This paper studies the complexity of distributed construction of purely additive spanners in the CONGEST model. We describe algorithms for building such spanners in several cases. Because of the need to simultaneously make decisions at far apart locations, the algorithms use additional mechanisms compared to their sequential counterparts. We complement our algorithms with a lower bound on the number of rounds required for computing pairwise spanners. The standard reductions from set-disjointness and equality seem unsuitable for this task because no specific edge needs to be removed from the graph. Instead, to obtain our lower bound, we define a new communication complexity problem that reduces to computing a sparse spanner, and prove a lower bound on its communication complexity using information theory. This technique significantly extends the current toolbox used for obtaining lower bounds for the CONGEST model, and we believe it may find additional applications.
Sparse spanners with a small multiplicative stretch are well-understood: Alth " @cite_39 in 1993 showed that any weighted graph @math on @math vertices has a spanner of size @math with multiplicative stretch @math , for every integer @math . Since then, several works @cite_33 @cite_52 @cite_2 @cite_46 @cite_56 @cite_48 @cite_50 @cite_41 @cite_42 have considered the problem of efficiently constructing sparse spanners with small stretch and have used spanners in the applications of computing approximate distances and approximate shortest paths efficiently.
@cite_33 @cite_41 @cite_48 @cite_42 @cite_52 @cite_39 @cite_56 @cite_50 @cite_2 @cite_46
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A spanner is a sparse subgraph of a given graph that preserves approximate distance between each pair of vertices. In precise words, a t-spanner of a graph G = (V,E), for any t ∈ N, is a subgraph (V,ES), ES ⊆ E such that, for any u, v ∈ V , their distance in the subgraph is at most t times their distance in the original graph. The parameter t is called the stretch associated with the t-spanner. The concept of spanner was defined formally by Peleg and Schaffer [13] though the associated notion was used implicitly by Awerbuch [3] in the context of network synchronizers. Computing tspanner of smallest size for a given graph is a well motivated combinatorial problem with numerous applications in the area of distributed systems, communication networks and all pairs approximate shortest paths (see [4, 13] and references therein). However, computing t-spanner of smallest size for a graph is NP-hard. In fact, for t > 2, it is NP-hard [6] even to approximate the smallest size of t-spanner of a graph with ratio O(2(1−μ) lnn) for any μ > 0. Having realized this fact, researchers have pursued another direction : to design an efficient algorithm which, for a given graph on n vertices, outputs a t-spanner whose size is of the order of the maximum size of the sparsest tspanner of a graph on n vertices. A 43 years old girth lower bound conjecture by Erdős [7] implies that there are graphs on n vertices whose 2k as well as (2k− 1)spanner will requireΩ(n1+1 k) edges. This conjecture We obtain the following results related to dynamic versions of the shortest-paths problem: Reductions that show that the incremental and decremental single-source shortest-paths problems, for weighted directed or undirected graphs, are, in a strong sense, at least as hard as the static all-pairs shortest-paths problem. We also obtain slightly weaker results for the corresponding unweighted problems. A randomized fully-dynamic algorithm for the all-pairs shortest-paths problem in directed unweighted graphs with an amortized update time of @math (we use @math to hide small poly-logarithmic factors) and a worst case query time is O(n3 4). A deterministic O(n2log n) time algorithm for constructing an O(log n)-spanner with O(n) edges for any weighted undirected graph on n vertices. The algorithm uses a simple algorithm for incrementally maintaining single-source shortest-paths tree up to a given distance. 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]. Let k ≥ 2 be an integer. We show that any undirected and unweighted graph G = (V, E) on n vertices has a subgraph G' = (V, E') with O(kn1+1 k) edges such that for any two vertices u, v ∈ V, if Δ G (u, v) = d, then Δ G' (u, v) = d+O(d1-1 k-1). Furthermore, we show that such subgraphs can be constructed in O(mn1 k) time, where m and n are the number of edges and vertices in the original graph. We also show that it is possible to construct a weighted graph G* = (V, E*) with O(kn1+1 (2k-1)) edges such that for every u, v ∈ V, if Δ G (u, v) = d, then Δ ≤ Δ G* (u, v) = d + O(d1-1 k-1). These are the first such results with additive error terms of the form o(d), i.e., additive error terms that are sublinear in the distance being approximated. Let G=(V,E) be an unweighted undirected graph on n vertices. A simple argument shows that computing all distances in G with an additive one-sided error of at most 1 is