A. Lopez-Ortiz and S. Schuierer. Generalized streets revisited. In Proc. 4th Annu. European Sympos. Algorithms, volume 1136 of Lecture Notes Comput. Sci., pages 546-- 558. Springer-Verlag, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....it is a street; for star shaped polygons many of such vertex pairs exist. Das et al. 6] have improved on this result by giving an optimal linear time algorithm. Datta and Icking [9] introduced generalized streets, a concept further generalized by Datta et al. 8] and by Lopez Ortiz and Schuierer [18]. Ghosh and Saluja [10] have described how to walk an unknown street incurring a minimum number of turns. Other research addressed the gap between the # 2 lower bound and the first upper bound of 5.72 known for the class of street polygons. The upper bound was lowered to 4.44 in Icking [11] then ....

A. Lopez-Ortiz and S. Schuierer. Generalized streets revisited. In Proc. 4th Annu. European Sympos. Algorithms, volume 1136 of Lecture Notes Comput. Sci., pages 546-- 558. Springer-Verlag, 1996.

....arbitrary obstacles having a total of n vertices [2] even if we restrict ourselves to searching in a simple polygon. Therefore, the on line search problem has been studied previously for the case where the geometry of the terrain is restricted to searching in special classes of simple polygons [4, 5, 8, 16, 17]. In this paper we study the continuous angular bisector (CAB) strategy to search in street polygons. In a street P the starting point s and the target t are located on the boundary of P and This research is supported by the DFG Project Diskrete Probleme , No. Ot 64 8 2. y Faculty of ....

A. Lopez-Ortiz and S. Schuierer. Generalized streets revisited. In M. Serna J. Diaz, editor, Proc. 4th ESA, pages 546-558. LNCS 1136, 1996.

....namely the generalized streets,orG streets for short. We give a strategy for searching in the rectilinear case which is 9 competitive in comparison to the L 1 shortest path and 9.06 competitive for L 2 . Both bounds are optimal, the lower bound of 9. 06 has been found by Lopez Ortiz and Schuierer [26], who also describe an 80 competitive strategy for searching in the non rectilinear case. Related results exist about competitive searching in star shaped polygons [23,24,28] competitiveness with respect to the link distance [9] and further generalized polygons [5] The rest of the paper is ....

....to w shows that the maximum is reached at w =9atwhich # 82 9.06 is the maximum value. The claim of the theorem follows by extending the analysis to every pair of such chords and from the fact that the L 2 shortest path must also visit the points p 1 and p 2 . Lopez Ortiz and Schuierer [24,26,29] remark that # 82 is also a lower bound, so it is in fact the optimal competitive factor. To see this, one can adapt the proof of the lower bound in Theorem 15 to the L 2 case by introducing caves also to the lower part of the polygon in Figure 7 and by arranging the caves in bow tie form with ....

A. Lopez-Ortiz and S. Schuierer. Generalized streets revisited. In Proc. 4th Annu. European Sympos. Algorithms, volume 1136 of Lecture Notes Comput. Sci., pages 546--558. Springer-Verlag, 1996.

....of simple polygons, so called Generalized streets (G streets, for short) and presented an on line strategy which achieves an optimal 9 competitive ratio (resp. p 82 competitive) in Manhattan metric L 1 ( resp. L 2 ) metric to search in an unknown rectilinear G street. Moreover, L opez Ortiz [10] has proposed a strategy with competitive ratio of 80 in L 1 metric to search in arbitrary oriented G streets. In addition, an even larger class of rectilinear simple polygons is given by the class of HV streets with the property that every boundary point is mutually weakly visible from a point on ....

....and presented an on line strategy that achieves a competitive lower bound of 2 p 5 2 p 2 ( 1:498) in the L 2 metric to search in unknown arbitrary streets. The interesting open question remains if there exist more general natural classes of simple polygons larger than HV streets [4, 10] which can be searched competitively. Furthermore, we want to find a strategy that achieves a better competitive ratio of 80 in L 1 metric for searching arbitrary G streets. A common research problem is to improve the competitive factor of 133 in L 2 metric for exploring an unknown simple ....

A. L'opez-Ortiz and S. Schuierer. Generalized Streets Revisited. In European Symposium of Algorithms. Springer Verlag, Lecture Notes in Computer Science, Vol. 1136, pp. 546-558, 1996.

.... G streets ) for which they prove bounds in the L 1 metric: 14.5 is an upper and lower bound on the competitive ratio for HV streets, while 19.97 is an upper bound for G streets. See also Schuierer [355] for lower bounds on the competitive ratio in G streets. L opez Ortiz and Schuierer [262] give a competitive strategy (with ratio 80) in arbitrarily oriented (nonrectilinear) G streets. Kleinberg [242] considers searching in general rectilinear simple polygons, obtaining a strategy with competitive ratio O(m) where m is the number of essential cuts, which may be much smaller than ....

