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

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. 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. (Algorithmica, 2017). We always found that Fast-Sparse-Spanner generated near-greedy t-spanners while being fast and memory-efficient. 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. 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. As a byproduct of our research, we design and engineer Fast-Stretch-Factor, a practical parallelizable algorithm that can measure the stretch-factor of any graph generated by Fast-Sparse-Spanner. Our experiments show that it is much faster than the naive Dijkstra-based stretch-factor measurement algorithm.