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**11 - 14**of**14**### Editors: Natacha Portier and Thomas Wilke

"... Abstract Given an undirected n-node unweighted graph G = (V, E), a spanner with stretch function f (·) is a subgraph H ⊆ G such that, if two nodes are at distance d in G, then they are at distance at most f (d) in H. Spanners are very well studied in the literature. The typical goal is to construct ..."

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Abstract Given an undirected n-node unweighted graph G = (V, E), a spanner with stretch function f (·) is a subgraph H ⊆ G such that, if two nodes are at distance d in G, then they are at distance at most f (d) in H. 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 u-v distance only for pairs (u, v) in a given set P ⊆ V × V . Such P-spanners were studied before [Coppersmith,Elkin'05] only in the special case that f (·) 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 P) and of the best known spanners (with the same f (·)). In more detail, for arbitrary P, we show that there exists a P-spanner of size O(n(|P| log n) 1/4 ) with Alternatively, for any ε > 0, there exists a P-spanner of size O(n|P| We also consider the relevant special case that there is a critical set of nodes S ⊆ V , and we wish to approximate either the distances within nodes in S or from nodes in S to any other node. We show that there exists an (S × S)-spanner of size O(n |S|) with f (d) = d + 2, and an (S × V )-spanner of size O(n |S| log n) with f (d) = d + 2 log n. All the mentioned pairwise spanners can be constructed in polynomial time. ACM Subject Classification G.2.2 Graphs Algorithms Keywords and phrases Introduction Let G = (V, E) be an undirected unweighted graph. A subgraph H of G is a spanner with stretch function f (·) if, given any two nodes s, t ∈ V at distance δ G (s, t) in G, the distance δ H (s, t) between the same two nodes in H is at most f (δ G (α and β are the multiplicative stretch and additive stretch of the spanner, respectively). If β = 0 the spanner is called multiplicative, and if α = 1 the spanner is called purely-additive. Spanners are very well studied in the literature (see Section 1.2). The typical goal is to achieve the sparsest possible spanner for a given stretch function f (·)

### Very Sparse Additive Spanners and Emulators

, 2015

"... We obtain new upper bounds on the additive distortion for graph emulators and spanners on relatively few edges. We introduce a new subroutine called “strip creation,” and we combine this subroutine with several other ideas to obtain the following results: 1. Every graph has a spanner on O(n1+) edges ..."

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We obtain new upper bounds on the additive distortion for graph emulators and spanners on relatively few edges. We introduce a new subroutine called “strip creation,” and we combine this subroutine with several other ideas to obtain the following results: 1. Every graph has a spanner on O(n1+) edges with Õ(n1/2−/2) additive distortion. 2. Every graph has an emulator on Õ(n1+) edges with Õ(n1/3−2/3) additive distortion whenever ∈ [0, 1 5 3. Every graph has a spanner on Õ(n1+) edges with Õ(n2/3−5/3) additive distortion whenever ∈ [0, 1 4 Our first spanner has the new best known asymptotic edge-error tradeoff for additive spanners whenever ∈ [0, 1 7 Our second spanner has the new best tradeoff whenever

### Bypassing Erdős’ Girth Conjecture: Hybrid Stretch and Sourcewise Spanners

, 2014

"... 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 co ..."

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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(n1+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 neigh-boring vertex pairs enjoy a multiplicative (2k − 1) stretch (hence, tight by the conjecture). We show that for every unweighted n-vertex graph G with m edges, there is a (polynomially constructible) k-hybrid spanner with O(k2 · n1+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 · n1+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).