Better Distance Preservers and Additive Spanners

Abstract

We make improvements to the upper bounds on several popular types of distance preserving graph sketches. These sketches are all various restrictions of the additive pairwise spanner problem, in which one is given an undirected unweighted graph G, a set of node pairs P , and an error allowance +β, and one must construct a sparse subgraphThe first part of our paper concerns pairwise distance preservers, which make the restriction β = 0 (i.e. distances must be preserved exactly). Our main result here is an upper bound of |H| = O(n 2/3 |P | 2/3 + n|P | 1/3 ) when G is undirected and unweighted. This improves on existing bounds whenever |P | = ω(n 3/4 ), and it is the first such improvement in the last ten years.We then devise a new application of distance preservers to graph clustering algorithms, and we apply this algorithm to subset spanners, which require P = S × S for some node subset S, and (standard) spanners, which require P = V × V . For both of these objects, our construction generalizes the best known bounds when the error allowance is constant, and we obtain the strongest polynomial error/sparsity tradeoff that has yet been reported (in fact, for subset spanners, ours is the first nontrivial construction that enjoys improved sparsity from a polynomial error allowance).We leave open a conjecture that O(n 2/3 |P | 2/3 + n) pairwise distance preservers are possible for undirected unweighted graphs. Resolving this conjecture in the affirmative would improve and simplify our upper bounds for all the graph sketches mentioned above. * gbodwin@stanford.edu †

Linear Size Distance Preservers

“…Perhaps surprisingly, Coppersmith and Elkin showed that nontrivial linear size distance preservers do indeed exist: in any undirected graph, one can preserve any p = O(n 1/2 ) pairwise distances using a subgraph on just O(n) edges. This unexpected fact has found several interesting applications: for example, Elkin and Pettie [26] used it to build the first linear-size log n stretch path reporting distance oracle, Bodwin and Vassilevska W. [13] have used it to build the current most accurate additive spanner of linear size, Pettie [38] used them as an ingredient in state-of-the-art constructions for mixed spanners, and they were employed by Elkin, Filtser, and Neiman [24] to build terminal subgraph spanners.…”

“…Figure 1: State of the art upper and lower bounds for pairwise distance preservers, for directed/undirected and weighted/unweighted graphs. The first upper bound for undirected unweighted graphs is due to [13], and the remaining bounds in this chart are all due to [19]. The hidden n o(1) factor in the unweighted lower bound has the form 2…”

“…The proof of Theorem 3.1 overcomes a somewhat surprising technical hurdle. It was shown in [13] that, if one uses standard "consistent" methods for breaking ties between equally short paths, then no theorem of this sort will be possible; instead, the lower bound on ω(p)-size distance preservers goes all the way up to o(n 2 ). In other words, any general construction of O(p) size preservers when p = o(n 2 ) (like the one offered by Theorem 3.1) must use some new unusual method of shortest path tiebreaking.…”

“…This line of work was implicit in many notable papers over the last 15 years, but it was first explicitly abstracted by Cygan, Grandoni, and Kavitha [20]. Later improved upper bounds came from Kavitha and Varma [31], Kavitha [30], and Bodwin and Williams [13], and new lower bounds came from Woodruff [41], Parter [32], and Abboud and Bodwin [1,2]. More generally, the field of (all pairs) spanners, in which all pairwise distances must be approximately preserved, has received quite a lot of research attention over the last few decades.…”

“…This notion is formalized as: Definition 3. ( [13]) Given a graph G, a shortest path tiebreaking scheme (or tiebreaking scheme for short) is a function π that maps ordered pairs of nodes (s, t) to a shortest path in G from s to t. When the underlying graph is clear from context, we will suppress the subscript G. For a set of node pairs P , we will commonly use π(P ) as a shorthand for…”