as hard as Boolean matrix multiplication. Building on recent work of [SIAM J. Comput., 28 (1999), pp. 1167--1181], we describe an @math -time algorithm APASP2 for computing all distances in G with an additive one-sided error of at most 2. Algorithm APASP2 is simple, easy to implement, and faster than the fastest known matrix-multiplication algorithm. Furthermore, for every even k>2, we describe an @math -time algorithm APASPk for computing all distances in G with an additive one-sided error of at most k. We also give an @math -time algorithm @math for producing stretch 3 estimated distances in an unweighted and undirected graph on n vertices. No constant stretch factor was previously achieved in @math time. We say that a weighted graph F=(V,E') k-emulates an unweighted graph G=(V,E) if for every @math we have @math . We show that every unweighted graph on n vertices has a 2-emulator with @math edges and a 4-emulator with @math edges. These results are asymptotically tight. Finally, we show that any weighted undirected graph on n vertices has a 3-spanner with @math edges and that such a 3-spanner can be built in @math time. We also describe an @math -time algorithm for estimating all distances in a weighted undirected graph on n vertices with a stretch factor of at most 3. From the Publisher: Discrete geometry investigates combinatorial properties of configurations of geometric objects. To a working mathematician or computer scientist, it offers sophisticated results and techniques of great diversity and it is a foundation for fields such as computational geometry or combinatorial optimization. This book is primarily a textbook introduction to various areas of discrete geometry. In each area, it explains several key results and methods, in an accessible and concrete manner. It also contains more advanced material in separate sections and thus it can serve as a collection of surveys in several narrower subfields. The main topics include: basics on convex sets, convex polytopes, and hyperplane arrangements; combinatorial complexity of geometric configurations; intersection patterns and transversals of convex sets; geometric Ramsey-type results; polyhedral combinatorics and high-dimensional convexity; and lastly, embeddings of finite metric spaces into normed spaces. Jiri Matousek is Professor of Computer Science at Charles University in Prague. His research has contributed to several of the considered areas and to their algorithmic applications. This is his third book. Thorup and Zwick showed that for any integer k≥ 1, it is possible to preprocess any positively weighted undirected graph G=(V,E), with |E|=m and |V|=n, in O(kmn @math ) expected time and construct a data structure (a (2k–1)-approximate distance oracle) of size O(kn @math ) capable of returning in O(k) time an approximation @math of the distance δ(u,v) from u to v in G that satisfies @math , for any two vertices u,v∈ V. They also presented a much slower O(kmn) time deterministic algorithm for constructing approximate distance oracle with the slightly larger size of O(kn @math log n). We present here a deterministic O(kmn @math ) time algorithm for constructing oracles of size O(kn @math ). Our deterministic algorithm is slower than the randomized one by only a logarithmic factor. Using our derandomization technique we also obtain the first deterministic linear time algorithm for constructing optimal spanners of weighted graphs. We do that by derandomizing the O(km) expected time algorithm of Baswana and Sen (ICALP’03) for constructing (2k–1)-spanners of size O(kn @math ) of weighted undirected graphs without incurring any asymptotic loss in the running time or in the size of the spanners produced. We study the s-sources almost shortest paths(abbreviated s-ASP) problem. Given an unweightedgraph G e (V,E),and a subset S s Vof s nodes, the goal is to compute almostshortest paths between all the pairs of nodes St V. We devise an algorithm withrunning timeO(mEmnrp s mn1 p z)for this problem that computes the pathsPu,wfor all pairs (u,w) iS t V such that thelength ofPu,wis at most (1 p e)dG(u,w)p b(z,r,e), andb(z,r,e) is constant whenz, r, and e are arbitrarily smallconstants. We also devise a distributed protocol for thes-ASP problem that computes the pathsP u,w as above, and has time and communication complexities ofO(s mDiam(G) pn1 pz 2) (respectively,O(s mDiam(G) log3n p n1p z 2 log n)) andO(mEmnr ps m n1p z) (respectively,O(mEmnr ps m n1p z pn1 p r pz(r m z 2) 2)) in thesynchronous (respectively asynchronous) setting. Our sequential algorithm, as well as the distributed protocol,is based on a novel algorithm for constructing (1 pe, b(z,r, e))-spannersof size O(n1p z), developed in this article. Thisalgorithm has running time ofO(mEmnr), which issignificantly faster than the previously known algorithm given inElkin and Peleg [2001], whose running time isO(n2p r). We also develop the firstdistributed protocol for constructing (1 pe,b)-spanners. The communication complexity ofthis protocol is near optimal.