Sparse Temporal Spanners with Low Stretch

Abstract

A temporal graph is an undirected graph G = ( V, E ) along with a function λ : E → N + that assigns a time-label to each edge in E . A path in G such that the traversed time-labels are non-decreasing is called temporal path . Accordingly, 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 measured as the number of its edges. In this work, we study the size-stretch trade-offs of temporal spanners. In particular we show that temporal cliques always admit a temporal (2 k − 1) − spanner with (cid:101) O ( kn 1+ 1 k ) edges, where k > 1 is an integer parameter of choice. Choosing k = (cid:98) log n (cid:99) , we obtain a temporal O (log n )-spanner with (cid:101) O ( n ) edges that has almost the same size (up to logarithmic factors) as the temporal spanner given in [Casteigts et al., JCSS 2021] which only preserves temporal connectivity. We then turn our attention to 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 (cid:101) O ( n/ log(1 + ε )) edges suffice to obtain a stretch of (1 + ε ), for any small ε > 0. This result is essentially tight in the following sense: there are temporal graphs G for which any temporal subgraph preserving exact distances from a single-source must use Ω( n 2 ) edges. Interestingly enough, our analysis can be extended to the case of additive stretch for which we prove an upper bound of (cid:101) O ( n 2 /β ) on the size of any temporal β -additive spanner, which we show to be 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.