geb. 051, D 79110 Freiburg, FRG, e mail: schuiere informatik.uni freiburg.de all points in P are visible from some point on the shortest path from s to t. 2 Previous Work The class of street polygons was rst introduced by Klein, and he was also the rst to present a search strategy for streets [9]. His strategy lad is based on the idea of minimizing the local absolute detour. He gives an upper bound on its competitive ratio of 1 3=2 ( 5:71) later improved by Icking to 1 =2 p 1 2 =4 ( 4:44) 6] A number of other strategies have been presented since by Kleinberg [10] ....

....14, 15] Semrau [18] Dasgupta et al. 3] and Kranakis and Spatharis [11] Unfortunately, the analyses of the last two results turned out to be erroneous. The currently best known competitive ratio is 1:51 [7] It is well known that there is no strategy with a competitive ratio less than p 2 [9]. In this paper we present the rst exact analysis of the strategy continuous angular bisector (CAB) which has been considered independently by several authors both in its continuous [3, 12, 15] and discrete form [9, 12] We show that the competitive ratio of CAB is 1:6837. The previously best ....

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R. Klein. Walking an unknown street with bounded detour. Computationl Geometry: Theory & Applications, 1:325-351, 1992.

....Introduction Searching for a target is an important and well studied problem in robotics. In many realistic situations the robot does not possess complete knowledge about its environment, for instance, the robot may not have a map of its surroundings, or the location of the target may be unknown [3 6, 9, 11, 12, 14, 16, 17]. 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 ....

R. Klein. Walking an unknown street with bounded detour. Comput. Geom. Theory Appl., 1:325-351, 1992.

....Introduction Searching for a target is an important and well studied problem in robotics. In many realistic situations the robot does not possess complete knowledge about its environment, for instance, the robot may not have a map of its surroundings, or the location of the target may be unknown [DI94, IK95, Kle92, LOS95, PY89]. Since the robot has to make decisions about the search based only on the part of its environment that it has explored before, the search of the robot can be viewed as an on line problem. One way to judge the performance of an on line search strategy is to compare the distance traveled by the ....

R. Klein. Walking an unknown street with bounded detour. Comput. Geom. Theory Appl., 1:325--351, 1992.

....In a street P the starting point s and the target t are located on the boundary of P and all points in P are visible from some point on the shortest path from s to t. The class of street polygons was first introduced by Klein, and he was also the first to present a search strategy for streets [15]. His strategy lad is based on the idea of minimizing the local absolute detour. He gives an upper bound on its competitive ratio of 1 3=2( 5:71) The upper bound on the competitive factor was later improved by Icking to 1 =2 p 1 2 =4 ( 4:44) 11] A number of other strategies have been ....

....al. 6] and Kranakis and Spatharis [17] Unfortunately, the analyses of the last two results turned out to be erroneous. The currently best known competitive ratio is 1:51 [12] Due to the simple lower bound example shown in Figure 1 there is no strategy with a competitive ratio less than p 2 [15]. If a strategy moves to the left or right before seeing t, then t can be placed on the opposite side, thus forcing the robot to travel more than p 2 times the diagonal. Interestingly, this example also provides a lower bound for randomized strategies. To see this consider the expected distance ....

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R. Klein. Walking an unknown street with bounded detour. Computationl Geometry: Theory & Applications, 1:325--351, 1992.

....Consider the motion planning problem for a robot walking in an unknown terrain. The robot can see only some part of its environment and has to base its decision where to go next only on what he has seen so far. However, the robot may have influence on future inputs by its current decisions. See [K91] for an example of an algorithm of this kind. Sleator and Tarjan [ST85] have introduced the concept of competitive analysis for online algorithms. See also [MMS88] They compare an online algorithm with the optimal offline algorithm solving the same problem rather than comparing any two online ....

