Computing the Greedy Spanner in Linear Space

TL;DR

This paper introduces a linear-space algorithm for computing the greedy spanner in Euclidean space, enabling the construction of high-quality spanners on much larger datasets than previously possible.

Contribution

The authors develop the first linear-space algorithm for the greedy spanner with a practical implementation that handles up to a million vertices.

Findings

Achieved the first implementation of a greedy spanner algorithm with near-quadratic running time.

Successfully computed greedy spanners for graphs with up to one million vertices.

Demonstrated significant improvements over previous algorithms in terms of space and scalability.

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 of n points use Omega(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 O(n)-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.