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36
Approximating shortest paths in anisotropic regions
 In Proc. 18th annual ACMSIAM symposium on Discrete Algorithm
, 2007
"... Our goal is to find an approximate shortest path for a point robot moving in a planar subdivision with n vertices. Let ρ � 1 be a real number. Distances in each face of this subdivision are measured by a convex distance function whose unit disk is contained in a concentric unit Euclidean disk, and c ..."
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Cited by 16 (2 self)
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Our goal is to find an approximate shortest path for a point robot moving in a planar subdivision with n vertices. Let ρ � 1 be a real number. Distances in each face of this subdivision are measured by a convex distance function whose unit disk is contained in a concentric unit Euclidean disk, and contains a concentric Euclidean disk with radius 1/ρ. Different convex distance functions may be used for different faces, and obstacles are allowed. These convex distance functions may be asymmetric. For any ε ∈ (0, 1) and for any two points vs and vd, we give an algorithm that finds a path �from vs to vd whose cost is at 2 ρ log ρ most (1 + ε) times the optimal. Our algorithm runs in O ε2 n3 log � � ρn ε time. This bound does not depend on any other parameters; in particular it does not depend on the minimum angle in the subdivision. We give applications to two special cases that have been considered before: the weighted region problem and motion planning in the presence of uniform flows. For the weighted region � problem with weights in [1, ρ] ∪ {∞}, the time ρ log ρ bound of our algorithm improves to O ε n3 log � � ρn
Approximate Euclidean shortest paths amid convex obstacles
 Proc. 20th ACMSIAM Sympos. Discrete Algorithms
"... We develop algorithms and data structures for the approximate Euclidean shortest path problem amid a set P of k convex obstacles in R 2 and R 3, with a total of n faces. The running time of our algorithms is linear in n, and the size and query time of our data structure are independent of n. Our app ..."
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Cited by 9 (3 self)
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We develop algorithms and data structures for the approximate Euclidean shortest path problem amid a set P of k convex obstacles in R 2 and R 3, with a total of n faces. The running time of our algorithms is linear in n, and the size and query time of our data structure are independent of n. Our approeach is to quickly compute a small sketch Q of P whose size is independent of n and then compute approximate shortest paths with respect to Q.
Realtime path planning in heterogeneous environments
 Computer Animation and Virtual Worlds (CAVW
"... Modern virtual environments can contain a variety of characters and traversable regions. Each character may have different preferences for the traversable region types. Pedestrians may prefer to walk on sidewalks, but they may occasionally need to traverse roads and dirt paths. By contrast, wild ani ..."
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Cited by 6 (1 self)
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Modern virtual environments can contain a variety of characters and traversable regions. Each character may have different preferences for the traversable region types. Pedestrians may prefer to walk on sidewalks, but they may occasionally need to traverse roads and dirt paths. By contrast, wild animals might try to stay in forest areas, but they are able to leave their protective environment when necessary. This paper presents a novel path planning method named MIRAN (Modified Indicative Routes and Navigation) that takes a character’s region preferences into account. Given an indicative route as a rough estimation of a character’s preferred route, MIRAN efficiently computes a visually convincing path that is smooth, keeps clearance from obstacles, avoids unnecessary detours, and allows local changes to avoid other characters. To the best of our knowledge, MIRAN is the first path planning method that supports the above features while using an exact representation of the navigable space. Experiments show that with our approach a wide range of different character behaviors can be simulated. It also overcomes problems that occur in previous path planning methods such as the Indicative Route Method. The resulting paths are wellsuited for realtime simulations and gaming applications.
Approximate Shortest Path Queries on Weighted Polyhedral Surfaces
"... We consider the classical geometric problem of determining shortest paths between pairs of points lying on a weighted polyhedral surface P consisting of n triangular faces. We present query algorithms that compute approximate distances and/or approximate (weighted) shortest paths. Our algorithm take ..."
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We consider the classical geometric problem of determining shortest paths between pairs of points lying on a weighted polyhedral surface P consisting of n triangular faces. We present query algorithms that compute approximate distances and/or approximate (weighted) shortest paths. Our algorithm takes as input an approximation parameter ε ∈ (0, 1) and a query time parameter q and builds a data structure which is then used for answering ǫapproximate distance queries in O(q) time. This algorithm is source point independent and improves significantly on the best previous solution. For the case where one of the query points is fixed we build a data structure that can answer ǫapproximate distance queries to any query point in P in O(log 1) time. This is an improveε ment upon the previously known solution for the Euclidean fixed source query problem. Our algorithm also generalizes the setting from previously studied unweighted polyhedral to weighted polyhedral surfaces of arbitrary genus. Our solutions are based on a novel graph separator algorithm introduced here which extends and generalizes previously known separator algorithms.
Algorithms for Approximate Shortest Path Queries on Weighted Polyhedral Surfaces
, 2008
"... We consider the well known geometric problem of determining shortest paths between pairs of points on a polyhedral surface P, where P consists of triangular faces with positive weights assigned to them. The cost of a path in P is defined to be the weighted sum of Euclidean lengths of the subpaths w ..."
