MPVX15 devised a distributed They actually showed a PRAM algorithm. The distributed algorithm with these properties is implicit in MPVX15 . algorithm in the CONGEST model, that given a parameter @math , constructs an @math -spanner of an input unweighted @math -vertex graph with @math expected edges in @math rounds of communication. In this paper we improve the result of MPVX15 , by showing a @math -round distributed algorithm in the same model, that constructs a @math -spanner with @math edges, with probability @math , for any @math . Moreover, when @math , our algorithm produces (still in @math rounds) ultra-sparse spanners, i.e., spanners of size @math , with probability @math . To our knowledge, this is the first distributed algorithm in the CONGEST or in the PRAM models that constructs spanners or skeletons (i.e., connected spanning subgraphs) that sparse. Our algorithm can also be implemented in linear time in the standard centralized model, and for large @math , it provides spanners that are sparser than any other spanner given by a known (near-)linear time algorithm. We also devise improved bounds (and algorithms realizing these bounds) for @math -spanners and emulators. In particular, we show that for any unweighted @math -vertex graph and any @math , there exists a @math -emulator with @math edges. All previous constructions of @math -spanners and emulators employ a superlinear number of edges, for all choices of parameters. Finally, we provide some applications of our results to approximate shortest paths' computation in unweighted graphs.
Efficient algorithms for constructing @math -spanners were also devised in @cite_50 @cite_43 @cite_47 @cite_35 @cite_24 . These algorithms are based, however, on different approaches than that of the current paper. The latter is based on @cite_6 . Specifically, the approach of @cite_50 @cite_43 is based on @cite_21 @cite_28 construction of pairwise covers and hopsets, i.e., the algorithm works top-down. It recurses in small clusters, and eliminates large ones. The approach of @cite_47 @cite_35 @cite_24 is based on @cite_61 collection of trees, used originally for distance oracles. Streaming algorithms for constructing multiplicative spanners were given in @cite_9 @cite_27 @cite_16 , and near-additive spanners in @cite_50 @cite_43 . Spanners and emulators with sublinear error were given in @cite_47 @cite_35 . Spanners with purely additive error and lower bounds concerning them were given in @cite_7 @cite_6 @cite_12 @cite_3 @cite_52 @cite_26 @cite_1 @cite_8 .
@cite_35 @cite_3 @cite_43 @cite_8 @cite_21 @cite_52 @cite_26 @cite_7 @cite_28 @cite_6 @cite_27 @cite_50 @cite_16 @cite_12 @cite_61 @cite_9 @cite_1 @cite_24 @cite_47
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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(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). A spanner is a sparse subgraph that approximately preserves the pairwise distances of the original graph. It is well known that there is a smooth tradeoff between the sparsity of a spanner and the quality of its approximation, so long as distance error is measured multiplicatively. A central open question in the field is to prove or disprove whether such a tradeoff exists also in the regime of additive error. That is, is it true that for all e>0, there is a constant ke such that every graph has a spanner on O(n1+e) edges that preserves its pairwise distances up to +ke? Previous lower bounds are consistent with a positive resolution to this question, while previous upper bounds exhibit the beginning of a tradeoff curve: all graphs have +2 spanners on O(n3 2) edges, +4 spanners on O(n7 5) edges, and +6 spanners on O(n4 3) edges. However, progress has mysteriously halted at the n4 3 bound, and despite significant effort from the community, the question has remained open for all 0 Our main result is a surprising negative resolution of the open question, even in a highly generalized setting. We show a new information theoretic incompressibility bound: there is no function that compresses graphs into O(n4 3 − e) bits so that distance information can be recovered within +no(1) error. As a special case of our theorem, we get a tight lower bound on the sparsity of additive spanners: the +6 spanner on O(n4 3) edges cannot be improved in the exponent, even if any subpolynomial amount of additive error is allowed. Our theorem implies new lower bounds for related objects as well; for example, the twenty-year-old +4 emulator on O(n4 3) edges also cannot be improved in the exponent unless the error allowance is polynomial. Central to our construction is a new type of graph product, which we call the Obstacle Product. Intuitively, it takes two graphs G,H and produces a new graph G H whose shortest paths structure looks locally like H but globally like G. Spanners, emulators, and approximate distance oracles can be viewed as lossy compression schemes that represent an unweighted graph metric in small space, say O(n1+Δ) bits. There is an inherent tradeoff between the sparsity parameter Δ and the stretch function f of the compression scheme, but the qualitative nature of this tradeoff has remained a persistent open problem. It has been known for some time that when Δ ≥ 1 3 there are schemes with constant additive stretch (distance d is stretched to at most f(d) = d + O(1)), and recent results of Abboud and Bodwin show that when Δ In this paper we show that the lower bound of Abboud and Bodwin is just the first step in a hierarchy of lower bounds that characterize the asymptotic behavior of the optimal stretch function f for sparsity parameter Δ ∈ (0,1 3). Specifically, for any integer k ≥ 2, any compression scheme with size [EQUATION] has a sublinear additive stretch function f: f(d) = d + Ω(d1−1 k). This lower bound matches Thorup and Zwick's (2006) construction of sublinear additive emulators. It also shows that Elkin and Peleg's (1 + ϵ, β)-spanners have an essentially optimal tradeoff between Δ, ϵ, and β, and that the sublinear additive spanners of Pettie (2009) and Chechik (2013) are not too far from optimal. To complement these lower bounds we present a new construction of (1 + ϵ, O(k ϵ)k−1)-spanners with size [EQUATION], where hk Our lower bound technique exhibits several interesting degrees of freedom in the framework of Abboud and Bodwin. By carefully exploiting these freedoms, we are able to obtain lower bounds for several related combinatorial objects. We get lower bounds on the size of (β, ϵ)-hopsets, matching Elkin and Neiman's construction (2016), and lower bounds on shortcutting sets for digraphs that preserve the transitive closure. Our lower bound simplifies Hesse's (2003) refutation of Thorup's conjecture (1992), which stated that adding a linear number of shortcuts suffices to reduce the diameter to polylogarithmic. Finally, we show matching upper and lower bounds for graph compression schemes that work for graph metrics with girth at least 2γ + 1. One consequence is that 's (2010) additive O(γ)-spanners with size [EQUATION] cannot be improved in the exponent. We present efficient algorithms for computing very sparse low distortion spanners in distributed networks and prove some non-trivial lower bounds on the trade-off 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 an O(log n)1+o(1)-time algorithm for finding a (2O(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 a 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 doubly 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.