Hershberger and Snoeyink. Computing minimum length paths of a given homotopy class. CGTA: Computational Geometry: Theory and Applications, 4, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....triangulation on the input map and use edges of this triangulation to construct a simplified map. 3. 2 Computational Geometry Map simplification (and the related problem of chain simplification where each map consists of only one chain) has been studied extensively by computational geometers [18, 7, 4, 15, 1]; recent work [9] has shown that map simplification is NP hard. Moreover, Estkowski [10] has also shown that it is hard to obtain a solution to this problem that approximates the optimal answer to within a polynomial factor. For chain simplification, the best known method [4] obtains the optimal ....

J. Hershberger and J. Snoeyink. Computing minimum length paths of a given homotopy class. Comput. Geom. Theory Appl., 4:63--98, 1994.

....are drawn as thickly as possible and proportional to the corresponding edge weights. Duncan et al. 7] and Efrat et al. 10] present an O(kn n ) algorithm for this problem, where n is the number of edges and k is the maximum of their input and output complexities. Hershberger and Snoeyink [12] give an algorithm for the homotopic shortest path problem. Their algorithm assumes a triangulation of size n of the polygonal domain, and finds a shortest path homotopic to a given path of k edges in time linear in k plus the number of triangles (with repetition) visited by the input path. This ....

....of the paths are not regarded as obstacles. Let k in be the number of edges in all the paths of # . Let k out be the number of edges in all the paths of #. Note that k in and k out can be arbitrarily large compared to n, and that k out nk in . The algorithm of Hershberger and Snoeyink [12] finds homotopic shortest paths in time O(nk in ) The deterministic algorithm presented in this paper runs in time O(k out k in log n n # n) and the randomized algorithm in time O(k out k in log n n(log n) These are improvements except when k in is quite small compared to n. Our ....

Hershberger and Snoeyink. Computing minimum length paths of a given homotopy class. CGTA: Computational Geometry: Theory and Applications, 4:63--97, 1994.

.... linear time using the algorithms of Chazelle [1] or Lee and Preparata [8] The latter algorithm is known as the funnel algorithm and it can be extended to river routing [2, 4, 9] In our setting, the shortest paths can be found in optimal O(nk) time using the algorithm of Hershberger and Snoeyink [5]. 1.2 Our Results We show how to solve the FED problem in O(nk n 3 ) time and O(n k) space. We describe the algorithm in the tradition of the homotopic wire routing where n is the number of wires and k is the maximum of the initial and final complexities of all the paths. We also show how to ....

....homotopic routing of maximum separation. We begin with the initial set of wires W and compute for each w # W the shortest path w # homotopic to w, which yields a set of shortest paths W # that is a homotopic shift of W . This can be done in O(nk) time using a result from Hershberger and Snoeyink [5]. The idea of the shortest path computation is to triangulate the region using the O(n) terminals and obstacles, yielding a triangulation of size O(n) A shortest path w # homotopic to a given path w can be computed in time O(Cw #w ) where Cw is the complexity of w and #w is the number of times ....

Hershberger and Snoeyink. Computing minimum length paths of a given homotopy class. CGTA: Computational Geometry: Theory and Applications, 4, 1994.

....we consider an feasible shortcut edge e to be an edge of G(s) only if e does not intersect s 0 (as before) and e is homotopically equivalent, with respect to Q, to the chain C(e) that it replaces. In order to test if C(e) is homotopically equivalent to e, we could apply the techniques of [13] to compute a shortest path homotopically equivalent to C(e) within a triangulation of the points Q; if the shortest path is a straight segment (e) then they are homotopically equivalent, and otherwise they are not. In order to avoid computing and storing a triangulation, and for simplicity of ....

J. Hershberger and J. Snoeyink. Computing minimum length paths of a given homotopy class. Comput. Geom. Theory Appl., 4:63-98, 1994.

....and needs O(n p n) preprocessing time and space. If we restrict ourselves to simple rectilinear polygons, the situation gets considerably simpler since it can be shown that there is always a rectilinear path between two points that is shortest w.r.t. both the rectilinear link and the L 1 metric [10, 11]. We call such a path a smallest path [12, 13] In this setting de Berg [10] presents a data structure that allows to answer the following types of queries. If we are given two arbitrary points inside a polygon, the rectilinear link distance and the L 1 distance between the two points can be ....

