Results 1  10
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106
Fast exact and approximate geodesics on meshes
 ACM Trans. Graph
, 2005
"... The computation of geodesic paths and distances on triangle meshes is a common operation in many computer graphics applications. We present several practical algorithms for computing such geodesics from a source point to one or all other points efficiently. First, we describe an implementation of th ..."
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Cited by 67 (0 self)
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The computation of geodesic paths and distances on triangle meshes is a common operation in many computer graphics applications. We present several practical algorithms for computing such geodesics from a source point to one or all other points efficiently. First, we describe an implementation of the exact “single source, all destination ” algorithm presented by Mitchell, Mount, and Papadimitriou (MMP). We show that the algorithm runs much faster in practice than suggested by worst case analysis. Next, we extend the algorithm with a merging operation to obtain computationally efficient and accurate approximations with bounded error. Finally, to compute the shortest path between two given points, we use a lowerbound property of our approximate geodesic algorithm to efficiently prune the frontier of the MMP algorithm, thereby obtaining an exact solution even more quickly.
Boundary recognition in sensor networks by topological methods
 In Proc. of the ACM/IEEE International Conference on Mobile Computing and Networking (MobiCom
, 2006
"... Wireless sensor networks are tightly associated with the underlying environment in which the sensors are deployed. The global topology of the network is of great importance to both sensor network applications and the implementation of networking functionalities. In this paper we study the problem of ..."
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Cited by 65 (16 self)
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Wireless sensor networks are tightly associated with the underlying environment in which the sensors are deployed. The global topology of the network is of great importance to both sensor network applications and the implementation of networking functionalities. In this paper we study the problem of topology discovery, in particular, identifying boundaries in a sensor network. Suppose a large number of sensor nodes are scattered in a geometric region, with nearby nodes communicating with each other directly. Our goal is to find the boundary nodes by using only connectivity information. We do not assume any knowledge of the node locations or interdistances, nor do we enforce that the communication graph follows the unit disk graph model. We propose a simple, distributed algorithm that correctly detects nodes on the boundaries and connects them into meaningful boundary cycles. We obtain as a byproduct the medial axis of the sensor field, which has applications in creating virtual coordinates for routing. We show by extensive simulation that the algorithm gives good results even for networks with low density. We also prove rigorously the correctness of the algorithm for continuous geometric domains.
Approximation algorithms for TSP with neighborhoods in the plane
 J. ALGORITHMS
, 2001
"... In the Euclidean TSP with neighborhoods (TSPN), we are given a collection of n regions (neighborhoods) and we seek a shortest tour that visits each region. As a generalization of the classical Euclidean TSP, TSPN is also NPhard. In this paper, we present new approximation results for the TSPN, incl ..."
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Cited by 63 (8 self)
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In the Euclidean TSP with neighborhoods (TSPN), we are given a collection of n regions (neighborhoods) and we seek a shortest tour that visits each region. As a generalization of the classical Euclidean TSP, TSPN is also NPhard. In this paper, we present new approximation results for the TSPN, including (1) a constantfactor approximation algorithm for the case of arbitrary connected neighborhoods having comparable diameters; and (2) a PTAS for the important special case of disjoint unit disk neighborhoods (or nearly disjoint, nearlyunit disks). Our methods also yield improved approximation ratios for various special classes of neighborhoods, which have previously been studied. Further, we give a lineartime O(1) approximation algorithm for the case of neighborhoods that are (innite) straight lines.
Exact and Approximate Distances in Graphs  a survey
 In ESA
, 2001
"... We survey recent and not so recent results related to the computation of exact and approximate distances, and corresponding shortest, or almost shortest, paths in graphs. We consider many different settings and models and try to identify some remaining open problems. ..."
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Cited by 57 (0 self)
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We survey recent and not so recent results related to the computation of exact and approximate distances, and corresponding shortest, or almost shortest, paths in graphs. We consider many different settings and models and try to identify some remaining open problems.
Modeling acoustics in virtual environments using the uniform theory of diffraction
 ACM Computer Graphics, SIGGRAPH’01 Proceedings
, 2001
"... Realistic modeling of reverberant sound in 3D virtual worlds provides users with important cues for localizing sound sources and understanding spatial properties of the environment. Unfortunately, current geometric acoustic modeling systems do not accurately simulate reverberant sound. Instead, they ..."
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Cited by 49 (8 self)
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Realistic modeling of reverberant sound in 3D virtual worlds provides users with important cues for localizing sound sources and understanding spatial properties of the environment. Unfortunately, current geometric acoustic modeling systems do not accurately simulate reverberant sound. Instead, they model only direct transmission and specular reflection, while diffraction is either ignored or modeled through statistical approximation. However, diffraction is important for correct interpretation of acoustic environments, especially when the direct path between sound source and receiver is occluded. The Uniform Theory of Diffraction (UTD) extends geometrical acoustics with diffraction phenomena: illuminated edges become secondary sources of diffracted rays that in turn may propagate through the environment. In this paper, we propose an efficient way for computing the acoustical effect of diffraction paths using the UTD for deriving secondary diffracted rays and associated diffraction coefficients. Our main contributions are: 1) a beam tracing method for enumerating sequences of diffracting edges efficiently and without aliasing in densely occluded polyhedral environments; 2) a practical approximation to the simulated sound field in which diffraction is considered only in shadow regions; and 3) a realtime auralization system demonstrating that diffraction dramatically improves the quality of spatialized sound in virtual environments.
