Better Distance Preservers and Additive Spanners

Wemake improvements to the current upper bounds on pairwise distance preservers and additive spanners. A distance preserver is a sparse subgraph that exactly preserves all distances in a given pair set P . We show that every undirected unweighted graph has a distance preserver on O(max{n|P |, n|P |}) edges, and we conjecture that O(n|P |+ n) is possible. An additive subset spanner is a sparse subgraph that preserves all distances in S×S for a node subset S up to a small additive error function. Our second contribution is a new application of distance preservers to graph clustering algorithms, and an application of this clustering algorithm to produce new subset spanners. Ours are the first subset spanners that benefit from a non-constant error allowance. For constant d, we show that subset spanners with +n error can be obtained at any of the following sparsity levels: 1. O(|S|n + n) 2. O(|S|n + n), so long as |S| ≥ Θ(n) 3. O(|S|n), so long as |S| ≤ O(n) 4. O(|S|n + n) under our distance preserver conjecture An additive spanner is a subset spanner on V × V . The existence of error sensitive subset spanners was an open problem that formed a bottleneck in additive spanner constructions. By resolving this problem, we are able to prove that for any constant d, there are additive spanners with +n error at any of the following sparsity levels: 1. O(n + n) 2. O(n + n) so long as d ≤ 3/13 3. O(n) so long as d ≥ 3/13 4. O(n + n) under our distance preserver conjecture If our distance preserver conjecture is true, then the fourth additive spanner is the best known for the entire range d ∈ (0, 3/7]. Otherwise, the first is the best known for d ∈ [1/3, 3/7], the second is the best known for d ∈ [3/13, 1/3], and and the third is the best known for d ∈ (0, 3/13]. As an auxiliary result, we prove that all graphs have +6 pairwise spanners on Õ(n|P |) edges. [email protected] [email protected]