J. Hershberger and J. Snoeyink, "Computing minimum length paths of a given homotopy class", in Algorithms and Data Structures, Proc. 2nd Workshop on Algorithms and Data Structures, WADS'91, eds. F. Dehne, J.-R. Sack, F. Santoro, (Springer Verlag, Heidelberg, 1991), pp. 331--342.

....produced by this method tended to conform to the subjective solutions produced by domain experts. Computational Geometry Map simplification (and the related problem of chain simplification where each map consists of only one chain) has been studied extensively by computational geometers [9, 4, 3, 8, 1]; recent work [6] has shown that map simplification is NP hard. Moreover, Estkowski [7] has also shown that that it is hard to obtain a solution to this problem that approximates the optimal answer to within a polynomial factor. For chain simplification, the best known method [3] obtains the ....

J. Hershberger and J. Snoeyink. Computing minimum length paths of a given homotopy class. Comput. Geom. Theory Appl., 4:63--98, 1994.

....c be a nonconvex face. For each cycle V of its boundary, we determine the subset W V = fw 1 ; w j g of reflex vertices plus the vertices whose y coordinates are locally minimum or maximum. Next, for each 1 i j, we apply the algorithm of de Berg et al. 4] or of Hershberger and Snoeyink [14] to compute the (geodesic) shortest path V i between w i and w i 1 homotopic to V i (relative to c) Roughly speaking, both of these algorithm triangulate the nonconvex face, traverse all triangles adjacent to V i , and maintain the shortest path V i . This gives the reduced cycle V for ....

....triangles adjacent to V i , and maintain the shortest path V i . This gives the reduced cycle V for V . The total time spent in computing shortest paths over all cycles of c is proportional to the number of edges in c, because each triangle is traversed only a constant number of times; see [4, 14] for a proof. After having computed reduced cycles for all components of the boundary of c, the body 1 In [18] Matousek gave an algorithm for computing a spanning path with low stabbing whose time complexity was O(n 3=2 log 2 n) But its running time can be improved to O(n 4=3 log 2 n) ....

Hershberger, J. and J. Snoeyink, Computing minimum length paths of a given homotopy class, Proc. 2nd Workshop on Algo. Data Struct. (1991), pp. 331--342.

....that outputs a path whose length is close to the shortest path length and which time complexity is polynomial in the complexity of representation of obstacles and in log 1 . Constructing the shortest path in a given homotopy class was considered for polygonal obstacles in the plane in [HS94] but within a rather restricted setting. Namely, the space admissible for paths is a boundary triangulated 2 manifold. Such a manifold is a simplicial complex in which all vertices are boundary vertices. Homotopy classes are represented by paths. This particular triangulation permits to ....

....Such a manifold is a simplicial complex in which all vertices are boundary vertices. Homotopy classes are represented by paths. This particular triangulation permits to construct straightforwardly a covering space and easily find the shortest path homotopic to the given one. The efforts of [HS94] are concentrated on other kind of problems related to unifying treatments of various metrics and improving particular data structures. Our result. We show that for the case of semi algebraic obstacles in the plane and for rather general way of representing classes of homotopy the problem of ....

J. Hershberger and J. Snoeyink. Computing minimum length paths of a given homotopy class. Computational Geometry, 4:63--97, 1994.

....of algebraic functions can find it in polytime. 1 Introduction. We consider the problem of constructing minimum link minimum turn polygon amidst semi algebraic obstacles in the plane either for a given homotopy class or globally. Usually this problem is motivated by robotics (see, e.g. HS94, MPA92, 1 Address: Dept. of Comput. Sci. Penn State University, University Park, PA 16802, USA. E mail: grigorie sol4.cse.psu.edu [ Supported by a grant of Minist ere de l Enseignement Sup erieur et de la Recherche de la France (December 1995 January 1996) and by NSF grant CCR 9424358. 2 ....

....a shortest path in R 3 even amidst polyhedral obstacles is known to be NP hard [CR87] 1.3 Previous results. The problem of minimizing the number of links of a polygonal path amidst polygonal obstacles and the total turn was studied in the case when obstacles are polygonal, see e.g. MRW92] HS94] CGM 95] and section 4.4.2 in [MS95] A more complicated problem of minimizing the length of minimum link paths is considered in [MPA92] In [HS94] the authors consider also the problem of constructing a minimum link path for a given class of homotopy, again for polygonal obstacles. ....