R. Klein. Walking an unknown street with bounded detour. 32nd FOCS, 304-313,1991

....Introduction Searching for a target is an important and well studied problem in robotics. In many realistic situations the robot does not possess complete knowledge about its environment, for instance, the robot may not have a map of its surroundings, or the location of the target may be unknown [DI94, IK95, Kle92, LOS95, PY89]. Since the robot has to make decisions about the search based only on the part of its environment that it has explored before, the search of the robot can be viewed as an on line problem. One way to judge the performance of an on line search strategy is to compare the distance traveled by the ....

R. Klein. Walking an unknown street with bounded detour. Comput. Geom. Theory Appl., 1:325--351, 1992.

....path in the polygon. The competitive ratio is the worst case ratio achieved over all possible problem instances. A strategy is called competitive if its competitive ratio is constant. In recent years, the competitive searching in unknown polygons has been intensively studied by many researchers [1, 3, 6, 2, 9, 5, 4] in computational geometry. Since it is impossible to competitively find a target in general polygons [6, 9] most of the work has focused on restricting the classes of polygons for which constant competitive ratios can be achieved. Klein [6] introduced a class of polygons called streets. He gave ....

....is called competitive if its competitive ratio is constant. In recent years, the competitive searching in unknown polygons has been intensively studied by many researchers [1, 3, 6, 2, 9, 5, 4] in computational geometry. Since it is impossible to competitively find a target in general polygons [6, 9], most of the work has focused on restricting the classes of polygons for which constant competitive ratios can be achieved. Klein [6] introduced a class of polygons called streets. He gave a 1 3 2 ( 5:72) competitive strategy for finding the target in a street and also proves a lower bound ....

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R. Klein. Walking an unknown street with bounded detour. In Proc. 32nd Annu. IEEE Sympos. Found. Comput. Sci., pages 304--313, 1991.

....# n) Co#man and Gilbert [12] study the performance of simple heuris NAVIGATING IN UNFAMILIAR GEOMETRIC TERRAIN 113 tics in the presence of randomly placed obstacles. Kalyanasundaram and Pruhs [16] and Mei and Igarashi [22] consider scenes in which all obstacles have bounded aspect ratios. Klein [18] has given a small constant upper bound on the ratio for scenes that are streets, a class of simple polygons. Earlier, Lumelsky and Stepanov [21] gave a simple navigation algorithm that guarantees R(S) to be bounded by d(S) plus the sum of the perimeters of all obstacles with no restrictions on ....

R. Klein, Walking an unknown street with bounded detour, in Proc. 32nd Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1991, pp. 303--313.

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R. Klein. Walking an unknown street with bounded detour. Comput. Geom. Theory Appl., 1:325--351, 1992.

.... by Papadimitriou and Yanakakis [10] Blum, Raghhavan, and Schieber [2] and Eades, Lin, and Wormald [5] for path planning in the presence of obstacles in the plane, by Deng, Kameda, and Papadimitriou [4] for learning the interior of a polygon that may have a bounded number of holes, and by Klein [7] for finding a path in the interior of special simple polygons called streets. ....

R. Klein. Walking an unknown street with bounded detour. Computational Geometry: Theory and Applications 1, 1992, pp. 325--351.

....the targetpoint on sight. In arbitrary simple polygons no strategy can guarantee a search path a constant times as long as the shortest path from start to goal at the most. Therefore the following sub class of polygons for which a constant performance ratio can be achieved was introduced by Klein [26,27]. A polygon P with two distinguished vertices s and t is called a street if the two boundary chains leading from s to t are mutually weakly visible, i.e. if each point on one of the chains can see at least one point of the other; see Fig. 3 for an example. The first competitive strategy [26,27] ....

....[26,27] A polygon P with two distinguished vertices s and t is called a street if the two boundary chains leading from s to t are mutually weakly visible, i.e. if each point on one of the chains can see at least one point of the other; see Fig. 3 for an example. The first competitive strategy [26,27] for searching a street had a competitive ratio not bigger than 5.72, i.e. the first upper bound for the ratio was given. Besides, it was shown that no strategy can achieve a competitive ratio of less than # 2 1.41, the ultimative lower bound to the ratio as turned out later. Since then, ....