A. L'opez-Ortiz and S. Schuierer. Generalized streets revisited. In J. Diaz and M. Serna, editors, Proc. 4th Annu. European Sympos. Algorithms, volume 1136 of Lecture Notes Comput. Sci., pages 546--558. Springer-Verlag, 1996.

....arbitrary obstacles having a total of n vertices [2] even if we restrict ourselves to searching in a simple polygon. Therefore, the on line search problem has been studied previously for the case where the geometry of the terrain is restricted to searching in special classes of simple polygons [4, 5, 8, 16, 17]. In this paper we study the continuous angular bisector (CAB) strategy to search in street polygons. 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. This research is ....

A. Lopez-Ortiz and S. Schuierer. Generalized streets revisited. In M. Serna J. Diaz, editor, Proc. 4th ESA, pages 546--558. LNCS 1136, 1996.

.... optimal competitive ratio is given by the minimum of the function 1 2a m = a 1) ln a) for a 1 [Gal80, KRT97, KMSY94] Searching on m rays has proven to be a very useful tool for searching in a number of classes of simple polygons, such as star shaped polygons [LOS97] generalized streets [DI94, LOS96], HV streets [DHS95] and streets [DHS95, Hip94] However, the proof of optimality for the above m way ray searching strategy relies on the unboundedness of the rays, that is, on the fact that the target can be placed arbitrarily far away from the starting point of the rays [BYCR93, Gal80] But, ....

....we consider polygons and the robot is equipped with a range nder, then it is possible to obtain an upper bound D on the distance to the target. In this case it is implicitly assumed that the strategy for searching on m rays remains optimal though no proof of this assumption has been presented yet [DHS95, DI94, LOS96]. In this paper we provide the rst lower bound proof for searching on m bounded rays; more precisely, we investigate the question if the knowledge of an upper bound on the distance to the target provides an advantage to the robot. Let C D m be the optimal competitive ratio to search on m rays ....

A. Lopez-Ortiz and S. Schuierer. Generalized streets revisited. In M. Serna J. Diaz, editor, Proc. 4th European Symposium on Algorithms, pages 546-558. LNCS 1136, 1996.

....research is supported by the DFG Project Diskrete Probleme , No. Ot 64 8 1. 2 The lower bound for searching in m rays has proved to be a very useful tool for proving lower bounds for searching in a number of classes of simple polygons, such as star shaped polygons [13] generalized streets [6, 15], HV streets [5] and streets [5, 8] In this paper we are interested in obtaining upper and lower bounds for the competitive ratio of parallel searching on m concurrent rays. This problem has been adressed before in two contexts. The rst context is the on line construction of hybrid algorithms ....

A. Lopez-Ortiz and S. Schuierer. Generalized streets revisited. In M. Serna J. Diaz, editor, Proc. 4th European Symposium on Algorithms, pages 546-558. LNCS 1136, 1996.

.... competitive ratio is given by the minimum of the function 1 2a m = a Gamma 1) ln a) for a 1 [Gal80, KRT93, KMS94] Searching on m rays has proven to be a very useful tool for searching in a number of classes of simple polygons, such as star shaped polygons [LOS97] generalized streets [DI94, LOS96], HV streets [DHS95] and streets [DHS95, Hip94] However, the proof of optimality for the above m way ray searching strategy relies on the unboundedness of the rays, that is, on the fact that the target can be placed arbitrarily far away from the starting point of the rays [BYCR93, Gal80] But, ....

....consider polygons and the robot is equipped with a range finder, then it is possible to obtain an upper bound D on the distance to the target. In this case it is implicitly assumed that the strategy for searching on m rays remains optimal though no proof of this assumption has been presented yet [DHS95, DI94, LOS96]. In this paper we provide the first lower bound proof for searching on m bounded rays; more precisely, we investigate the question if the knowledge of an upper bound on the distance to the target provides an advantage to the robot. Let C D m be the optimal competitive ratio to search on m rays ....

A. L'opez-Ortiz and S. Schuierer. Generalized streets revisited. In M. Serna J. Diaz, editor, Proc. 4th European Symposium on Algorithms, pages 546--558. LNCS 1136, 1996.

....obstacles having a total of n vertices [4] even if we restrict ourselves to searching in a simple polygon. Therefore, the on line search problem has been studied previously in various contexts where the geometry of the obstacles is restricted such as searching in special classes of simple polygons [7, 8, 13, 22, 23], among rectangles [2, 3, 4, 5, 24, 25] convex polygons [14] and on the real line [1, 9, 10] This research is supported by the DFG Project Diskrete Probleme , No. Ot 64 8 2. y Faculty of Computer Science, University of New Brunswick, NB, E3B 5A3, Canada, e mail: alopez o unb.ca z ....

....the direction it is facing always bisects its visibility angle. 2 It is somewhat surprising that CAB can be analysed exactly as CAB consists of hyperbolic arcs whose length cannot always be expressed in a closed form. 1 The same observation also holds for biased strategies as introduced in [22]. 2 The visibility angle is defined below. 2 s t Figure 2: A street polygon. The importance of CAB is threefold: ffl it compares favourably to most other strategies proposed [15, 16, 19, 20, 21] ffl it is a C 1 continuous strategy in large parts of the polygon, as opposed to all others ....

