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STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Geographic routing without location information
Proceedings of the 9th annual international conference on Mobile computing and networking
The Geometric Dilation of Finite Point Sets
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Geometric Spanner Networks
Sparse geometric graphs with small dilation
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Light orthogonal networks with constant geometric dilation
Journal of Discrete Algorithms
Succinct Greedy Geometric Routing in the Euclidean Plane
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
Some Results on Greedy Embeddings in Metric Spaces
Discrete & Computational Geometry
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Discrete & Computational Geometry - Special Issue: 25th Annual Symposium on Computational Geometry; Guest Editor: John Hershberger
Improved upper bound on the stretch factor of delaunay triangulations
Proceedings of the twenty-seventh annual symposium on Computational geometry
Geometric Approximation Algorithms
Geometric Approximation Algorithms
Competitive routing in the half-θ6-graph
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
On succinct convex greedy drawing of 3-connected plane graphs
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Monotone drawings of graphs with fixed embedding
GD'11 Proceedings of the 19th international conference on Graph Drawing
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In this paper we introduce self-approaching graph drawings. A straight-line drawing of a graph is self-approaching if, for any origin vertex s and any destination vertex t, there is an st-path in the graph such that, for any point q on the path, as a point p moves continuously along the path from the origin to q, the Euclidean distance from p to q is always decreasing. This is a more stringent condition than a greedy drawing (where only the distance between vertices on the path and the destination vertex must decrease), and guarantees that the drawing is a 5.33-spanner. We study three topics: (1) recognizing self-approaching drawings; (2) constructing self-approaching drawings of a given graph; (3)constructing a self-approaching Steiner network connecting a given set of points. We show that: (1) there are efficient algorithms to test if a polygonal path is self-approaching in ℝ2 and ℝ3, but it is NP-hard to test if a given graph drawing in ℝ3 has a self-approaching uv-path; (2) we can characterize the trees that have self-approaching drawings; (3) for any given set of terminal points in the plane, we can find a linear sized network that has a self-approaching path between any ordered pair of terminals.