Experimental Study of Geometric t-Spanners

Farshi, Mohammad; Gudmundsson, Joachim

doi:10.1007/11561071_50

Cited by 19 publications

(38 citation statements)

References 15 publications

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“…Similar to [7] we have used three kinds of distributions from which we draw our points: a uniform distribution, a gamma distribution with shape parameter 0.75, and a distribution consisting of √ n uniformly distributed pointsets of √ n uniformly distributed points. The results from the gamma distribution were nearly identical to those of the uniform pointset, so we did not include them.…”

Section: Resultsmentioning

confidence: 99%

“…With the four optimizations described in the following sections, the algorithm attains running times that are a small constant slower than the algorithm introduced in [7] called FG-greedy, which is considered the fastest practical algorithm known in literature.…”

Section: Making the Algorithm Practicalmentioning

confidence: 99%

“…This line of research resulted in a O(n 2 log n) algorithm [2] for metric spaces of bounded doubling dimension (and therefore also for Euclidean spaces). A different algorithm which has proven to work well in practice is the so called FG-Greedy algorithm introduced in [7]. Although a Θ(n 3 log n) bound was proven for this algorithm, experiments show that it tends to have near-quadratic behaviour in practice.…”

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confidence: 99%

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Computing the Greedy Spanner in Linear Space

et al. 2015

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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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Section: Resultsmentioning

confidence: 99%

Section: Making the Algorithm Practicalmentioning

confidence: 99%

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confidence: 99%

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Computing the Greedy Spanner in Linear Space

et al. 2015

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“…O(n 3 log n) time, O(n 2 ) space algorithm that works well in practice [6] (time bounds assume fixed t). While the linear space algorithm allows us to compute greedy spanners on much larger sets than before -quadratic space usage limits us to about 13.000 points -the space usage of the algorithm is still quite large (especially for low t) and the algorithm requires extensive tweaking to have acceptable performance.…”

Section: Introductionmentioning

confidence: 99%

“…We present a generalized argument in this paper in the form of a framework for analyzing greedy spanner algorithms. We use this framework to obtain time bounds for the Improved-Greedy algorithm [6] (which also uses a caching strategy) and a variation on that algorithm for which no cubic worst-cases are known, explaining its performance in practice and providing upper bounds that match known lower bounds [2]. The framework generalizes the (ad-hoc) arguments used to prove that the state-of-the-art sub-cubic-time algorithms are nearquadratic.…”

Section: Introductionmentioning

confidence: 99%

A Framework for Computing the Greedy Spanner

Bouts

Brink

Buchin

2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

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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

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