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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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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.
Geodesic Paths On 3D Surfaces: Survey and Open Problems
, 904
"... This survey gives a brief overview of theoretically and practically relevant algorithms to compute geodesic paths and distances on threedimensional surfaces. The survey focuses on polyhedral threedimensional surfaces. 1 ..."
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This survey gives a brief overview of theoretically and practically relevant algorithms to compute geodesic paths and distances on threedimensional surfaces. The survey focuses on polyhedral threedimensional surfaces. 1
On Flat Polyhedra deriving from Alexandrov’s Theorem
, 2010
"... We show that there is a straightforward algorithm to determine if the polyhedron guaranteed to exist by Alexandrov’s gluing theorem is a degenerate flat polyhedron, and to reconstruct it from the gluing instructions. The algorithm runs in O(n 3) time for polygons of n vertices. ..."
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We show that there is a straightforward algorithm to determine if the polyhedron guaranteed to exist by Alexandrov’s gluing theorem is a degenerate flat polyhedron, and to reconstruct it from the gluing instructions. The algorithm runs in O(n 3) time for polygons of n vertices.
Approximating Geodesic Distances on 2Manifolds in R 3
"... We present an algorithm for approximating geodesic distances on 2manifolds in R 3. Our algorithm works on an εsample of the underlying manifold and computes approximate geodesic distances between all pairs of points in this sample. The approximation error is multiplicative and depends on the densi ..."
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We present an algorithm for approximating geodesic distances on 2manifolds in R 3. Our algorithm works on an εsample of the underlying manifold and computes approximate geodesic distances between all pairs of points in this sample. The approximation error is multiplicative and depends on the density of the sample. For an εsample S, the algorithm has a nearoptimal running time of O ( S  2 log S  ) , an optimal space requirement of O ( S  2) , and approximates the geodesic distances up to a factor of 1 − O ( √ ε) and (1 − O (ε)) −1. 1
Combinatorial Optimization of Cycles and Bases
 PROCEEDINGS OF SYMPOSIA IN APPLIED MATHEMATICS
"... We survey algorithms and hardness results for two important classes of topology optimization problems: computing minimumweight cycles in a given homotopy or homology class, and computing minimumweight cycle bases for the fundamental group or various homology groups. ..."
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We survey algorithms and hardness results for two important classes of topology optimization problems: computing minimumweight cycles in a given homotopy or homology class, and computing minimumweight cycle bases for the fundamental group or various homology groups.