On sparse spanners of weighted graphs
Discrete & Computational Geometry
Approximating s-t minimum cuts in Õ(n2) time
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Spanners and emulators with sublinear distance errors
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Ramsey partitions and proximity data structures
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Approximate distance oracles for unweighted graphs in expected O(n2) time
ACM Transactions on Algorithms (TALG)
Streaming algorithm for graph spanners---single pass and constant processing time per edge
Information Processing Letters
Graph sparsification by effective resistances
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Proceedings of the forty-first annual ACM symposium on Theory of computing
A simple linear time algorithm for computing a (2k - 1)-spanner of o(n1+1/k) size in weighted graphs
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
Additive spanners and (α, β)-spanners
ACM Transactions on Algorithms (TALG)
Approaching Optimality for Solving SDD Linear Systems
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
A general framework for graph sparsification
Proceedings of the forty-third annual ACM symposium on Theory of computing
How user behavior is related to social affinity
Proceedings of the fifth ACM international conference on Web search and data mining
Spectral sparsification of graphs: theory and algorithms
Communications of the ACM
Hi-index | 0.02 |
In this paper we introduce a new notion of distance between nodes in a graph that we refer to as robust connectivity. Robust connectivity between a pair of nodes u and v is parameterized by a threshold k and intuitively captures the number of paths between u and v of length at most k. Using this new notion of distances, we show that any black box algorithm for constructing a spanner can be used to construct a spectral sparsifier. We show that given an undirected weighted graph G, simply taking the union of spanners of a few (polylogarithmically many) random subgraphs of G obtained by sampling edges at different probabilities, after appropriate weighting, yields a spectral sparsifier of G. We show how this be done in Õ(m) time, producing a sparsifier with Õ(n/ε2) edges. While the cut sparsifiers of Benczur and Karger are based on weighting edges according to (inverse) strong connectivity, and the spectral sparsifiers are based on resistance, our method weights edges using the robust connectivity measure. The main property that we use is that this new measure is always greater than the resistance when scaled by a factor of O(k) (k is chosen to be O(log n)), but, just like resistance and connectivity, has a bounded sum, i.e. Õ(n), over all the edges of the graph.