The Norms of Graph Spanners

A $t$-spanner of a graph $G$ is a subgraph $H$ in which all distances are preserved up to a multiplicative $t$ factor. A classical result of Alth\"ofer et al. is that for every integer $k$ and every graph $G$, there is a $(2k-1)$-spanner of $G$ with at most $O(n^{1+1/k})$ edges. But for some settings the more interesting notion is not the number of edges, but the degrees of the nodes. This spurred interest in and study of spanners with small maximum degree. However, this is not necessarily a robust enough objective: we would like spanners that not only have small maximum degree, but also have "few" nodes of "large" degree. To interpolate between these two extremes, in this paper we initiate the study of graph spanners with respect to the $\ell_p$-norm of their degree vector, thus simultaneously modeling the number of edges (the $\ell_1$-norm) and the maximum degree (the $\ell_{\infty}$-norm). We give precise upper bounds for all ranges of $p$ and stretch $t$: we prove that the greedy $(2k-1)$-spanner has $\ell_p$ norm of at most $\max(O(n), O(n^{(k+p)/(kp)}))$, and that this bound is tight (assuming the Erd\H{o}s girth conjecture). We also study universal lower bounds, allowing us to give "generic" guarantees on the approximation ratio of the greedy algorithm which generalize and interpolate between the known approximations for the $\ell_1$ and $\ell_{\infty}$ norm. Finally, we show that at least in some situations, the $\ell_p$ norm behaves fundamentally differently from $\ell_1$ or $\ell_{\infty}$: there are regimes ($p=2$ and stretch $3$ in particular) where the greedy spanner has a provably superior approximation to the generic guarantee

Congested Clique Algorithms for Graph Spanners

Graph spanners are sparse subgraphs that faithfully preserve the distances in the original graph up to small stretch. Spanner have been studied extensively as they have a wide range of applications ranging from distance oracles, labeling schemes and routing to solving linear systems and spectral sparsification. A k-spanner maintains pairwise distances up to multiplicative factor of k. It is a folklore that for every n-vertex graph G, one can construct a (2k-1) spanner with O(n^{1+1/k}) edges. In a distributed setting, such spanners can be constructed in the standard CONGEST model using O(k^2) rounds, when randomization is allowed. In this work, we consider spanner constructions in the congested clique model, and show: - a randomized construction of a (2k-1)-spanner with O~(n^{1+1/k}) edges in O(log k) rounds. The previous best algorithm runs in O(k) rounds; - a deterministic construction of a (2k-1)-spanner with O~(n^{1+1/k}) edges in O(log k +(log log n)^3) rounds. The previous best algorithm runs in O(k log n) rounds. This improvement is achieved by a new derandomization theorem for hitting sets which might be of independent interest; - a deterministic construction of a O(k)-spanner with O(k * n^{1+1/k}) edges in O(log k) rounds