An ¦ α § β ¨-spanner of an unweighted graph G is a subgraph H that approximates distances in G in the following sense. For any two vertices u § v: δH ¦ u § v¨� © αδG ¦ u § v¨� � β, where δG is the distance w.r.t. G. It is well known that there exist (multiplicative) ¦ 2k � 1 § 0 ¨-spanners of size O ¦ n 1 � 1 � k ¨ and that there exist (purely additive) ¦ 1 § 2 ¨-spanners of size O ¦ n 3 � 2 ¨. However no other ¦ 1 § O ¦ 1¨� ¨-spanners are known to exist. In this paper we develop a couple new techniques for constructing ¦ α § β ¨-spanners. The first result is a purely additive ¦ 1 § 6 ¨-spanner of size O ¦ n 4 � 3 ¨. Our 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. This general approach should lead to new spanner constructions. The second result is a truly simple linear time construction of ¦ k § k � 1 ¨-spanners with size O ¦ n 1 � 1 � k ¨. In a distributed network the algorithm terminates in a constant number of rounds and has expected size O ¦ n 1 � 1 � k ¨. The new idea here is primarily in the analysis of the construction. We show that a few simple and local rules for picking spanner edges induce seemingly coordinated global behavior.

For an unweighted undirected graph G = (V, E), and a pair of positive integers α ≥ 1, β ≥ 0, a subgraph G ′ = (V, H), H ⊆ E, is called an (α, β)-spanner of G if for every pair of vertices u, v ∈ V, distG ′(u, v) ≤ α · distG(u, v) + β. It was shown in [20] that for any ɛ> 0, κ = 1, 2,..., there exists an integer β = β(ɛ, κ) such that for every n-vertex graph G there exists a (1 + ɛ, β)-spanner G ′ with O(n 1+1/κ) edges. An efficient distributed protocol for constructing (1+ ɛ, β)-spanners was devised in [18]. The running time and the communication complexity of that protocol are O(n 1+ρ) and O(|E|n ρ), respectively, where ρ is an additional control parameter of the protocol that affects only the additive term β. In this paper we devise a protocol with a drastically improved running time (O(n ρ) as opposed to O(n 1+ρ)) for constructing (1 + ɛ, β)-spanners. Our protocol has the same communication complexity as the protocol of [18], and it constructs spanners with essentially the same properties as the spanners that are constructed by the protocol of [18]. We also show that our protocol for constructing (1+ɛ, β)spanners can be adapted to the streaming model, and devise a streaming algorithm that uses a constant number of passes and O(n 1+1/κ · log n) bits of space for computing allpairs-almost-shortest-paths of length at most by a multiplicative factor (1 + ɛ) and an additive term of β greater than the shortest paths. Our algorithm processes each edge in time O(n ρ), for an arbitrarily small ρ> 0. The only

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 δH(u, v) ≤ δG(u, v) + d. For ln n every k = O (), we construct a graph G on n vertices ln ln n for which any additive spanner of G with distortion 2k − 1 has Ω ( 1 k n1+1/k) edges. This matches the lower bound previously known only to hold under a 1963 conjecture of Erdös. 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 δG(u, v) ≤ δH(u, v) ≤ δG(u, v) + d. We show a lower bound of Ω ( 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 Ω ( 1 k n1+1/k−c(1+2/(k−1)) ) for addi-tive spanners and Ω ( 1 k 2 n 1+1/k−c(1+2/(k−1)) ) for emulators. For k = 2 these can be improved to Ω(n 3/2−c). This partially answers a question of Baswana et al (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.

Abstract. A spanner of an undirected unweighted graph is a subgraph that approximates the distance metric of the original graph with some specified accuracy. Specifically, we say H ⊆ G is an f-spanner of G if any two vertices u, v at distance d in G are at distance at most f(d) in H. There is clearly some tradeoff between the sparsity of H and the distortion function f, though the nature of this tradeoff is still poorly understood. In this paper we present a simple, modular framework for constructing sparse spanners that is based on interchangable components called connection schemes. By assembling connection schemes in different ways we can recreate the additive 2- and 6-spanners of Aingworth et al. and Baswana et al. and improve on the (1+ɛ, β)-spanners of Elkin and Peleg, the sublinear additive spanners of Thorup and Zwick, and the (non constant) additive spanners of Baswana et al. Our constructions rival the simplicity of all comparable algorithms and provide substantially better spanners, in some cases reducing the density doubly exponentially. 1

This paper presents compact roundtrip routing schemes with local tables of size Õ( √ n) and stretch 6 for any directed network with arbitrary edge weights; and with local tables of size Õ(ǫ−1 n 2/k) and stretch min((2 k/2 − 1)(k + ǫ),16k 2 + 8k − 8), for any directed network with polynomially-sized edges, both in the topology-independent node-name model. 1 These are the first topology-independent results that apply to routing in directed networks.

