P. K. Agarwal, B. Aronov, J. O'Rourke, and C. A. Schevon. Star unfolding of a polytope with applications. SIAM J. Comput., 26:1689--1713, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....planar spanners, with tradeo s among the approximation factor and the preprocessing time, storage space, and query time. For the problem of shortest paths on a polyhedral surface, some results are also known on the two point query problem, at least for the case of convex polytopes. Agarwal et al. [1] have also shown that two point queries can be answered in time O( p n=m 1=4 ) log n) with O(n 6 m 1 ) preprocessing time and storage, for any choice of 1 m n 2 , and 0. Har Peled [12] obtains results for the approximate two point query problem: He gives an O(n) time algorithm ....

....Section 4, we will also de ne the SPMequivalence decomposition. 3 Method I: Mapping to Higher Dimensions In this section, we show how the two point query problem can be solved in optimal time (O(log n) by mapping it into a four dimensional point location query. A related approach was used in [1] for the case of shortest path queries on a convex surface. First, we note that a shortest path (s; t) from s to t either consists of the single segment st (if s sees t) or consists of a polygonal chain (s; v i ; v j ; t) whose bend points occur at obstacle vertices, where v i is the ....

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P. K. Agarwal, B. Aronov, J. O'Rourke, and C. A. Schevon. Star unfolding of a polytope with applications. SIAM J. Comput., 26:1689-1713, 1997.

....is Touch 3D [11] which supports nonconvex polyhedra by using multiple pieces when needed. It is known that if we allow cuts across the faces as well as along the edges, then every convex polyhedron has an unfolding. Two such unfoldings are known. The simplest to describe is the star unfolding [1, 2], which cuts from a generic point on the polyhedron along shortest paths to each of the vertices. The second is the source unfolding [12, 16] which cuts along points with more than one shortest path to a generic source point. There has been little theoretical work on unfolding nonconvex ....

Pankaj K. Agarwal, Boris Aronov, Joseph O'Rourke, and Catherine A. Schevon. Star unfolding of a polytope with applications. SIAM Journal on Computing, 26(6):1689-- 1713, December 1997.

....P (s; t) of an approximate shortest path between s and t on P , in O( log n) 1:5 1= 3 ) time. In contrast, the fastest known algorithm for computing a data structure that supports queries of computing the length of the exact shortest path between any pair of points on P is due to [AAOS96]; it requires O(n 6 m 1 ffi ) space and preprocessing time, for any ffi 0, and answers a 2 query in O( p n=m 1=4 ) log m) time, where 1 m n 2 (the constants of proportionality depends on ffi) Approximate geodesic diameter. We present in Section 4 an algorithm that computes, for a ....

....0 1, two points s; t 2 P such that d P (s; t) 1 Gamma )D P , where D P = max s;t2 P d P (s; t) is the geodesic diameter of P . The running time of the algorithm is O(n 1= 6 ) In contrast, the fastest known algorithm for the exact geodesic diameter takes O(n 8 log n) time (see [AAOS96]) The paper is organized as follows. Section 2 introduces the required terminology and establishes some initial properties, mostly taken or adapted from [AHPSV96] In Section 3 we describe the algorithm for answering two point approximate shortest path queries. In Section 4 we derive an ....

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P.K. Agarwal, B. Aronov, J. O'Rourke, and C. Schevon. Star unfolding of a polytope with applications. SIAM J. Comput., 1996, to appear.

....This metric has a nice property: shortest paths can be computed in polynomial time. If paths were allowed to stray from the polyhedral surface, the problem of discovering a shortest path becomes NP hard [2] Thus, over the years there has been considerable interest in this distance metric (e.g. [1,3,5,7,8,9,10]) One approach to generating a geometric MST is to first compute some (small) superset of MST edges, such that edges of this superset which define an MST can be selected quickly. For example, given m sites in the plane and using an L 2 distance metric, an algorithm might first compute a Delaunay ....

Pankaj, K., Aronov, B., O'Rourke, J., and Schevon, C. A. Star unfolding of a polytope with applications. SIAM J. Comput. 26, 1997, 1689-1713.