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J. Hershberger and J. Snoeyink. Computing minimum length paths of a given homotopy class. Computational Geometry, 4:63--97, 1994.

.... where t 1 ; Delta Delta Delta ; t 2n are T junctions, consists of an optimal completion of the optimal connections within the arcs included in [t 1 ; Delta Delta Delta ; t 2n ] Geodesic paths joining compatible Tjunctions are computed by means of Hershberger and Snoeyink s funnel algorithm [4] after a triangulation of the polygonal line to which the occlusion boundary reduces. Once the best level lines configuration has been computed, the disocclusion process is completed by means of a simple geodesic propagation within the occlusion of the gray level values of the restored level ....

J. Hershberger and J. Snoeyink, "Computing minimum length paths of a given homotopy class", Comput. Geom Theory Appl., Vol. 4, pp. 63-98, 1994.

....a radius equal to the given threshold. The solution is a polygonal chain with minimum links, which is homotopic to the input line into such region. Linear algorithms for computing homotopy paths inside simple polygons were proposed by Suri [Sur86] Ghosh [Gho91] and Hershberger and Snoeyink [Her94]. All such algorithms follow a greedy approach. Guibas et al. also study problem (i) as a problem of ordered stabbing [Gui93] Only disks centered at vertices of the input chain are considered: the solution must be a minimum link chain stabbing such discs in order. For such a problem, they give an ....

Hershberger, J., Snoeyink, J., Computing minimum length paths of a given homotopy class, Computational Geometry - Theory and Applications, 4, 1994, pp.63-97.

....have been added to the sequence of polygonal paths, we have j j = j Pij 2k Gamma 2 2j 0 j 2k Gamma 2 4j 0 j: The time complexity of the above algorithm is computed as follows. Step 1 takes O(n) time[4] Step 2 takes O(n log n) time. Step 3 takes O(n) time[9] Step 4 takes O(kn)[5, 6, 12], where k = jM j. By Corollary 2, k m, where m is the link length of an optimal path. Hence, the time taken by this step is O(mn) The cost of building the complete graph G in Step 5 is O(k 2 ) The total time required by Step 6 is dominated by that of finding a minimum cost spanning tree, ....

J. Hershberger and J. Snoeyink, "Computing minimum length paths of a given homotopy class," Computational Geometry: Theory and Applications, 4, 2, (June 1994), 63-97.

....to do funnel splitting efficiently (since, in this case, we cannot discard one side of each split funnel) Then, after storing the SPM(s) in an appropriate O(n) size point location data structure (see, e.g. 180] single source queries can be answered in O(log n) time. Hershberger and Snoeyink [203] have substantially simplified the original algorithm of [186] The above result can be strengthened even further to the case of two point queries. Guibas and Hershberger [185] have shown how a simple polygon can be preprocessed in time O(n) into a data structure of size O(n) to support ....

....points. A special case of the shortest path problem in polygonal domains is that in which the homotopy type of the desired path is specified, e.g. by giving the sequence (possibly with repetitions) of the N visited triangles, in some triangulation of P . In this case, Hershberger and Snoeyink [203] have shown how to compute a shortest path of the given homotopy type in time O(N ) using a generalization of the linear time methods in simple polygons. This problem is of interest in applications to VLSI routing problems; see [122, 162, 256] To compute a shortest path in general polygonal ....

J. Hershberger and J. Snoeyink. Computing minimum length paths of a given homotopy class. Comput. Geom. Theory Appl., 4:63--98, 1994.

....solved in O(n) time. Their method can be viewed as an extension of the linear time method of Suri [54] which computes a minimum link path inside a simple polygon, to the problem of finding a minimum link monotone polygonal chain that stabs a given set of line segments. Hershberger and Snoeyink [32] show how to further generalize this method to find in O(n) time a minimum link path of a particular homotopy type in a non simple polygon, and Guibas, Hershberger, Mitchell, and Snoeyink [29] show how to generalize this method even further to find in O(n) time a minimum link stabber for any given ....