Touring a Sequence of Polygons
 In Proc. 35th Annu. ACM Sympos. Theory Comput
, 2003
"... Given a sequence of k polygons in the plane, a start point s, and a target point, t, we seek a shortest path that starts at s, visits in order each of the polygons, and ends at t. If the polygons are disjoint and convex, we give an algorithm running in time O(kn log(n/k)), where n is the total numbe ..."
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Cited by 31 (4 self)
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Given a sequence of k polygons in the plane, a start point s, and a target point, t, we seek a shortest path that starts at s, visits in order each of the polygons, and ends at t. If the polygons are disjoint and convex, we give an algorithm running in time O(kn log(n/k)), where n is the total number of vertices specifying the polygons. We also extend our results to a case in which the convex polygons are arbitrarily intersecting and the subpath between any two consecutive polygons is constrained to lie within a simply connected region; the algorithm uses O(nk log n) time. Our methods are simple and allow shortest path queries from s to a query point t to be answered in time O(k log n + m), where m is the combinatorial path length. We show that for nonconvex polygons this "touring polygons" problem is NPhard.
On the complexity of approximating TSP with neighborhoods and related problems
 Computational Complexity
"... We prove that various geometric covering problems, related to the Travelling Salesman Problem cannot be efficiently approximated to within any constant factor unless P = NP. This includes the GroupTravelling Salesman Problem (TSP with Neighborhoods) in the Euclidean plane, the GroupSteinerTree in ..."
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Cited by 30 (2 self)
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We prove that various geometric covering problems, related to the Travelling Salesman Problem cannot be efficiently approximated to within any constant factor unless P = NP. This includes the GroupTravelling Salesman Problem (TSP with Neighborhoods) in the Euclidean plane, the GroupSteinerTree in the Euclidean plane and the Minimum Watchman Tour and the Minimum Watchman Path in 3D. It resolves three open problems presented in the comprehensive survey of Mitchell [Mit00], improves a previously known approximation hardness factor of 2041 2040 [GL00, dBGK+ 02] for the first problem, and it is the first approximation hardness factor for the other problems. Some inapproximability factors are also shown for special cases of the above problems, where the size of the sets is bounded. GroupTSP and GroupSteinerTree where each neighbourhood is connected are also considered. It is shown that approximating these variants to within any constant factor smaller than 2, is NPhard. For the GroupTravelling Salesman and GroupSteinerTree Problems in dimension d, we show an innapproximability factor of O(log d−1 d HyperGraph VertexCover.
Approximate shortest path on a polyhedral surface and its applications
 ComputerAided Design
, 2000
"... A new algorithm is proposed for calculating the approximate shortest path on a polyhedral surface. The method mainly uses Dijkstra’s algorithm and is based on selective refinement of the discrete graph of a polyhedron. Although the algorithm is an approximation, it has the significant advantages of ..."
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Cited by 28 (1 self)
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A new algorithm is proposed for calculating the approximate shortest path on a polyhedral surface. The method mainly uses Dijkstra’s algorithm and is based on selective refinement of the discrete graph of a polyhedron. Although the algorithm is an approximation, it has the significant advantages of being fast, easy to implement, high approximation accuracy, and numerically robust. The approximation accuracy and computation time are compared between this approximation algorithm and the extended Chen & Han (ECH) algorithm that can calculate the exact shortest path for nonconvex polyhedra. The approximation algorithm can calculate shortest paths within 0.4 % accuracy to roughly 1001000 times faster than the ECH algorithm in our examples. Two applications are discussed of the approximation algorithm to geometric modeling.
Algorithms for Rapidly Dispersing Robot Swarms in Unknown Environments
, 2002
"... We develop and analyze algorithms for dispersing a swarm of primitive robots in an unknown environment, R. The primary objective is to minimize the makespan, that is, the time to fill the entire region. An environment is composed of pixels that form a connected subset of the integer grid. There is a ..."
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Cited by 28 (6 self)
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We develop and analyze algorithms for dispersing a swarm of primitive robots in an unknown environment, R. The primary objective is to minimize the makespan, that is, the time to fill the entire region. An environment is composed of pixels that form a connected subset of the integer grid. There is at most one robot per pixel and robots move horizontally or vertically at unit speed. Robots enter R by means of k ≥ 1 door pixels. Robots are primitive finite automata, only having local communication, local sensors, and a constantsized memory. We first give algorithms for the singledoor case...
Computing Homotopic Shortest Paths Efficiently
"... We give algorithms to find shortest paths homotopic to given disjoint paths that wind amongst n point obstacles in the plane. Our deterministic algorithm runs in time O(k log n+n √n), and the randomized version in time O(k log n+n(log n)1(1+ε)), where k is the input plus output sizes o ..."
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Cited by 24 (2 self)
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We give algorithms to find shortest paths homotopic to given disjoint paths that wind amongst n point obstacles in the plane. Our deterministic algorithm runs in time O(k log n+n √n), and the randomized version in time O(k log n+n(log n)1(1+ε)), where k is the input plus output sizes of the paths.