....16] currently, the best known algorithm due to Chen and Han [3] runs in O(n 2 ) time. 1 The best known algorithm for the query problem, where both source and destination are unknown, requires O(n 6 m 1 ffi ) for ffi 0 and 1 m n 2 , to answer queries in O( n=m 1=4 ) log n) time [2]. Mitchell and Papadimitriou [9] introduced the Weighted Region Problem and an algorithm that computes a shortest weighted cost path between two points in a planar subdivision; it requires O(n 8 log n) time in the worst case. They state that their algorithm applies to non convex polyhedral ....

P.K. Agarwal, B. Aronov, J. O'Rourke, and C. Schevon, "Star Unfolding of a Polytope with Applications", to appear in SIAM Journal on Computing, 1997.

....planar spanners, with tradeoffs among the approximation factor and the preprocessing time, storage space, and query time. For the problem of shortest paths on a polyhedral surface, some results are also known on the two point query problem, at least for the case of convex polytopes. Agarwal et al. [1] have also shown that two point queries can be answered in time O( p n=m 1=4 ) log n) with O(n 6 m 1 ffi ) preprocessing time and storage, for any choice of 1 m n 2 , and ffi 0. Har Peled [12] obtains results for the approximate two point query problem: He gives an O(n) time ....

....Section 4, we will also define the SPMequivalence decomposition. 3 Method I: Mapping to Higher Dimensions In this section, we show how the two point query problem can be solved in optimal time (O(log n) by mapping it into a four dimensional point location query. A related approach was used in [1] for the case of shortest path queries on a convex surface. First, we note that a shortest path (s; t) from s to t either consists of the single segment st (if s sees t) or consists of a polygonal chain (s; v i ; v j ; t) whose bend points occur at obstacle vertices, where v i is the ....

[Article contains additional citation context not shown here]

P. K. Agarwal, B. Aronov, J. O'Rourke, and C. A. Schevon. Star unfolding of a polytope with applications. SIAM J. Comput., 26:1689--1713, 1997.

....19] currently, the best known algorithm due to Chen and Han [3] runs in O(n 2 ) time. The best known algorithm for the query problem, where both source and destination are unknown, requires O(n 6 m 1 ffi ) for ffi 0 and 1 m n 2 , to answer queries in O( n=m 1=4 ) log n) time [2]. Mitchell and Papadimitriou [12] introduced the weighted region problem and presented an algorithm that computes a shortest weighted cost path between two points in a planar subdivision; it requires O(n 8 log n) time in the worst case. They state that their algorithm applies to non convex ....

P.K. Agarwal, B. Aronov, J. O'Rourke, and C. Schevon, "Star Unfolding of a Polytope with Applications", to appear in SIAM Journal on Computing, 1997.

....surface P. The surface is composed of triangular regions (faces) in which each region has an associated positive weight. The cost of travel through each region is the distance traveled times its weight. The computation of Euclidean shortest paths on nonconvex polyhedra has been investigated by [5, 2, 1]; currently, the best known algorithm due to Chen and Han [2] runs in O(n 2 ) time. Mitchell and Papadimitriou [6] introduced the Weighted Region Problem and an algorithm that computes a shortest weighted cost path between two points in a planar subdivision; it requires O(n 8 log n) time in ....

P.K. Agarwal, B. Aronov, J. O'Rourke, and C. Schevon, "Star Unfolding of a Polytope with Applications ", to appear in SIAM Journal on Computing, 1997.

....set of edge sequences corresponding to shortest paths on the surface of a convex polytope P in 3 . In particular, Mount [300] has shown that the worst case number of distinct edge sequences that correspond to a shortest path between some pair of points is Theta(n 4 ) Further, Agarwal et al. [3] have shown that the exact set of such sequences can be computed in time O(n 6 fi(n) log n) where fi(n) o(log n) A simpler O(n 6 ) algorithm can compute a small superset of the sequences [3] The number of maximal edge sequences for shortest paths is Theta(n 3 ) as shown by ....

