# Sparse Temporal Spanners with Low Stretch

A temporal graph is an undirected graph G=(V,E) along with a function that assigns a time-label to each edge in E. A path in G with non-decreasing time-labels is called temporal path and the distance from u to v is the minimum length (i.e., the number of edges) of a temporal path from u to v. A temporal α-spanner of G is a (temporal) subgraph H that preserves the distances between any pair of vertices in V, up to a multiplicative stretch factor of α. The size of H is the number of its edges. In this work we study the size-stretch trade-offs of temporal spanners. We show that temporal cliques always admit a temporal (2k-1)-spanner with Õ(kn^1+1/k) edges, where k>1 is an integer parameter of choice. Choosing k=⌊log n⌋, we obtain a temporal O(log n)-spanner with Õ(n) edges that has almost the same size (up to logarithmic factors) as the temporal spanner in [Casteigts et al., JCSS 2021] which only preserves temporal connectivity. We then consider general temporal graphs. Since Ω(n^2) edges might be needed by any connectivity-preserving temporal subgraph [Axiotis et al., ICALP'16], we focus on approximating distances from a single source. We show that Õ(n/log(1+ε)) edges suffice to obtain a stretch of (1+ε), for any small ε>0. This result is essentially tight since there are temporal graphs for which any temporal subgraph preserving exact distances from a single-source must use Ω(n^2) edges. We extend our analysis to prove an upper bound of Õ(n^2/β) on the size of any temporal β-additive spanner, which is tight up to polylogarithmic factors. Finally, we investigate how the lifetime of G, i.e., the number of its distinct time-labels, affects the trade-off between the size and the stretch of a temporal spanner.

READ FULL TEXT