.... total number of events must be O(n) hence, the total time to enumerate all the geodesically critical values is O(n log n) Given these geodesically critical values it is then a simple manner to determine the consecutive pair that contains ffl by using the method of Hershberger and Snoeyink [32], Guibas et al. 29] or Hakimi and Schmeichel [30] to drive a binary search among the set of geodesicallycritical values. Using one of these simple algorithms, all of which are based upon the greedy method, to test if a given ffl is smaller than ffl takes O(n) time. This implies that we can ....

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J. Hershberger and J. Snoeyink, "Computing minimum length paths of a given homotopy class," in Proc. 2nd Workshop Algorithms Data Struct., Lecture Notes in Computer Science, vol. 519, Springer-Verlag, 331--342, 1991.

....values to O(n) and thus obtained an O(n log n) algorithm through several applications of pipelined parametric searching techniques. For a survey of results when A and C are not constrained to be x monotone, see the paper of Eu and Toussaint [ET94] and related work by Hershberger and Snoeyink [HS94] and Guibas, Hershberger, Mitchell, and Snoeyink [GHMS93] In general, algorithms for the min # problem have linear complexity, while those for the min ffl have quadratic complexity. Next let us consider the problem variant where the vertices of A have to be a subset of the vertices of C. Now we ....

J. Hershberger and J. Snoeyink. Computing minimum length paths of a given homotopy class. Comput. Geom. Theory Appl., 4:63--98, 1994.

...., the parents of x in T a and T b , along with the distances from x to a and b. We first triangulate the polygon in linear time using Chazelle s algorithm [5] and then build the shortest path trees T a and T b also in linear time using an algorithm of Guibas et al. 7] or Hershberger and Snoeyink [11]. Lemma 3.1 ( 7, 11] After linear time preprocessing, the distances d(x; a) and d(x; b) for any vertex x 2 P , can be computed in O(1) time. Given two arbitrary vertices x; y 2 P , we let ff(x; y) denote the lowest common ancestor of x and y in T a . Similarly, fi(x; y) denotes the lowest ....

....in T a and T b , along with the distances from x to a and b. We first triangulate the polygon in linear time using Chazelle s algorithm [5] and then build the shortest path trees T a and T b also in linear time using an algorithm of Guibas et al. 7] or Hershberger and Snoeyink [11] Lemma 3. 1 ([7, 11]) After linear time preprocessing, the distances d(x; a) and d(x; b) for any vertex x 2 P , can be computed in O(1) time. Given two arbitrary vertices x; y 2 P , we let ff(x; y) denote the lowest common ancestor of x and y in T a . Similarly, fi(x; y) denotes the lowest common ancestor of x and ....

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J. Hershberger and J. Snoeyink. Computing minimum length paths of a given homotopy class. In Proceedings of the 2nd Workshop on Algorithms and Data Structures, pages 331--342. Springer-Verlag, 1991. Lecture Notes in Computer Science 519.

....is to compute a shortest path between two points avoiding all the obstacles. Due to its simple formulation and obvious applications in routing and robotics, the problem has drawn the attention of many researchers in computational geometry; we mention only a few papers most relevant to our work [4, 13, 14, 17, 18, 19, 25]. The problem of computing shortest paths in the presence of a single obstacle has received special attention, due to its applications in various geometric problems involving a simple polygon [4, 13, 14, 17] The roles of free space and obstacle space have traditionally been reversed in this ....

.... computational geometry; we mention only a few papers most relevant to our work [4, 13, 14, 17, 18, 19, 25] The problem of computing shortest paths in the presence of a single obstacle has received special attention, due to its applications in various geometric problems involving a simple polygon [4, 13, 14, 17]. The roles of free space and obstacle space have traditionally been reversed in this special case: the interior of the polygon represents the free space and the boundary of the polygon represents an impenetrable obstacle. After several years of The authors were at DEC Systems Research Center, ....

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J. Hershberger and J. Snoeyink. Computing minimum length paths of a given homotopy class. Comp. Geom.: Theory and Appl., 4:63--97, 1994.

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J. Hershberger and J. Snoeyink. Computing minimum length paths of a given homotopy class. Computational Geometry: Theory and Applications, 1993.

