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.
There are several algorithms for computing multiplicative and additive spanners in weighted and unweighted graphs. In unweighted graphs, for any integer @math , Halperin and Zwick @cite_1 gave a linear time algorithm to compute a multiplicative @math -spanner of size @math , where @math is the number of vertices. Note that for @math one obtains a spanner with multiplicative stretch @math and with size @math : we will use this type of spanner in Theorem . Analogous results are also known for weighted graphs @cite_10 @cite_18 @cite_13 .
@cite_13 @cite_10 @cite_18 @cite_1
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An (α, β)-spanner of an unweighted graph G is a subgraph H that distorts distances in G up to a multiplicative factor of α and an additive term β. It is well known that any graph contains a (multiplicative) (2k−1, 0)-spanner of size O(n1+1 k) and an (additive) (1,2)-spanner of size O(n3 2). However no other additive spanners are known to exist. In this article we develop a couple of new techniques for constructing (α, β)-spanners. Our first result is an additive (1,6)-spanner of size O(n4 3). The construction algorithm can be understood as an economical agent that assigns costs and values to paths in the graph, purchasing affordable paths and ignoring expensive ones, which are intuitively well approximated by paths already purchased. We show that this path buying algorithm can be parameterized in different ways to yield other sparseness-distortion tradeoffs. Our second result addresses the problem of which (α, β)-spanners can be computed efficiently, ideally in linear time. We show that, for any k, a (k,k−1)-spanner with size O(kn1+1 k) can be found in linear time, and, further, that in a distributed network the algorithm terminates in a constant number of rounds. Previous spanner constructions with similar performance had roughly twice the multiplicative distortion. 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 @math -spanner of a graph G is a subgraph H such that @math for every pair of vertices u,w, where distG'(u,w) denotes the distance between two vertices u and v in G'. It is known that every graph G has a polynomially constructible @math -spanner (also known as multiplicative @math -spanner) of size @math for every integer @math , and a polynomially constructible (1,2)-spanner (also known as additive 2-spanner) of size @math . This paper explores hybrid spanner constructions (involving both multiplicative and additive factors) for general graphs and shows that the multiplicative factor can be made arbitrarily close to 1 while keeping the spanner size arbitrarily close to O(n), at the cost of allowing the additive term to be a sufficiently large constant. More formally, we show that for any constant @math there exists a constant @math such that for every @math -vertex graph G there is an efficiently constructible @math -spanner of size @math . Let G(V, E) be an undirected weighted graph with |V| = n, and |E| = m. A t-spanner of the graph G(V, E) is a sub-graph G(V, ES) such that the distance between any pair of vertices in the spanner is at most t times the distance between the two in the given graph. A 1963 girth conjecture of Erdos implies that Ω(n1+1 k) edges are required in the worst case for any (2k - 1)-spanner, which has been proved for k = 1, 2, 3, 5. There exist polynomial time algorithms that can construct spanners with the size that matches this conjectured lower bound, and the best known algorithm takes O(mn1 k) expected running time. In this paper, we present an extremely simple linear time randomized algorithm that constructs (2k - 1)-spanner of size matching the conjectured lower bound. Our algorithm requires local information for computing a spanner, and thus can be adapted suitably to obtain efficient distributed and parallel algorithms.