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Cited by 5 (0 self)
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We consider the well known geometric problem of determining shortest paths between pairs of points on a polyhedral surface P, where P consists of triangular faces with positive weights assigned to them. The cost of a path in P is defined to be the weighted sum of Euclidean lengths of the subpaths within each face of P. We present query algorithms that compute approximate distances and/or approximate shortest paths on P. Our allpairs query algorithms take as input an approximation parameter ε ∈ (0,1) and a query time parameter q, in a certain range, and builds a data structure APQ(P,ε;q), which is then used for answering εapproximate distance queries in O(q) time. As a building block of the APQ(P,ε;q) data structure, we develop a single source query data structure SSQ(a;P,ε) that can answer εapproximate distance queries from a fixed point a to any query point on P in logarithmic time. Our algorithms answer shortest path queries in weighted surfaces, which is an important extension, both theoretically and practically, to the extensively studied Euclidean distance case. In addition, our algorithms improve upon previously known query algorithms for shortest paths on surfaces. The algorithms are based on a novel graph separator algorithm introduced and analyzed here, which extends and generalizes previously known separator algorithms.
An Approximation Algorithm for Shortest Descending Paths
, 2007
"... A path from s to t on a polyhedral terrain is descending if the height of a point p never increases while we move p along the path from s to t. No efficient algorithm is known to find a shortest descending path (SDP) from s to t in a polyhedral terrain. We give a simple approximation algorithm that ..."
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Cited by 4 (2 self)
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A path from s to t on a polyhedral terrain is descending if the height of a point p never increases while we move p along the path from s to t. No efficient algorithm is known to find a shortest descending path (SDP) from s to t in a polyhedral terrain. We give a simple approximation algorithm that solves the SDP problem on general terrains. Our algorithm discretizes the terrain with O(n2 � 2 n X X/ǫ) Steiner points so that after an O ǫ log � � nX ǫtime preprocessing phase for a given vertex s, we can determine a (1 + ǫ)approximate SDP from s to any point v in O(n) time if v is either a vertex of the terrain or a Steiner point, and in O(nX/ǫ) time otherwise. Here n is the size of the terrain, and X is a parameter of the geometry of the terrain. 1
klink shortest paths in weighted subdivisions
 In Proceedings of the 9th International Workshop on Algorithms and Data Structures
, 2005
"... Abstract. We study the shortest path problem in weighted polygonal subdivisions of the plane, with the additional constraint of an upper bound, k, on the number of links (segments) in the path. We prove structural properties of optimal paths and utilize these results to obtain approximation algorith ..."
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Abstract. We study the shortest path problem in weighted polygonal subdivisions of the plane, with the additional constraint of an upper bound, k, on the number of links (segments) in the path. We prove structural properties of optimal paths and utilize these results to obtain approximation algorithms that yield a path having O(k) links and weighted length at most (1 + ɛ) times the weighted length of an optimal klink path, for any fixed ɛ> 0. Some of our results make use of a new solution for the 1link case, based on computing optimal solutions for a special sumoffractionals (SOF) problem. We have implemented a system, based on the CORE library, for computing optimal 1link paths; we experimentally compare our new solution with a previous method for 1link optimal paths based on a pruneandsearch scheme. 1
CNOP: a package for constrained network optimization
 IN PROC. 3RD INT. WORKSHOP ON ALGORITHM ENGINEERING AND EXPERIMENTS (ALENEX 01), LNCS 2153
, 2001
"... We present a generic package for resource constrained network optimization problems. We illustrate the flexibility and the use of our package by solving four applications: route planning, curve approximation, minimum cost reliability constrained spanning trees and the table layout problem. ..."
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We present a generic package for resource constrained network optimization problems. We illustrate the flexibility and the use of our package by solving four applications: route planning, curve approximation, minimum cost reliability constrained spanning trees and the table layout problem.
Querying Approximate Shortest Paths in Anisotropic Regions
, 2007
"... We present a data structure for answering approximate shortest path queries in a planar subdivision from a fixed source. Let ρ � 1 be a real number. Distances in each face of this subdivision are measured by a possibly asymmetric convex distance function whose unit disk is contained in a concentric ..."
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Cited by 3 (0 self)
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We present a data structure for answering approximate shortest path queries in a planar subdivision from a fixed source. Let ρ � 1 be a real number. Distances in each face of this subdivision are measured by a possibly asymmetric convex distance function whose unit disk is contained in a concentric unit Euclidean disk, and contains a concentric Euclidean disk with radius 1/ρ. Different convex distance functions may be used for different faces, and obstacles are allowed. Let ε be any number strictly between 0 and 1. Our data structure returns a (1 + ε) approximation of the shortest path cost from the fixed source to a query destination in O(log ρn) time. Afterwards, a (1 + ε)approximate shortest path can be reported in time linear in itsc omplexity. The data structure uses O ( ρ2n 4 ε2 log ρn) space and can be built in ε O ( ρ2n 4 ε2 (log ρn ε)2) time. Our time and space bounds do not depend on any geometric parameter of the subdivision such as the minimum angle.