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.
Compared to the preservers in @cite_9 , we achieve sparser pairwise spanners with additive stretch @math for @math , and a sparser subsetwise spanners for @math . Interestingly, our sourcewise spanners are always sparser than the pairwise preservers from @cite_9 .
@cite_9
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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). 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. 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. 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. 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. We introduce and study the notions of pairwise and sourcewise preservers. Given an undirected N-vertex graph G = (V,E) and a set P of pairs of vertices, let G' = (V,H), H E, be called a pairwise preserver of G with respect to P if for every pair u,w P, distG'(u,w) = distG(u,w). For a set S V of sources, a pairwise preserver of G with respect to the set of all pairs P = (S 2) of sources is called a sourcewise preserver of G with respect to S. We prove that for every undirected possibly weighted N-vertex graph G and every set P of P = O(N1 2) pairs of vertices of G, there exists a linear-size pairwise preserver of G with respect to P. Consequently, for every subset S V of S = O(N1 4) sources, there exists a linear-size sourcewise preserver of G with respect to S. On the negative side we show that neither of the two exponents (1 2 and 1 4) can be improved even when the attention is restricted to unweighted graphs. Our lower bounds involve constructions of dense convexly independent sets of vectors with small Euclidean norms. We believe that the link between the areas of discrete geometry and spanners that we establish is of independent interest and might be useful in the study of other problems in the area of low-distortion embeddings. 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, namely ( O(n( n)^ ) ) , where ( = (1 + 5 ) 2 ) is the golden ratio. As the distance increases the multiplicative distortion of a Fibonacci spanner 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. This paper considers additive and purely additive spanners. We present a new purely additive spanner of size O(n7 5) with additive stretch 4. This construction fills in the gap between the two existing constructions for purely additive spanners, one for 2-additive spanner of size O(n3 2) and the other for 6-additive spanner of size O(n4 3), and thus answers a main open question in this area. In addition, we present a construction for additive spanners with O(n1+δ) edges and additive stretch of O(n1 2−3δ 2) for any 3 17 ≤ δ < 1 3, improving the stretch of the existing constructions from O(n1−3δ) to O(√n1−3δ). Finally, we show that our (1,n1 2−3δ 2)-spanner construction can be tweaked to give a sublinear additive spanner of size O(n1+3 17) with additive stretch O(√distance). We present an O(nlogn)-approximation algorithm for the problem of finding the sparsest spanner of a given directed graph G on n vertices. A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph. More precisely, given a graph G=(V,E) with nonnegative edge lengths d:E->R^>=^0 and a stretchk>=1, a subgraph H=(V,E"H) is a k-spanner of G if for every edge (s,t)@?E, the graph H contains a path from s to t of length at most [email protected]?d(s,t). The previous best approximation ratio was [email protected]?(n^2^ ^3), due to Dinitz and Krauthgamer (STOC @?11). We also improve the approximation ratio for the important special case of directed 3-spanners with unit edge lengths from [email protected]?(n) to O(n^1^ ^3logn). The best previously known algorithms for this problem are due to Berman, Raskhodnikova and Ruan (FSTTCS @?10) and Dinitz and Krauthgamer. The approximation ratio of our algorithm almost matches Dinitz and [email protected]?s lower bound for the integrality gap of a natural linear programming relaxation. Our algorithm directly implies an O(n^1^ ^3logn)-approximation for the 3-spanner problem on undirected graphs with unit lengths. An easy O(n)-approximation algorithm for this problem has been the best known for decades. Finally, we consider the Directed Steiner Forest problem: given a directed graph with edge costs and a collection of ordered vertex pairs, find a minimum-cost subgraph that contains a path between every prescribed pair. We obtain an approximation ratio of O(n^2^ ^3^+^@e) for any constant @e>0, which improves the O(n^@[email protected]?min(n^4^ ^5,m^2^ ^3)) ratio due to Feldman, Kortsarz and Nutov ([email protected]?12). We introduce and study the notions of pair-wise and source-wise preservers.Given an undirected N-vertex graph G = (V, E) and a subset P of pairs of vertices, let G' = (V, H), H ⊆ E, be called a pair-wise preserver of G with respect to P if for every pair u, w ∈ P, dist G' (u, w) = dist G (u, w). For a set S ⊆ V of sources, a pair-wise preserver of G with respect to the set of all pairs P = (S 2) of sources is called a source-wise preserver of G with respect to S.We prove that for every undirected possibly weighted N-vertex graph G and every subset P of P = O(N1 2) pairs of vertices of G, there exists a linear-size pair-wise preserver of G with respect to P. Consequently, for every subset S ⊆ V of S = O(N1 4) sources, there exists a linear-size source-wise preserver of G with respect to S. On the negative side we show that neither of the two exponents (1 2 and 1 4) can be improved even when the attention is restricted to unweighted graphs.Our lower bounds involve constructions of dense convexly independent sets of vectors with small Euclidean norms. We believe that the link between the areas of Discrete Geometry and spanners that we establish is of independent interest, and might be useful in the study of other problems in the area of low-distortion embeddings. 