This paper studies the complexity of distributed construction of purely additive spanners in the CONGEST model. We describe algorithms for building such spanners in several cases. Because of the need to simultaneously make decisions at far apart locations, the algorithms use additional mechanisms compared to their sequential counterparts. We complement our algorithms with a lower bound on the number of rounds required for computing pairwise spanners. The standard reductions from set-disjointness and equality seem unsuitable for this task because no specific edge needs to be removed from the graph. Instead, to obtain our lower bound, we define a new communication complexity problem that reduces to computing a sparse spanner, and prove a lower bound on its communication complexity using information theory. This technique significantly extends the current toolbox used for obtaining lower bounds for the CONGEST model, and we believe it may find additional applications.
For unweighted graphs, one seeks spanners where the stretch is purely additive and as mentioned earlier, an almost tight bound of @math is known for how sparse a purely additive spanner can be. Bollob ' a @cite_31 were the first to study a variant of pairwise preservers called distance preservers , where the set of relevant pairs is @math , for a given parameter @math . Coppersmith and Elkin @cite_30 showed pairwise preservers of size @math and @math for any @math . For @math , the bound of @math for pairwise preservers has very recently been improved to @math by Bodwin and Williams @cite_49 .
@cite_30 @cite_31 @cite_49
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Given an undirected @math -node unweighted graph @math , a spanner with stretch function @math is a subgraph @math such that, if two nodes are at distance @math in @math , then they are at distance at most @math in @math . Spanners are very well studied in the literature. The typical goal is to construct the sparsest possible spanner for a given stretch function. In this paper we study pairwise spanners, where we require to approximate the @math - @math distance only for pairs @math in a given set @math . Such @math -spanners were studied before [Coppersmith,Elkin'05] only in the special case that @math is the identity function, i.e. distances between relevant pairs must be preserved exactly (a.k.a. pairwise preservers). Here we present pairwise spanners which are at the same time sparser than the best known preservers (on the same @math ) and of the best known spanners (with the same @math ). In more detail, for arbitrary @math , we show that there exists a @math -spanner of size @math with @math . Alternatively, for any @math , there exists a @math -spanner of size @math with @math . We also consider the relevant special case that there is a critical set of nodes @math , and we wish to approximate either the distances within nodes in @math or from nodes in @math to any other node. We show that there exists an @math -spanner of size @math with @math , and an @math -spanner of size @math with @math . All the mentioned pairwise spanners can be constructed in polynomial time. Let G=(V,E) be an undirected unweighted graph on n vertices. A subgraph H of G is called an (all-pairs) purely additive spanner with stretch β if for every (u,v)∈V×V, d i s t H (u,v)≤d i s t G (u,v) + β. The problem of computing sparse spanners with small stretch β is well-studied. Here we consider the following variant: we are given ( P V V ) and we seek a sparse subgraph H where d i s t H (u,v)≤d i s t G (u,v) + β for each ((u,v) P ). That is, distances for pairs outside ( P ) need not be well-approximated in H. Such a subgraph is called a pairwise spanner with additive stretch β and our goal is to construct such subgraphs that are sparser than all-pairs spanners with the same stretch. We show sparse pairwise spanners with additive stretch 4 and with additive stretch 6. We also consider the following special cases: ( P = S V ) and ( P = S T ), where S⊆V and T⊆V, and show sparser pairwise spanners for these cases. An (α,β)-spanner of an n-vertex graph G = (V,E) is a subgraph H of G satisfying that dist(u, v, H) ≤ α·dist(u, v, G) + β for every pair (u, v) ∈ V ×V, where dist(u,v,G′) denotes the distance between u and v in G′ ⊆ G. It is known that for every integer k ≥ 1, every graph G has a polynomially constructible (2k − 1,0)-spanner of size O(n 1 + 1 k ). This size-stretch bound is essentially optimal by the girth conjecture. Yet, it is important to note that any argument based on the girth only applies to adjacent vertices. It is therefore intriguing to ask if one can “bypass” the conjecture by settling for a multiplicative stretch of 2k − 1 only for neighboring vertex pairs, while maintaining a strictly better multiplicative stretch for the rest of the pairs. We answer this question in the affirmative and introduce the notion of k-hybrid spanners, in which non neighboring vertex pairs enjoy a multiplicative k stretch and the neighboring vertex pairs enjoy a multiplicative (2k − 1) stretch (hence, tight by the conjecture). We show that for every unweighted n-vertex graph G, there is a (polynomially constructible) k-hybrid spanner with O(k 2 ·n 1 + 1 k ) edges. This should be compared against the current best (α,β) spanner construction of [5] that obtains (k,k − 1) stretch with O(k ·n 1 + 1 k ) edges. An alternative natural approach to bypass the girth conjecture is to allow ourself to take care only of a subset of pairs S ×V for a given subset of vertices S ⊆ V referred to here as sources. Spanners in which the distances in S ×V are bounded are referred to as sourcewise spanners. Several constructions for this variant are provided (e.g., multiplicative sourcewise spanners, additive sourcewise spanners and more). An additive spanner of an unweighted undirected graph G with distortion d is a subgraph H such that for any two vertices u, v G, we have _H ( u,v ) = _G ( u,v ) + d. For every k = O ( n n ), we construct a graph G on n vertices for which any additive spanner of G with distortion 2k - 1 has ( 1 k n^ 1 + 1 k ) edges. This matches the lower bound previously known only to hold under a 1963 conjecture of Erdos. We generalize our lower bound in a number of ways. First, we consider graph emulators introduced by Dor, Halperin, and Zwick (FOCS, 1996), where an emulator of an unweighted undirected graph G with distortion d is like an additive spanner except H may be an arbitrary weighted graph such that _G ( u,v ) _H ( u,v ) _G ( u,v ) + d. We show a lower bound of ( 1 k^2 n^ 1 + 1 k ) edges for distortion-(2k - 1) emulators. These are the first non-trivial bounds for k 3. Second, we parameterize our bounds in terms of the minimum degree of the graph. Namely, for minimum degree n^ 1 k+c for any c 0, we prove a bound of ( 1 k n^ 1 + 1 k - c(1 + 2 (k - 1)) ) for additive spanners and ( 1 k^2 n^ 1 + 1 k - c(1 + 2 (k - 1)) ) for emulators. For k = 2 these can be improved to ( n^ 3 2 - c ). This partially answers a question of (SODA, 2005) for additive spanners. Finally, we continue the study of pair-wise and source-wise distance preservers defined by Coppersmith and Elkin (SODA, 2005) by considering their approximate variants and their relaxation to emulators. We prove the first lower bounds for such graphs.