Congested Clique Algorithms for Graph Spanners
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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: (1) A randomized construction of a (2k-1)-spanner with O(n^1+1/k) edges in O( k) rounds. The previous best algorithm runs in O(k) rounds. (2) A deterministic construction of a (2k-1)-spanner with O(n^1+1/k) edges in O( k +( n)^3) rounds. The previous best algorithm runs in O(k n) rounds. This improvement is achieved by a new derandomization theorem for hitting sets which might be of independent interest. (3) A deterministic construction of a O(k)-spanner with O(k · n^1+1/k) edges in O( k) rounds.
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