On Strong Diameter Padded Decompositions
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Given a weighted graph G=(V,E,w), a partition of V is Δ-bounded if the diameter of each cluster is bounded by Δ. A distribution over Δ-bounded partitions is a β-padded decomposition if every ball of radius γΔ is contained in a single cluster with probability at least e^-β·γ. The weak diameter of a cluster C is measured w.r.t. distances in G, while the strong diameter is measured w.r.t. distances in the induced graph G[C]. The decomposition is weak/strong according to the diameter guarantee. Formerly, it was proven that K_r free graphs admit weak decompositions with padding parameter O(r), while for strong decompositions only O(r^2) padding parameter was known. Furthermore, for the case of a graph G, for which the induced shortest path metric d_G has doubling dimension d, a weak O(d)-padded decomposition was constructed, which is also known to be tight. For the case of strong diameter, nothing was known. We construct strong O(r)-padded decompositions for K_r free graphs, matching the state of the art for weak decompositions. Similarly, for graphs with doubling dimension d we construct a strong O(d)-padded decomposition, which is also tight. We use this decomposition to construct (O(d),Õ(d))-sparse cover scheme for such graphs. Our new decompositions and cover have implications to approximating unique games, the construction of light and sparse spanners, and for path reporting distance oracles.
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