....We can then attempt to find a minimum link representative of the homotopy class. Section 3 makes the definitions for such a homotopy method more precise. Its four subsections contain the following results: Sec. 3. 1 We briefly outline minimum link path algorithms developed in a companion paper [14] and apply them to approximate paths and polygons. These are greedy algorithms that, after the region R has been triangulated, find a path in time proportional to the number of triangles that the path passes through. Sec. 3.2 In contrast, we show that the problem of computing a minimum link ....

....is a continuous map Gamma: S 1 Theta [0; 1] R such that Gamma(x; 0) ff(x) and Gamma(x; 1) fi(x) ffl Two subdivisions ff and fi in R are homotopic in R if ff can be deformed to fi within R. 3. 1 Computing minimum link paths and polygons of a given homotopy type In a companion paper [14], we investigate the problem of computing minimum link paths and closed curves of a given homotopy class in triangulated polygons. Theorem 3.1 (From [14] A minimum link path ff 0 that is homotopic to a chain ff can be computed in time proportional to the number of links of ff and number ....

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John Hershberger and Jack Snoeyink. Computing minimum length paths of a given homotopy class. Technical Report RUU--CS--90--37, Utrecht Unversity, Department of Computer Science, December 1990.

....Consider the problem of computing minimum link paths: Given a simple polygon P containing two points s and t, compute a piecewise linear path from s to t that does not intersect the exterior of P and has the minimum number of line segments. Suri [24, 25] Ghosh [6] and Hershberger and Snoeyink [10] are all guilty of linear time algorithms for this problem. In Section 2 however, we give a simple instance in which representing path vertices with rational coordinates requires Theta(n 2 log n) bits, even though the input polygon uses Theta(n log n) bits. The same problem can be observed ....

....3.1 When path vertices are restricted to input data and first derived points, F (P; s; t) 2 (P; s; t) Given P , we can compute, in O(jP j) time, a path from s to t with at most 2 (P; s; t) segments. Proof : The Euclidean shortest path ff from s to t in P can be obtained in linear time [3, 10, 15]. We can traverse the path ff and label each vertex as a left or right turn. We call an edge qu of ff an inflection edge if the labels of q and u differ; edges incident to s and to t are also called inflection edges. Because ff turns only at reflex vertices of P , the line through qu is derived ....

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J. Hershberger and J. Snoeyink. Computing minimum length paths of a given homotopy class. Computational Geometry: Theory and Applications, 4:63--97, 1994.

....from one region to another to form a simply connected surface (due to Lemma 2.2) in space that only Compliant Motion in the Plane 21 overlaps itself when projected to the plane. Thus, the start triangles and trapezoids give a subdivision of a subset of the universal covering space of the polygon [21] the stop triangles and trapezoids give an alternate subdivision of the same subset. We concentrate on layers in this section, but the surface model simplifies some of the applications in Section 5. To analyze the complexity of this amended algorithm, we need to bound the number of events that ....

....2ffl. We compute the boundary of the non directional backprojection. Specifically, we compute a closed curve whose interior, as defined by its winding number, is exactly the non directional backprojection. The curve is a simple Jordan curve on the universal covering space of the interior of P [21]. That is, the interior of always lies to the left of when is traversed counterclockwise, and contains no island of P in its interior. If k = 1 (the simple polygon case) then the universal covering space is just P itself, and is a simple curve made up of line segments and circular arcs. If ....

[Article contains additional citation context not shown here]

J. Hershberger and J. Snoeyink, Computing minimum length paths of a given homotopy class, in Proceedings of the 2nd Workshop on Algorithms and Data Structures, SpringerVerlag, 1991, pp. 331--342. Lecture Notes in Computer Science 519.

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Hershberger and Snoeyink. Computing minimum length paths of a given homotopy class. CGTA: Computational Geometry: Theory and Applications, 4, 1994.

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J. Hershberger and J. Snoeyink, Computing minimum length paths of a given homotopy class. Computational Geometry -- Theory and Applications 4 (1994), pp. 63--97.

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J. Hershberger and J. Snoeyink, "Computing minimum length paths of a given homotopy class," Computational Geometry: Theory and Applications, 4,2 June 1994, 63-97.

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John Hershberger and Jack Snoeyink. Computing minimum length paths of a given homotopy class. In Proc. 2nd WADS Springer-Verlag LNCS 519, volume 2, pages 331--342, 1991.

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