TY - JOUR
T1 - Fast constructions of lightweight spanners for general graphs
AU - Elkin, Michael
AU - Solomon, Shay
N1 - Publisher Copyright:
© 2016 ACM.
PY - 2016/4
Y1 - 2016/4
N2 - It is long known that for every weighted undirected n-vertex m-edge graph G = (V, E, ω), and every integer k ≥ 1, there exists a ((2k- 1) · (1 + ϵ))-spanner with O(n1+1/k) edges and weight O(k · n1/k · ω(MST(G)), for an arbitrarily small constant ϵ > 0. (Here ω(MST(G)) stands for the weight of the minimum spanning tree of G.) To our knowledge, the only algorithms for constructing sparse and lightweight spanners for general graphs admit high running times. Most notable in this context is the greedy algorithm of Althöfer et al. [1993], analyzed by Chandra et al. [1992], which requires O(m· (n1+1/k + n · log n)) time. In this article, we devise an efficient algorithm for constructing sparse and lightweight spanners. Specifically, our algorithm constructs ((2k - 1) · (1 + ϵ))-spanners with O(k · n1+1/k) edges and weight O(k · n1/k) · ω(MST(G)), where ϵ > 0 is an arbitrarily small constant. The running time of our algorithm is O(k ·m+ min{n · log n,m· α(n)}). Moreover, by slightly increasing the running time we can reduce the other parameters. These results address an open problem by Roditty and Zwick [2004].
AB - It is long known that for every weighted undirected n-vertex m-edge graph G = (V, E, ω), and every integer k ≥ 1, there exists a ((2k- 1) · (1 + ϵ))-spanner with O(n1+1/k) edges and weight O(k · n1/k · ω(MST(G)), for an arbitrarily small constant ϵ > 0. (Here ω(MST(G)) stands for the weight of the minimum spanning tree of G.) To our knowledge, the only algorithms for constructing sparse and lightweight spanners for general graphs admit high running times. Most notable in this context is the greedy algorithm of Althöfer et al. [1993], analyzed by Chandra et al. [1992], which requires O(m· (n1+1/k + n · log n)) time. In this article, we devise an efficient algorithm for constructing sparse and lightweight spanners. Specifically, our algorithm constructs ((2k - 1) · (1 + ϵ))-spanners with O(k · n1+1/k) edges and weight O(k · n1/k) · ω(MST(G)), where ϵ > 0 is an arbitrarily small constant. The running time of our algorithm is O(k ·m+ min{n · log n,m· α(n)}). Moreover, by slightly increasing the running time we can reduce the other parameters. These results address an open problem by Roditty and Zwick [2004].
KW - Graph spanners
KW - Light spanners
KW - Sparse graphs
UR - http://www.scopus.com/inward/record.url?scp=84968735339&partnerID=8YFLogxK
U2 - 10.1145/2836167
DO - 10.1145/2836167
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AN - SCOPUS:84968735339
VL - 12
JO - ACM Transactions on Algorithms
JF - ACM Transactions on Algorithms
SN - 1549-6325
IS - 3
M1 - 29
ER -