Abstract
The greedy spanner is the highest quality geometric spanner (in e.g. edge count and weight, both in theory and practice) known to be computable in polynomial time. Unfortunately, all known algorithms for computing it on n points take varOmega (n^2) time, limiting its applicability on large data sets. We propose a novel algorithm design which uses the observation that for many point sets, the greedy spanner has many ‘short’ edges that can be determined locally and usually quickly. To find the usually few remaining ‘long’ edges, we use a combination of already determined local information and the well-separated pair decomposition. We give experimental results showing large to massive performance increases over the state-of-the-art on nearly all tests and real-life data sets. On the theoretical side we prove a near-linear expected time bound on uniform point sets and a near-quadratic worst-case bound. Our bound for point sets drawn uniformly and independently at random in a square follows from a local characterization of t-spanners we give on such point sets. We give a geometric property that holds with high probability, which in turn implies that if an edge set on these points has t-paths between pairs of points ‘close’ to each other, then it has t-paths between all pairs of points. This characterization gives an O(n log ^2 n log ^2 log n) expected time bound on our greedy spanner algorithm, making it the first subquadratic time algorithm for this problem on any interesting class of points. We also use this characterization to give an O((n + |E|) log ^2 n log log n) expected time algorithm on uniformly distributed points that determines whether E is a t-spanner, making it the first subquadratic time algorithm for this problem that does not make assumptions on E.
Highlights
A Euclidean graph on a set of n points in the Euclidean plane is a weighted graph with geometric distances as edge weights
If a shortest route in the graph is at most t times longer than the direct geometric distance between its endpoints, we say these endpoints have a t-path: a Euclidean graph is a t-spanner if all pairs of points have t-paths
A considerable amount of research has been done on the topic of spanners [15,20] since they were introduced in network design [21] and in geometry [10]
Summary
A Euclidean graph on a set of n points in the Euclidean plane is a weighted graph with geometric distances as edge weights. For any t > 1, we can efficiently find a t-spanner with O n t −1 edges in the Euclidean plane [20] These ‘approximations’ have very few edges compared to the complete Euclidean graph, while approximately maintaining distances. This makes them a useful tool in many areas. Bounded degree spanners are used in wireless network design [14], where for example points of high degree tend to have problems with interference. By using such a bounded degree spanner the problem of interference is reduced while the connectivity is maintained. Spanners have been used as components in various geometric and distributed algorithms
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