# Planar spanners and approximate shortest path queries among obstacles in the plane

## Abstract

We consider the problem of finding an obstacle-avoiding path between two points *s* and *t* in the plane, amidst a set of disjoint polygonal obstacles with a total of *n* vertices. The length of this path should be within a small constant factor *c* of the length of the shortest possible obstacle-avoiding *s-t* path measured in the *L*_{ p }-metric. Such an approximate shortest path is called a *c-short path*, or a short path with *stretch factor c*. The goal is to preprocess the obstacle-scattered plane by creating an efficient data structure that enables fast reporting of a *c*-short path (or its length). In this paper, we give a family of algorithms for the above problem that achieve an interesting trade-off between the stretch factor, the query time and the preprocessing bounds. Our main results are algorithms that achieve logarithmic length query time, after subquadratic time and space preprocessing.

## Keywords

Short Path Planar Graph Short Path Problem Path Query Short Path Tree## Preview

Unable to display preview. Download preview PDF.

## References

- 1.S. Arya, G. Das, D.M. Mount, J.S. Salowe and M. Smid, “Euclidean spanners: short, thin, and lanky”,
*Proc. 27th ACM STOC*, 1995, pp. 489–498.Google Scholar - 2.S. Arya, D.M. Mount, N.S. Netanyahu, R. Silverman and A. Wu, “An optimal algorithm for approximate nearest neighbor searching”,
*Proc. 5th ACM-SIAM Symp. on Discrete Algorithms*, 1994, pp. 573–582.Google Scholar - 3.M. J. Atallah and D. Z. Chen, “Parallel rectilinear shortest paths with rectangular obstacles”,
*Comp. Geometry: Theory and Appl.*, 1:2 (1991), pp.79–113.Google Scholar - 4.M. J. Atallah and D. Z. Chen, “On parallel rectilinear obstacle-avoiding paths”,
*Computational Geometry: Theory and Applications*, 3 (1993), pp. 307–313.Google Scholar - 5.D. Z. Chen. “On the all-pairs Euclidean short path problem”,
*Proc. 6th Annual ACM-SIAM Symp. on Discrete Algorithms*, San Francisco, 1995, pp. 292–301.Google Scholar - 6.D. Z. Chen and K. S. Klenk, “Rectilinear short path queries among rectangular obstacles”,
*Proc. 7th Can. Conf. on Comp. Geometry*, 1995, pp. 169–174.Google Scholar - 7.D. Z. Chen, K. S. Klenk, and H.-Y. T. Tu, “Rectilinear shortest path queries among weighted obstacles in the rectilinear plane”,
*Proc. 11th Annual ACM Symp. on Computational Geometry*, 1995, pp. 370–379.Google Scholar - 8.L. P. Chew, “Constrained Delaunay triangulations”,
*Algorithmica*, 4 (1989), pp. 97–108.CrossRefGoogle Scholar - 9.L. P. Chew, “There are planar graphs almost as good as the complete graph”,
*J. of Computer and System Sciences*, 39 (1989), pp. 205–219.Google Scholar - 10.L. P. Chew, “Planar graphs and sparse graphs for efficient motion planning in the plane”, Computer Science Tech Report, PCS-TR90-146, Dartmouth College.Google Scholar
- 11.L. P. Chew and R. L. Drysdale, “Voronoi diagrams based on convex distance functions”,
*Proc. 1st Annual ACM Symp. on Comp. Geometry*, 1985, pp. 235–244.Google Scholar - 12.Y.-J. Chiang, F. P. Preparata, and R. Tamassia, “A unified approach to dynamic point location, ray shooting, and shortest paths in planar maps”,
*Proc. of the 4th ACM-SIAM Symp. on Discrete Algorithms*, 1993, pp. 44–53.Google Scholar - 13.K. L. Clarkson, “Approximation algorithms for shortest path motion planning”,
*Proc. 19th Annual ACM Symp. Theory of Computing*, 1987, pp. 56–65.Google Scholar - 14.H. Djidjev, G. Pantziou and C. Zaroliagis, “On-line and dynamic algorithms for shortest path problems”,
*Proc. 12th Symp. on Theor. Aspects of Comp. Sc.*, LNCS 900, Springer-Verlag, 1995, pp. 193–204.Google Scholar - 15.D. P. Dobkin, S. J. Friedman, and K. J. Supowit, “Delaunay graphs are almost as good as complete graphs”,
*Discrete & Comp. Geometry*, 5 (1990), pp. 399–407.Google Scholar - 16.H. ElGindy and P. Mitra, “Orthogonal shortest route queries among axes parallel rectangular obstacles”,
*Int. J. of Comp. Geometry and Appl.*, 4 (1) (1994), 3–24.Google Scholar - 17.G. Frederickson, “Fast algorithms for shortest paths in planar graphs, with applications”,
*SIAM J. on Computing*, 16 (1987), pp.1004–1022.Google Scholar - 18.G.N. Frederickson, “Using cellular graph embeddings in solving all pairs shortest path problems”,
*J. of Algorithms*, 19 (1995), pp. 45–85.Google Scholar - 19.M. Goodrich, “Planar separators and parallel polygon triangulation”,
*Proc. 24th ACM Symp. on Theory of Comp.*, 1992, pp.507–516.Google Scholar - 20.M.T. Goodrich and R. Tamassia, “Dynamic ray shooting and shortest paths via balanced geodesic triangulations”,
*Proc. 9th Annual ACM Symp. on Computational Geometry*, 1993, pp. 318–327.Google Scholar - 21.L.J. Guibas and J. Hershberger, “Optimal shortest path queries in a simple polygon”,
*Proc. 3rd Annual ACM Symp. on Computational Geometry*, 1987, pp. 50–63.Google Scholar - 22.L.J. Guibas, J. Hershberger, D. Leven, M. Sharir, and R.E. Tarjan, “Linear time algorithms for visibility and shortest paths problems inside triangulated simple polygons”,
*Algorithmica*, 2 (1987), pp. 209–233.CrossRefGoogle Scholar - 23.P. Klein, S. Rao, M. Rauch and S. Subramanian, “Faster shortest-path algorithms for planar graphs”,
*Proc. 26th ACM Symp. on Theory of Comp.*, 1994, pp.27–37.Google Scholar - 24.R. Lipton and R. Tarjan, “A separator theorem for planar graphs”,
*SIAM J. Applied Mathematics*, 36 (1979), pp.177–189.Google Scholar - 25.B. Schieber and U. Vishkin, “On finding lowest common ancestors: Simplification and parallelization”,
*SIAM J. Computing*, 17:6 (1988), pp.1253–1262.Google Scholar