.... to a shortest path between some pair of points is Theta(n 4 ) Further, Agarwal et al. [3] have shown that the exact set of such sequences can be computed in time O(n 6 fi(n) log n) where fi(n) o(log n) A simpler O(n 6 ) algorithm can compute a small superset of the sequences [3]. The number of maximal edge sequences for shortest paths is Theta(n 3 ) as shown by Schevon and O Rourke [352] Some of these results depend on a careful study of the star unfolding with respect to a point p on the boundary, P , of P . The star unfolding is the (nonoverlapping [34] cell ....

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P. K. Agarwal, B. Aronov, J. O'Rourke, and C. A. Schevon. Star unfolding of a polytope with applications. SIAM J. Comput., 26:1689--1713, 1997.

....to compute an optimal path in polynomial time. As a result, this special case of the shortest path problem has received considerable attention in computational geometry. Within mathematics as well, the unfolding of convex polyhedra is a rich topic with a distinguished history see for instance [1]. Our main result is a very simple and practical algorithm for computing an approximation to the shortest path. Formally, let P denote a convex polytope with n vertices in R 3 , and let d P (p; q) denote the length of a shortest path on the surface of P between the points p and q. The ....

P. K. Agarwal, B. Aronov, J. O'Rourke, and C. Schevon. Star unfolding of a polytope with applications. Technical Report 031, Dept. Comput. Sci., Smith College, Northampton, MA, July 1993.

....shortest path between s and t on P , 1= time. The algorithm is presented in Section 3.2.1. In contrast, the fastest known algorithm for computing a data structure that supports queries of computing the length of the exact shortest path between any pair of points on P is due to [AAOS97] it requires O(n ) space and preprocessing time, for any ffi 0, and answers a query in O( log m) time, where 1 m n (the constants of proportionality depend on ffi) The results of Section 3.2.1 appeared in [HP99a] Approximate geodesic diameter. We present in Section 3.3 an ....

....and preprocessing time, for any ffi 0, and answers a query in O( log m) time, where 1 m n (the constants of proportionality depend on ffi) The results of Section 3.2.1 appeared in [HP99a] Approximate geodesic diameter. We present in Section 3. 3 an algorithm that computes, AAOS97] The results of Section 3.3 appeared in [HP99a] Approximate shortest path maps. The exact algorithms of [MMP87, SS86] receive as input a convex polytope or a polyhedral surface P, and a fixed source point s on P, and compute a map (i.e. a subdivision of P) of complexity Theta(n ) that ....

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P.K. Agarwal, B. Aronov, J. O'Rourke, and C.A. Schevon. Star unfolding of a polytope with applications. SIAM J. Comput., 26:1679--1713, 1997.

....commun. 1987. See [D1528] 2 www.ifor.math.ethz.ch staff fukuda unfoldhome unfoldopen.html. nonoverlapping unfoldings are guaranteed in at least two ways. Fix a source point x 2 P . The star unfolding with respect to x cuts P along the n shortest paths from x to each of the n vertices of P [AAOS97] This is clearly a tree (in fact a star) and clearly spans the vertices. That this unfolding does not overlap is by no means obvious, but has been established [AO92] Fig. 1 shows an example. Fig. 1. The star unfolding of a polytope of n = 18 vertices. A second unfolding, more difficult to ....

P. K. Agarwal, B. Aronov, J. O'Rourke, and C. A. Schevon. Star unfolding of a polytope with applications. SIAM J. Comput., 26:1689--1713, 1997.

.... runs in singly exponential time [RS89, Sha87] This has motivated researchers to develop efficient approximation algorithms [Pap85, Cla87] and to study interesting special cases [MMP87, Sha87] An earlier version of this paper was presented at the Second Scandinavian Workshop on Algorithm Theory [AAOS90]. Part of the work was carried out when the first two authors were at Courant Institute of Mathematical Sciences, New York University and later at Dimacs (Center for Discrete Mathematics and Theoretical Computer Science) a National Science Foundation Science and Technology Center ....

