Euclidean spanners are important data structures in geometric algorithm design, because they provide a means of approximating the complete Eu-clidean graph with only O(n) edges, so that the shortest path length between each pair of points is not more than a constant factor longer than the Euclidean distance between the points. In many applications of… (More)
Given a set S of n points in the plane, we give an O(n log n)-time algorithm that constructs a plane t-spanner for S, with t ≈ 10, such that the degree of each point of S is bounded from above by 27, and the total edge length is proportional to the weight of a minimum spanning tree of S. Previously, no algorithms were known for constructing plane t-spanners… (More)
Given a triangulation G, whose vertex set V is a set of n points in the plane, and given a real number γ with 0 < γ < π, we design an O(n)-time algorithm that constructs a connected spanning subgraph G of G whose maximum degree is at most 14 + 2π/γ. If G is the Delaunay triangulation of V , and γ = 2π/3, we show that G is a t-spanner of V (for some constant… (More)
This is the preliminary version of a chapter that will appear in the Handbook on Computational Geometry, edited by J.-R. Sack and J. Urrutia. A comprehensive overview is given of algorithms and data structures for proximity problems on point sets in IR D. In particular, the closest pair problem, the exact and approximate post-ooce problem, and the problem… (More)
Given a set S of n points in the plane, a Manhattan network on S is a (not necessarily planar) rectilinear network G with the property that for every pair of points in S the network G contains a path between them whose length is equal to the Manhattan distance between the points. A Manhattan network on S can be thought of as a graph G = (V, E) where the… (More)
We show that the Yao graph Y 4 in the L 2 metric is a spanner with stretch factor 8 √ 2(29+ 23 √ 2). Enroute to this, we also show that the Yao graph Y ∞ 4 in the L∞ metric is a plane spanner with stretch factor 8.
We consider the problem of finding an obstacle-avoiding path between two points s and t in the plane, amidst a set of disjoint polyg-onal 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 Lv-metric. Such an approximate… (More)