Abstract not found

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 ε pathcanbereportedintimelinearinitscomplexity. 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.

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 sub-paths within each face of P. We present query algorithms that compute approximate distances and/or approximate shortest paths on P. Our all-pairs 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 ε-approximate 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 ε-approximate 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.

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.

We present an algorithm for approximating geodesic distances on 2-manifolds in R 3. Our algorithm works on an ε-sample 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 ε-sample S, the algorithm has a near-optimal 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

We survey algorithms and hardness results for two important classes of topology optimization problems: computing minimum-weight cycles in a given homotopy or homology class, and computing minimum-weight cycle bases for the fundamental group or various homology groups.