Abstract
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 $$n$$n points use $$\varOmega (n^2)$$Ω(n2) 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 $$\mathbb {R}^d$$Rd running in $$O(n^2 \log ^2 n)$$O(n2log2n) 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.
Highlights
A t-spanner on a set of points, usually in the Euclidean plane, is a graph on these points that is a ‘t-approximation’ of the complete graph, in the sense that shortest routes in the graph are at most t times longer than the direct geometric distance
There exists a large number of constructions of t-spanners that can be parameterized with arbitrary t > 1. They have different strengths and weaknesses: some are fast to construct but of low quality (Θ-graph, which has no guarantees on its total weight), others are slow to construct but of high quality, some have an extremely low diameter and some are fast to construct in higher dimensions
We tried using the same range trees to perform the optimization of the previous paragraph only to well-separated pairs ‘close by’ our current well-separated pair. Both optimizations turned out to give a speed increase and in particular the second retained most of its effectiveness even though we only tried it on close-by pairs, but the overhead of range trees was vastly greater than the gain—in particular the space usage of range trees made the algorithm use about as much space as the original greedy algorithms
Summary
A t-spanner on a set of points, usually in the Euclidean plane, is a graph on these points that is a ‘t-approximation’ of the complete graph, in the sense that shortest routes in the graph are at most t times longer than the direct geometric distance. The greedy spanner is one of the first spanner algorithms that was considered, and it has been subject to a considerable amount of research regarding its properties and more recently regarding computing it efficiently This line of research resulted in a O(n2 log n) algorithm [2] for metric spaces of bounded doubling dimension (and for Euclidean spaces). Among the many spanner algorithms known, the greedy spanner is of special interest because of its exceptional quality: its size, weight and degree are asymptotically optimal, and in practice better than those of any other spanner construction algorithms with reasonable running times It produces spanners with about ten times as few edges, twenty times smaller total weight and six times smaller maximum degree than its closest wellknown competitor, the Θ-graph, on uniform pointsets.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