A. L'opez-Ortiz and S. Schuierer. Generalized streets revisited. In M. Serna, J. Diaz, editor, Proc. 4th European Symposium on Algorithms, pages 546--558. LNCS 1136, Springer, 1996. 13

....Institut fur Informatik, Am Flughafen 17, Geb. 051, D 79110 Freiburg, Germany email: schuiere informatik.uni freiburg.de 1 Searching on m rays has proven to be a very useful tool for searching in a number of classes of simple polygons, such as star shaped polygons [LOS97] generalized streets [DI94, LOS96], HV streets [DHS95] and streets [DHS95, Hip94] Hipke et al. consider the maximal reach of a strategy to search on the line if the competitive ratio of the strategy is given [HIKL97] The reach of a strategy X with competitive ratio C is the maximum distance D such that a target placed at a ....

A. L'opez-Ortiz and S. Schuierer. Generalized streets revisited. In M. Serna J. Diaz, editor, Proc. 4th European Symposium on Algorithms, pages 546--558. LNCS 1136, 1996.

....# 2 # 1.414. Finally, in Section 4 we show that # 2remainsa lower bound for searching in orthogonal streets even if the location of the goal is knowninadvance. Competitive on line searching has also been investigated in many other settings such as searching in other classes of simple polygons [6, 7, 11, 18, 20], among rectangles [2, 3, 4, 5, 21, 22] convex polygons [12] and on the real line [1, 8, 9] 2 Searching for a goal in a street In our model the room is a simple polygon P in the plane, the robot is just a point moving inside the polygon, and the start position s and the goal t are two of P s ....

A. Lopez-Ortiz and S. Schuierer. Generalized streets revisited. In J. Diaz and M. Serna, editors, Proc. 4th Annu. European Sympos. Algorithms,volume 1136 of Lecture Notes Comput. Sci., pages 546--558. Springer-Verlag, 1996.

....also present a different strategy which is a modification of Dijkstra s shortest path algorithm with a competitive ratio of 2n Gamma 7. Our algorithms work in any simple polygon. This should be contrasted with previous search algorithms that require restricted classes of polygons as their input [DHS95, DI94, Kle92, Kle94, LOS96, LOS97]. The paper is organized as follows. In the next section we introduce some definitions. In Section 3 we present the algorithm to search in a geometric tree and analyse its competitive ratio. Section 4 shows how the algorithm can be applied to searching in a simple polygon. It also introduces a ....

A. L'opez-Ortiz and S. Schuierer. Generalized streets revisited. In J. Diaz and M. Serna, editors, Proc. 4th European Symposium on Algorithms, pages 546--558. LNCS 1136, 1996.

....contains the This research is supported by the DFG Project Diskrete Probleme , No. Ot 64 8 2. goal g whose distance to the origin is unknown. The robot can only detect g if it stands on top of it. Many problems of searching in more complex geometries can be reduced to searching on m rays [DI94, LOS97, LOS96, DHS95]. In the deterministic case Baeza Yates et al. BYCR93] and Gal [Gal80] present an optimal search strategy that achieves a competitive ratio of 1 2m m = m Gamma 1) m Gamma1 . In the randomized case Kao, Reif, and Tate [KRT93] present an optimal randomized algorithm to search on m rays and ....

A. L'opez-Ortiz and S. Schuierer. Generalized streets revisited. In M. Serna J. Diaz, editor, Proc. 4th European Symposium on Algorithms, pages 546--558. LNCS 1136, 1996.

.... Cm achieved by this strategy is given by 1 2 m m (m Gamma 1) m Gamma1 : The lower bound for searching in m rays has proved to be a very useful tool for proving lower bounds for searching in a number of classes of simple polygons, such as star shaped polygons [LO96] generalized streets [DI94, LOS96], HV streets [DHS95] and streets [DHS95, Hip94] In this paper we are interested in obtaining upper and lower bounds for the competitive ratio of parallel searching on m concurrent rays. This problem has been adressed before in two contexts. The first context is the on line construction of ....

A. L'opez-Ortiz and S. Schuierer. Generalized streets revisited. In M. Serna J. Diaz, editor, Proc. 4th European Symposium on Algorithms, pages 546--558. LNCS 1136, 1996.

....ratio of the distance traveled by the robot to the optimal distance from s to t is called the competitive ratio of the search strategy. There are several known classes of polygons that admit search strategies for some targets with a constant competitive ratio, most notably streets [7] G streets [4, 10], HVstreets [3] and streets [3] However, the existence of a constant competitive searching strategy for these classes of polygons is strongly dependent on the position of the target. A natural question is to find a class of polygons which the robot may search at a constant competitive ratio ....

A. L'opez-Ortiz and S. Schuierer. "Generalized streets revisited", In J. Diaz and M. Serna, editors, Proc. 4th European Symposium on Algorithms, LNCS 1136, pages 546--558. Springer Verlag, 1996.

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