This paper concerns the efficient construction of sparse and low stretch spanners for unweighted arbitrary graphs with n nodes. All previous deterministic distributed algorithms, for constant stretch spanner of o(n²) edges, have a running time Ω(n^ɛ) for some constant ɛ > 0 depending on the stretch. Our deterministic distributed algorithms construct constant stretch spanners of o(n²) edges in o(n^ɛ) time for any constant ɛ > 0. More precisely, in the Linial’s free model, we construct in n O(1/ √ log n) time, for every graph, a 5-spanner of O(n 3/2) edges. The result is extended to O(k 2.322)-spanners with O(n 1+1/k) edges for every parameter k � 1. If the minimum degree of the graph is Ω(√n), then, in the same time complexity, a 9-spanner with O(n) edges can be constructed.

Given a directed graph G = (V, E) and an integer k ≥ 1, a k-transitive-closure-spanner (k-TC-spanner) of G is a directed graph H = (V, EH) that has (1) the same transitive-closure as G and (2) diameter at most k. Transitive-closure spanners are a common abstraction for applications in access control, property testing and data structures. We show a connection between 2-TC-spanners and local monotonicity reconstructors. A local monotonicity reconstructor, introduced by Saks and Seshadhri (SIAM Journal on Computing, 2010), is a randomized algorithm that, given access to an oracle for an almost monotone function f: [m] d → R, can quickly evaluate a related function g: [m] d → R which is guaranteed to be monotone. Furthermore, the reconstructor can be implemented in a distributed manner. We show that an efficient local monotonicity reconstructor implies a sparse 2-TC-spanner of the directed hypergrid (hypercube), providing a new technique for proving lower bounds for local monotonicity reconstructors. Our connection is,

Abstract. An (α, β)-spanner of a graph G is a subgraph H that approximates distances in G within a multiplicative factor α and an additive error β, ensuring that for any two nodes u, v, dH(u, v) ≤ α ·dG(u, v)+β. This paper concerns algorithms for the distributed deterministic construction of a sparse (α, β)-spanner H for a given graph G and distortion parameters α and β. It first presents a generic distributed algorithm that in constant number of rounds constructs, for every n-node graph and integer k ≥ 1, an (α, β)-spanner of O(βn 1+1/k) edges, where α and β are constants depending on k. For suitable parameters, this algorithm provides a (2k − 1, 0)-spanner of at most kn 1+1/k edges in k rounds, matching the performances of the best known distributed algorithm by Derbel et al. (PODC ’08). For k = 2 and constant ε> 0, it can also produce a (1+ε,2−ε)-spanner of O(n 3/2) edges in constant time. More interestingly, for every integer k> 1, it can construct in constant time a (1 + ε, O(1/ε) k−2)-spanner of O(ε −k+1 n 1+1/k) edges. Such deterministic

We present efficient algorithms for computing very sparse low distortion spanners in distributed networks and prove some non-trivial lower bounds on the tradeoff between time, sparseness, and distortion. All of our algorithms assume a synchronized distributed network, where relatively short messages may be communicated in each time step. Our first result is a fast distributed algorithm for finding an O(2 log ∗ n log n)-spanner with size O(n). Besides being nearly optimal in time and distortion, this algorithm appears to be the first that constructs an O(n)-size skeleton without requiring unbounded length messages or time proportional to the diameter of the network. Our second result is a new class of efficiently constructible (α, β)-spanners called Fibonacci spanners whose distortion improves with the distance being approximated. At their sparsest Fibonacci spanners can have nearly linear size O(n(log log n) φ) where φ = 1+ √ 5 2 is the golden ratio. As the distance increases the Fibonacci spanner’s multiplicative distortion passes through four discrete stages, moving from logarithmic to log-logarithmic, then into a period where it is constant, tending to 3, followed by another period tending to 1. On the lower bound side we prove that many recent sequential spanner constructions have no efficient counterparts in distributed networks, even if the desired distortion only needs to be achieved on the average or for a tiny fraction of the vertices. In particular, any distance preservers, purely additive spanners, or spanners with sublinear additive distortion must either be very dense, slow to construct, or have very weak guarantees on distortion. 1

Given a directed graph G = (V, E) and an integer k ≥ 1, a k-transitive-closure-spanner (k-TCspanner) of G is a directed graph H = (V, EH) that has (1) the same transitive-closure as G and (2) diameter at most k. Transitive-closure spanners were introduced in [7] as a common abstraction for applications in access control, property testing and data structures. In this work we study the number of edges in the sparsest 2-TC-spanners for the directed hypercube {0, 1} d and hypergrid {1, 2,..., m} d with the usual partial order, ≼, defined by: x1... xd ≼ y1...yd if and only if xi ≤ yi for all i ∈ {1,..., d}. We show that the number of edges in the sparsest 2-TCspanner of the hypercube is 2 cd+Θ(log d) , where c ≈ 1.1620. We also present upper and lower bounds on the size of the sparsest 2-TC-spanner of the directed hypergrid. Our first pair of upper and lower bounds for the hypergrid is in terms of an expression with binomial coefficients. The bounds differ by a factor of O(d 2m) and, in particular, give tight (up to a poly(d) factor) bounds for constant m. We also give a second set of bounds, which show that the number of edges in the sparsest 2-TC-spanner of the hypergrid is at most md log d ( { m and at least Ω max md logd m