2010
Algorithms for approximate shortest path queries on weighted polyhedral surfaces
Publication
Publication
Discrete and Computational Geometry , Volume 44  Issue 4 p. 762 801
We consider the wellknown 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, in a certain range, and build 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 singlesource 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.
Additional Metadata  

Query algorithm, Separator, Shortest path, Weighted polyhedral surface  
dx.doi.org/10.1007/s0045400992040  
Discrete and Computational Geometry  
Organisation  Carleton University 
Aleksandrov, L. (Lyudmil), Djidjev, H.N. (Hristo N.), Guo, H. (Hua), Maheshwari, A, Nussbaum, D, & Sack, J.R. (2010). Algorithms for approximate shortest path queries on weighted polyhedral surfaces. Discrete and Computational Geometry, 44(4), 762–801. doi:10.1007/s0045400992040
