Engineering an algorithm for constructing low-stretch geometric graphs with near-greedy average degrees

Abstract

We design and engineer Fast-Sparse-Spanner, a simple and practical (fast and memory-efficient) algorithm for constructing sparse low stretch factor geometric graphs on large pointsets in the plane. To our knowledge, this is the first practical algorithm to construct fast low stretch factor graphs on large pointsets with average degrees (hence, the number of edges) competitive with that of greedy spanners, the sparsest known class of Euclidean geometric spanners. Although theoretically not guaranteed to produce t-spanners, we always found in our rigorous experiments that Fast-Sparse-Spanner generated near-greedy size t-spanners.

To evaluate our implementation in terms of computation speed, memory usage, and quality of output, we performed extensive experiments with synthetic and real-world pointsets, and by comparing it to our closest competitor Bucketing, the fastest known greedy spanner algorithm for pointsets in the plane, devised by Alewijnse et al. (2017) [5]. Our experiment with constructing a 1.1-spanner on a large synthetic pointset with 128K points uniformly distributed within a square shows more than a 41-fold speedup with roughly a third of the memory usage of that of Bucketing, but with only a 3% increase in the average degree of the resulting graph. When ran on a pointset with a million points drawn from the same distribution, we observed a 130-fold speedup, with roughly a fourth of the memory usage of that of Bucketing, and just a 6% increase in the average degree. In terms of diameter, the graphs generated by Fast-Sparse-Spanner beat greedy spanners in most cases (have substantially lower diameter) while maintaining near-greedy average degree. Further, our algorithm can be easily parallelized to take advantage of parallel environments.

We share the implementations via GitHub for broader uses and future research.

GitHub repository. https://github.com/ghoshanirban/FSS.