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The greedy spanner is a high-quality spanner: its total weight, edge count and maximal degree are asymptotically optimal and in practice significantly better than for any other spanner with reasonable construction time. Unfortunately, all known algorithms that compute the greedy spanner on n points use Ω(n 2 ) space, which is impractical on large instances. To the best of our knowledge, the largest instance for which the greedy spanner was computed so far has about 13,000 vertices. We present a linear-space algorithm that computes the same spanner for points in R d running in O(n 2 log 2 n) time for any fixed stretch factor and dimension. We discuss and evaluate a number of optimizations to its running time, which allowed us to compute the greedy spanner on a graph with a million vertices. To our knowledge, this is also the first algorithm for the greedy spanner with a near-quadratic running time guarantee that has actually been implemented.

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The greedy spanner is a high-quality spanner: its total weight, edge count and maximal degree are asymptotically optimal and in practice significantly better than for any other spanner with reasonable construction time. Unfortunately, all known algorithms that compute the greedy spanner on n points use Ω(n 2 ) space, which is impractical on large instances. To the best of our knowledge, the largest instance for which the greedy spanner was computed so far has about 13,000 vertices. We present a linear-space algorithm that computes the same spanner for points in R d running in O(n 2 log 2 n) time for any fixed stretch factor and dimension. We discuss and evaluate a number of optimizations to its running time, which allowed us to compute the greedy spanner on a graph with a million vertices. To our knowledge, this is also the first algorithm for the greedy spanner with a near-quadratic running time guarantee that has actually been implemented.

The highest quality geometric spanner (e.g. in terms of edge count, both in theory and in practice) known to be computable in polynomial time is the greedy spanner. The state-of-the-art in computing this spanner are a O(n 2 log n) time, O(n 2 ) space algorithm and a O(n 2 log 2 n) time, O(n) space algorithm, as well as the 'improved greedy' algorithm, taking O(n 3 log n) time in the worst case and O(n 2 ) space but being faster in practice thanks to a caching strategy.We identify why this caching strategy gives speedups in practice. We formalize this into a framework and give a general efficiency lemma. From this we obtain many new time bounds, both on old algorithms and on new algorithms we introduce in this paper. Interestingly, our bounds are in terms of the well-separated pair decomposition, a data structure not actually computed by the caching algorithms.Specifically, we show that the 'improved greedy' algorithm has a O(n 2 log n log Φ) running time (where Φ is the spread of the point set) and a variation has a O(n 2 log 2 n) running time. We give a variation of the linear space stateof-the-art algorithm and an entirely new algorithm with a O(n 2 log n log Φ) running time, both of which improve its space usage by a factor O(1/(t − 1)), where t is the dilation of the spanner.We present experimental results comparing all the above algorithms. The experiments show that our new algorithm is much more space efficient than the existing linear space algorithm -up to 200 times when using low t -while being comparable in running time and much easier to implement. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from Permissions@acm.org. Categories and Subject Descriptors