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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 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 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.

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A graph separator is a set of vertices or edges whose removal divides an input graph into components of bounded size. This paper describes new algorithms for computing separators in planar graphs as well as techniques that can be used to speed up the implementation of graph partitioning algorithms and improve the partition quality. In particular, we consider planar graphs with costs and weights on the vertices, where weights are used to estimate the sizes of the partitions and costs are used to estimate the size of the separator. We show that in these graphs one can always find a small cost separator (consisting of vertices or edges) that partitions the graph into components of bounded weight. We describe implementations of the partitioning algorithms and discuss results of our experiments.