Spanners of Complete $k$-Partite Geometric Graphs

TL;DR

This paper presents efficient algorithms for constructing sparse spanners with low stretch factors in complete k-partite geometric graphs, establishing bounds on their optimality.

Contribution

The paper introduces two algorithms for creating low-stretch, sparse spanners in complete k-partite geometric graphs, with proven optimality bounds.

Findings

First algorithm produces a (5+ε)-spanner with O(n) edges in O(n log n) time.

Second algorithm produces a (3+ε)-spanner with O(n log n) edges in O(n log n) time.

Spanners with fewer than 3 stretch factor and O(n log n) edges do not exist for certain k ranges.

Abstract

We address the following problem: Given a complete $k$-partite geometric graph $K$ whose vertex set is a set of $n$ points in $\mathbb{R}^d$, compute a spanner of $K$ that has a ``small'' stretch factor and ``few'' edges. We present two algorithms for this problem. The first algorithm computes a $(5+\epsilon)$-spanner of $K$ with O(n) edges in $O(n \log n)$ time. The second algorithm computes a $(3+\epsilon)$-spanner of $K$ with $O(n \log n)$ edges in $O(n \log n)$ time. The latter result is optimal: We show that for any $2 \leq k \leq n - \Theta(\sqrt{n \log n})$, spanners with $O(n \log n)$ edges and stretch factor less than 3 do not exist for all complete $k$-partite geometric graphs.