Abstract
We consider the classical geometric problem of determining shortest paths between pairs of points lying on a weighted polyhedral surface P consisting of n triangular faces. We present query algorithms that compute approximate distances and/or approximate (weighted) shortest paths. Our algorithm takes as input an approximation parameter ε∈(0,1) and a query time parameter \(\mathfrak{q}\) and builds a data structure which is then used for answering ε-approximate distance queries in \(O(\mathfrak{q})\) time. This algorithm is source point independent and improves significantly on the best previous solution. For the case where one of the query points is fixed we build a data structure that can answer ε-approximate distance queries to any query point in P in \(O(\log\frac{1}{\varepsilon})\) time. This is an improvement upon the previously known solution for the Euclidean fixed source query problem. Our algorithm also generalizes the setting from previously studied unweighted polyhedral to weighted polyhedral surfaces of arbitrary genus. Our solutions are based on a novel graph separator algorithm introduced here which extends and generalizes previously known separator algorithms.
Research supported by NSERC, SUN Microsystems, and a P.E.O. Scholar Award. The first and the second author are Adjunct Professors at Carleton University.
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Agarwal, P.K., Aronov, B., O’Rourke, J., Schevon, C.A.: Star unfolding of a polytope with applications. SIAM J. Comput. 26(6), 1689–1713 (1997)
Agarwal, P.K., Har-Peled, S., Sharir, M., Varadarajan, K.R.: Approximate shortest paths on a convex polytope in three dimensions. J. ACM 44, 567–584 (1997)
Aleksandrov, L., Djidjev, H.: Linear algorithms for partitioning embedded graphs of bounded genus. SIAM J. Disc. Math. 9(1), 129–150 (1996)
Aleksandrov, L., Djidjev, H., Guo, H., Maheshwari, A.: Partitioning planar graphs with costs and weights. In: Mount, D.M., Stein, C. (eds.) ALENEX 2002. LNCS, vol. 2409, pp. 98–110. Springer, Heidelberg (2002)
Aleksandrov, L., Lanthier, M., Maheshwari, A., Sack, J.-R.: An ε-approximation algorithm for weighted shortest path queries on polyhedral surfaces. In: Proc. 14th Euro CG Barcelona, pp. 19–21 (1998)
Aleksandrov, L., Maheshwari, A., Sack, J.-R.: Determining approximate shortest paths on weighted polyhedral surfaces. J. ACM 52(1), 25–53 (2005)
Chazelle, B., Liu, D., Magen, A.: Sublinear geometric algorithms. SIAM J. Comput. 35, 627–646 (2006)
Chen, J., Han, Y.: Shortest paths on a polyhedron. In: Proc. 6th ACM Symposium on Computational Geometry, pp. 360–369 (1990); also IJCGA 6, 127–144 (1996)
Chiang, Y.-J., Mitchell, J.S.B.: Two-point euclidean shortest path queries in the plane. In: Proc. 10th ACM-SODA, Philadelphia, USA, pp. 215–224 (1999)
Djidjev, H.N.: Linear algorithms for graph separation problems. In: Karlsson, R., Lingas, A. (eds.) SWAT 1988. LNCS, vol. 318, pp. 643–645. Springer, Heidelberg (1988)
Frederickson, G.N.: Fast algorithms for shortest paths in planar graphs. SIAM J. Comput. 16, 1004–1022 (1987)
Gilbert, J.R., Hutchinson, J.P., Tarjan, R.E.: A separator theorem for graphs of bounded genus. J. Algorithms 5, 391–407 (1984)
Har-Peled, S.: Approximate shortest paths and geodesic diameters on convex polytopes in three dimensions. DCG 21, 216–231 (1999)
Har-Peled, S.: Constructing approximate shortest path maps in three dimensions. SIAM J. Comput. 28(4), 1182–1197 (1999)
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Aleksandrov, L., Djidjev, H.N., Guo, H., Maheshwari, A., Nussbaum, D., Sack, JR. (2006). Approximate Shortest Path Queries on Weighted Polyhedral Surfaces. In: Královič, R., Urzyczyn, P. (eds) Mathematical Foundations of Computer Science 2006. MFCS 2006. Lecture Notes in Computer Science, vol 4162. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11821069_9
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DOI: https://doi.org/10.1007/11821069_9
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