R. Klein. Walking an unknown street with bounded detour. Comput. Geom. Theory Appl., 1:325--351, 1992.

....as an approximation in this case. How much do we know about the competitive complexities of basic navigation tasks Apart from the wall problem mentioned above, there are only very few tasks whose complexities are precisely known. One of them is the street problem originally presented by Klein [26], whose complexity has finally been determined in Icking et al. 22] and independently by Semrau and Schuierer [40] This problem is briefly reviewed in Sect. 2 of this paper. In general it is quite di#cult to precisely determine the competitive complexity of a task. Then we can at least try to ....

....the targetpoint on sight. In arbitrary simple polygons no strategy can guarantee a search path a constant times as long as the shortest path from start to goal at the most. Therefore the following sub class of polygons for which a constant performance ratio can be achieved was introduced by Klein [26,27]. A polygon P with two distinguished vertices s and t is called a street if the two boundary chains leading from s to t are mutually weakly visible, i.e. if each point on one of the chains can see at least one point of the other; see Fig. 3 for an example. The first competitive strategy [26,27] ....

[Article contains additional citation context not shown here]

R. Klein. Walking an unknown street with bounded detour. In Proc. 32nd Annu. IEEE Sympos. Found. Comput. Sci., pages 304--313, 1991.

....leading from s to t are mutually weakly visible, i.e. if each point on one of the chains can see at least one point of the other; see Figure 1 for an example. Equivalently, from each s to t path inside P each point of the polygon is at least once visible. s t SP L R Figure 1: A street. In [13, 14] Klein provided the first competitive strategy for searching for the target point, t, of a street, starting from s. He proved an upper bound of 5.72 for the ratio of the length of the robot s path over the length of the shortest path from s to t in P . Also, it was shown that no strategy can ....

....via Internet, we learned that Schuierer and Semrau [22] have simultaneaously and independently studied the same strategy. However, their analytic approach is quite di#erent from our proof. 2 Definitions and known properties We briefly repeat necessary definitions and known facts, mostly from [14]. A simple polygon P is considered as a room, the edges are opaque walls. Two points are mutually visible, i.e. see each other, if the connecting line segment is contained within P . As usual, two sets of points are said to be mutually weakly visible if each point of one set can see at least one ....

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R. Klein. Walking an unknown street with bounded detour. Comput. Geom. Theory Appl., 1:325--351, 1992.

....i vers i tat Hagen, Praktische Informatik VI, D 58084 Hagen, Germany. This research was supported by the Deutsche Forschungsgemeinschaft, grant no. Kl 655 8 3. 1 The question arose if there are subclasses of polygons for which a constant performance ratio can be achieved. At FOCS 91, Klein [13] introduced the concept of streets. A polygon P with two distinguished vertices s and t is called a street if the two boundary chains leading from s to t are mutually weakly visible, i.e. if each point on one of the chains can see at least one point of the other; see Figure 1 for an example. ....

....leading from s to t are mutually weakly visible, i.e. if each point on one of the chains can see at least one point of the other; see Figure 1 for an example. Equivalently, from each s to t path inside P each point of the polygon is at least once visible. s t SP L R Figure 1: A street. In [13, 14] Klein provided the first competitive strategy for searching for the target point, t, of a street, starting from s. He proved an upper bound of 5.72 for the ratio of the length of the robot s path over the length of the shortest path from s to t in P . Also, it was shown that no strategy can ....

R. Klein. Walking an unknown street with bounded detour. In Proc. 32nd Annu. IEEE Sympos. Found. Comput. Sci., pages 304--313, 1991.

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R. KLEIN, Walking an unknown street with bounded detour, Comput. Geom., 1 (1992), pp. 325-- 351.

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R. Klein, Walking an unknown street with bounded detour, Computational Geometry Theory and Applications, 1 (1992), pp. 325--351.

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R. Klein. Walking an unknown street with bounded detour. Computational Geometry Theory and Applications, 1:325-351, 1992.

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Rolf Klein. Walking an Unknown Street with Bounded Detour. Computational Geometry: Theory and Applications, 1:325--351, 1992.

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