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)
Aimed at an audience of researchers and graduate students in computational geometry and algorithm design, this book uses the Geometric Spanner Network Problem to showcase a number of useful algorithmic techniques, data structure strategies, and geometric analysis techniques with many applications, practical and theoretical. The authors present rigorous… (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)
Bloom filters are a randomized data structure for membership queries dating back to 1970. Bloom filters sometimes give erroneous answers to queries, called false positives. Bloom analyzed the probability of such erroneous answers, called the false-positive rate, and Bloom's analysis has appeared in many publications throughout the years. We show that… (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.
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)