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. The goal of (stable) sparse recovery is to recover a @math -sparse approximation @math of a vector @math from linear measurements of @math . Specifically, the goal is to recover @math such that @math for some constant @math and norm parameters @math and @math . It is known that, for @math or @math , this task can be accomplished using @math non-adaptive measurements CRT06:Stable-Signal and that this bound is tight DIPW, FPRU, PW11 . In this paper we show that if one is allowed to perform measurements that are adaptive , then the number of measurements can be considerably reduced. Specifically, for @math and @math we show A scheme with @math measurements that uses @math rounds. This is a significant improvement over the best possible non-adaptive bound. A scheme with @math measurements that uses two rounds. This improves over the best possible non-adaptive bound. To the best of our knowledge, these are the first results of this type. We present a streaming algorithm for constructing sparse spanners and show that our algorithm significantly outperforms the state-of-the-art algorithm for this task (due to ). Specifically, the processing time per edge of our algorithm is O(1), and it is drastically smaller than that of the algorithm of , and all other efficiency parameters of our algorithm are no greater (and some of them are strictly smaller) than the respective parameters of the state-of-the-art algorithm. We also devise a fully dynamic centralized algorithm maintaining sparse spanners. This algorithm has incremental update time of O(1), and a nontrivial decremental update time. To our knowledge, this is the first fully dynamic centralized algorithm for maintaining sparse spanners that provides nontrivial bounds on both incremental and decremental update time for a wide range of stretch parameter t. For every fixed constant @math , we design an algorithm for computing the @math -sparse Walsh-Hadamard transform of an @math -dimensional vector @math in time @math . Specifically, the algorithm is given query access to @math and computes a @math -sparse @math satisfying @math , for an absolute constant @math , where @math is the transform of @math and @math is its best @math -sparse approximation. Our algorithm is fully deterministic and only uses non-adaptive queries to @math (i.e., all queries are determined and performed in parallel when the algorithm starts). An important technical tool that we use is a construction of nearly optimal and linear lossless condensers which is a careful instantiation of the GUV condenser (Guruswami, Umans, Vadhan, JACM 2009). Moreover, we design a deterministic and non-adaptive @math compressed sensing scheme based on general lossless condensers that is equipped with a fast reconstruction algorithm running in time @math (for the GUV-based condenser) and is of independent interest. Our scheme significantly simplifies and improves an earlier expander-based construction due to Berinde, Gilbert, Indyk, Karloff, Strauss (Allerton 2008). Our methods use linear lossless condensers in a black box fashion; therefore, any future improvement on explicit constructions of such condensers would immediately translate to improved parameters in our framework (potentially leading to @math reconstruction time with a reduced exponent in the poly-logarithmic factor, and eliminating the extra parameter @math ). Finally, by allowing the algorithm to use randomness, while still using non-adaptive queries, the running time of the algorithm can be improved to @math . We develop a new technique for constructing sparse graphs that allow us to prove near-linear lower bounds on the round complexity of computing distances in the CONGEST model. Specifically, we show an @math lower bound for computing the diameter in sparse networks, which was previously known only for dense networks [, SODA 2012]. In fact, we can even modify our construction to obtain graphs with constant degree, using a simple but powerful degree-reduction technique which we define. Moreover, our technique allows us to show @math lower bounds for computing @math -approximations of the diameter or the radius, and for computing a @math -approximation of all eccentricities. For radius, we are unaware of any previous lower bounds. For diameter, these greatly improve upon previous lower bounds and are tight up to polylogarithmic factors [, SODA 2012], and for eccentricities the improvement is both in the lower bound and in the approximation factor [Holzer and Wattenhofer, PODC 2012]. Interestingly, our technique also allows showing an almost-linear lower bound for the verification of @math -spanners, for @math . We study the recovery of sparse vectors from subsampled random convolutions via @math -minimization. We consider the setup in which both the subsampling locations as well as the generating vector are chosen at random. For a subgaussian generator with independent entries, we improve previously known estimates: if the sparsity @math is small enough, i.e., @math , we show that @math measurements are sufficient to recover @math -sparse vectors in dimension @math with high probability, matching the well-known condition for recovery from standard Gaussian measurements. If @math is larger, then essentially @math measurements are sufficient, again improving over previous estimates. Our results are shown via the so-called robust null space property which is weaker than the standard restricted isometry property. Our method of proof involves a novel combination of small ball estimates with chaining techniques which should be of independent interest. 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. Roadmap spanners provide a way to acquire sparse data structures that efficiently answer motion planning queries with probabilistic completeness and asymptotic near-optimality. The current SPARS method provides these properties by building two graphs in parallel: a dense asymptotically-optimal roadmap based on PRM* and its spanner. This paper shows that it is possible to relax the conditions under which a sample is added to the spanner and provide guarantees, while not requiring the use of a dense graph. A key aspect of SPARS is that the probability of adding nodes to the roadmap goes to zero as iterations increase, which is maintained in the proposed extension. The paper describes the new algorithm, argues its theoretical properties and evaluates it against PRM* and the original SPARS algorithm. The experimental results show that the memory requirements of the method upon construction are dramatically reduced, while returning competitive quality paths with PRM*. There is a small sacrifice in the size of the final spanner relative to SPARS but the new method still returns graphs orders of magnitudes smaller than PRM*, leading to very efficient online query resolution. 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 trade-off between the sparsity of H and the distortion function f, though the nature of the optimal trade-off is still poorly understood. In this article 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 [1999] and [2009], and give spanners whose multiplicative distortion quickly tends toward 1. Our results rival the simplicity of all previous algorithms and provide substantial improvements (up to a doubly exponential reduction in edge density) over the comparable spanners of Elkin and Peleg [2004] and Thorup and Zwick [2006].