TY - GEN

T1 - The norms of graph spanners

AU - Chlamtáč, Eden

AU - Dinitz, Michael

AU - Robinson, Thomas

N1 - Publisher Copyright: © Eden Chlamtáč, Michael Dinitz, and Thomas Robinson; licensed under Creative Commons License CC-BY

PY - 2019/7/1

Y1 - 2019/7/1

N2 - 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 Althöfer et al. is that for every integer k and every graph G, there is a (2k − 1)-spanner of G with at most O(n1+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 `p-norm of their degree vector, thus simultaneously modeling the number of edges (the `1-norm) and the maximum degree (the `∞-norm). We give precise upper bounds for all ranges of p and stretch t: we prove that the greedy (2k− 1)-spanner has `p norm of at k+p most max(O(n), O(n kp )), and that this bound is tight (assuming the Erdős 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 `1 and `∞ norm. Finally, we show that at least in some situations, the `p norm behaves fundamentally differently from `1 or `∞: there are regimes (p = 2 and stretch 3 in particular) where the greedy spanner has a provably superior approximation to the generic guarantee.

AB - 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 Althöfer et al. is that for every integer k and every graph G, there is a (2k − 1)-spanner of G with at most O(n1+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 `p-norm of their degree vector, thus simultaneously modeling the number of edges (the `1-norm) and the maximum degree (the `∞-norm). We give precise upper bounds for all ranges of p and stretch t: we prove that the greedy (2k− 1)-spanner has `p norm of at k+p most max(O(n), O(n kp )), and that this bound is tight (assuming the Erdős 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 `1 and `∞ norm. Finally, we show that at least in some situations, the `p norm behaves fundamentally differently from `1 or `∞: there are regimes (p = 2 and stretch 3 in particular) where the greedy spanner has a provably superior approximation to the generic guarantee.

KW - Approximations

KW - Spanners

UR - http://www.scopus.com/inward/record.url?scp=85069198114&partnerID=8YFLogxK

U2 - https://doi.org/10.4230/LIPIcs.ICALP.2019.40

DO - https://doi.org/10.4230/LIPIcs.ICALP.2019.40

M3 - Conference contribution

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019

A2 - Baier, Christel

A2 - Chatzigiannakis, Ioannis

A2 - Flocchini, Paola

A2 - Leonardi, Stefano

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019

Y2 - 9 July 2019 through 12 July 2019

ER -