....interest in its own right. The second problem studied in this paper is that of computing the geodesic diameter of P , i.e. the maximum distance along P between any two points on P . O Rourke and Schevon [OS89] gave an O(n 14 log n) time procedure for determining the geodesic diameter of P . In [AAOS90], we presented a simpler and faster algorithm whose running time is O(n 10 ) An even faster O(n 8 log n) algorithm is presented in the current version of the paper. The third problem involves answering queries of the form: Given x; y 2 P , determine the distance between x and y along P . ....

P. K. Agarwal, B. Aronov, J. O'Rourke, and C. Schevon. Star unfolding of a polytope with applications. In J. R. Gilbert and R. Karlsson, editors, Proc. of 2nd Scandanavian Workshop on Algorithm Theory, pages 251--263. Springer-Verlag, July 1990. Lecture Notes in Computer Science, Vol. 447.

....[35, 102, 135] A shortest path on the surface of a convex polytope can be represented by the sequence of edges that it crosses, and we refer to such a sequence of edges as a shortestpath edge sequence. It is known that there are O(n 4 ) shortest path edge sequences [104] Agarwal et al. [3] have shown that the exact set of all shortest path edge sequences can be computed in time O(n 5 s (n) log n) for some constant s 0, improving a previous algorithm by Schevon and O Rourke [126] Davenport Schinzel Sequences September 1, 1995 Miscellaneous Applications 34 Baltsan and Sharir ....

P. K. Agarwal, B. Aronov, J. O'Rourke, and C. Schevon, Star unfolding of a polytope with applications, Technical Report 031, Dept. Comput. Sci., Smith College, Northampton, MA, July 1993.

....(1) cut open and unfold a convex polyhedron to a simple planar polygon; and (2) fold and glue a simple planar polygon into a convex polyhedron. A convex polyhedron can always be unfolded, at least when we allow cuts across the faces of the polyhedron. One such unfolding is the star unfolding [2]. When cuts are limited to the edges of the polyhedron, it is not known whether every convex polyhedron has an unfolding [3] although there is software that has never failed to produce an unfolding, e.g. HyperGami [6] The problem of folding a polygon to form a convex polyhedron is partly ....

P. K. Agarwal, B. Aronov, J. O'Rourke, and C. A. Schevon. Star unfolding of a polytope with applications. SIAM Journal on Computing, 26(6):1689--1713, Dec. 1997.

....of vertices of P or Q, letting the context determine which. We will also freely employ two types of paths on the surface of a polytope: geodesics, which unfold (or develop ) to straight lines, and shortest paths, geodesics which are in addition shortest paths between their endpoints. See, e.g. AAOS97] for details and basic properties. 2.1 Perimeter Halving As a straightforward application of Aleksandrov s theorem, we prove that every convex polygon folds to a polytope. We will see in Section 4.2 that the converse does not hold. For two points x; y 2 P , define (x; y) be the open interval ....

P. K. Agarwal, B. Aronov, J. O'Rourke, and C. A. Schevon. Star unfolding of a polytope with applications. SIAM J. Comput., 26(6):1689--1713, December 1997.

No context found.

P. K. Agarwal, B. Aronov, J. O'Rourke, and C. A. Schevon. Star unfolding of a polytope with applications. SIAM J. Comput., 26:1689--1713, 1997.

No context found.

P. Agarwal, B. Aronov, J. O'Rourke, and C. Schevon, " Star unfolding of a polytope with applications", Proc. of the Scandanavian Workshop on Algorithm Theory. LNCS 447, 1990.

No context found.

P. Agarwal, B. Aronov, J. O'Rourke, and C. Schevon. Star unfolding of a polytope with applications. Proc. of the Scandanavian Workshop on Algorithm Theory. LNCS 447, 1990.

No context found.

Pankaj K. Agarwal, B. Aronov, J. O'Rourke, and C. Schevon. Star unfolding of a polytope with applications. Technical Report 031, Dept. Comput. Sci., Smith College, (Northampton, MA, 1993).

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