Given a weighted graph @math , a partition of @math is @math -bounded if the diameter of each cluster is bounded by @math . A distribution over @math -bounded partitions is a @math -padded decomposition if every ball of radius @math is contained in a single cluster with probability at least @math . The weak diameter of a cluster @math is measured w.r.t. distances in @math , while the strong diameter is measured w.r.t. distances in the induced graph @math . The decomposition is weak strong according to the diameter guarantee. Formerly, it was proven that @math free graphs admit weak decompositions with padding parameter @math , while for strong decompositions only @math padding parameter was known. Furthermore, for the case of a graph @math , for which the induced shortest path metric @math has doubling dimension @math , a weak @math -padded decomposition was constructed, which is also known to be tight. For the case of strong diameter, nothing was known. We construct strong @math -padded decompositions for @math free graphs, matching the state of the art for weak decompositions. Similarly, for graphs with doubling dimension @math we construct a strong @math -padded decomposition, which is also tight. We use this decomposition to construct @math -sparse cover scheme for such graphs. Our new decompositions and cover have implications to approximating unique games, the construction of light and sparse spanners, and for path reporting distance oracles.
Given a capacitated graph @math and a set of terminals @math , how should we produce a graph @math only on the terminals @math so that every (multicommodity) flow between the terminals in @math could be supported in @math with low congestion, and vice versa? (Such a graph @math is called a flow sparsifier for @math .) What if we want @math to be a “simple” graph? What if we allow @math to be a convex combination of simple graphs? Improving on results of Moitra [Proceedings of the 50th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 2009, pp. 3--12] and Leighton and Moitra [Proceedings of the 42nd ACM Symposium on Theory of Computing, ACM, New York, 2010, pp. 47--56], we give efficient algorithms for constructing (a) a flow sparsifier @math that maintains congestion up to a factor of @math , where @math ; (b) a convex combination of trees over the terminals @math that maintains congestion up to a factor of @math ; (c) for a planar graph... The metric Ramsey problem asks for the largest subset S of a metric space that can be embedded into an ultrametric (more generally into a Hilbert space) with a given distortion. Study of this problem was motivated as a non-linear version of Dvoretzky theorem. Mendel and Naor [MN07] devised the so called Ramsey Partitions to address this problem, and showed the algorithmic applications of their techniques to approximate distance oracles and ranking problems. In this paper we study the natural extension of the metric Ramsey problem to graphs, and introduce the notion of Ramsey Spanning Trees. We ask for the largest subset S ⊆ V of a given graph G = (V, E), such that there exists a spanning tree of G that has small stretch for S. Applied iteratively, this provides a small collection of spanning trees, such that each vertex has a tree providing low stretch paths to all other vertices. The union of these trees serves as a special type of spanner, a tree-padding spanner. We use this spanner to devise the first compact stateless routing scheme with O(1) routing decision time, and labels which are much shorter than in all currently existing schemes. We first revisit the metric Ramsey problem, and provide a new deterministic construction. We prove that for every k, any n-point metric space has a subset S of size at least n1--1 k which embeds into an ultrametric with distortion 8k. We use this result to obtain the state-of-the-art deterministic construction of a distance oracle. Building on this result, we prove that for every k, any n-vertex graph G = (V, E) has a subset S of size at least n1--1 k, and a spanning tree of G, that has stretch O(k log log n) between any point in S and any point in V.  We prove that any weighted graph @math with @math points and @math edges has a spanning tree @math such that @math . Moreover, such a tree... We study the problem of constructing universal Steiner trees for undirected graphs. Given a graph @math and a root node @math , we seek a single spanning tree @math of minimum stretch , where the stretch of @math is defined to be the maximum ratio, over all terminal sets @math , of the cost of the minimal sub-tree @math of @math that connects @math to @math to the cost of an optimal Steiner tree connecting @math to @math in @math . Universal Steiner trees (USTs) are important for data aggregation problems where computing the Steiner tree from scratch for every input instance of terminals is costly, as for example in low energy sensor network applications. We provide a polynomial time UST construction for general graphs with @math -stretch. We also give a polynomial time polylogarithmic-stretch construction for minor-free graphs. One basic building block of our algorithms is a hierarchy of graph partitions, each of which guarantees small strong diameter for each cluster and bounded neighbourhood intersections for each node. We show close connections between the problems of constructing USTs and building such graph partitions. Our construction of partition hierarchies for general graphs is based on an iterative cluster merging procedure, while the one for minor-free graphs is based on a separator theorem for such graphs and the solution to a cluster aggregation problem that may be of independent interest even for general graphs. To our knowledge, this is the first sub polynomial-stretch ( @math for any @math ) UST construction for general graphs, and the first polylogarithmic-stretch UST construction for minor-free graphs. [43] devised a distributed1 algorithm in the CONGEST model, that given a parameter k = 1, 2, ..., constructs an O(k)-spanner of an input unweighted n-vertex graph with O(n1+1 k) expected edges in O(k) rounds of communication. In this paper we improve the result of [43], by showing a k-round distributed algorithm in the same model, that constructs a (2k − 1)-spanner with O(n1+1 k ϵ) edges, with probability 1 − ϵ, for any ϵ > 0. Moreover, when k = ω(log n), our algorithm produces (still in k rounds) ultra-sparse spanners, i.e., spanners of size n(1 + o(1)), with probability 1 − o(1). 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 k, 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 (1 + ϵ, β)-spanners and emulators. In particular, we show that for any unweighted n-vertex graph and any ϵ > 0, there exists a [EQUATION]-emulator with O(n) edges. All previous constructions of (1 + ϵ, β)-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. We use exponential start time clustering to design faster parallel graph algorithms involving distances. Previous algorithms usually rely on graph decomposition routines with strict restrictions on the diameters of the decomposed pieces. We weaken these bounds in favor of stronger local probabilistic guarantees. This allows more direct analyses of the overall process, giving: Linear work parallel algorithms that construct spanners with O(k) stretch and size O(n1+1 k) in unweighted graphs, and size O(n1+1 k log k) in weighted graphs. Hopsets that lead to the first parallel algorithm for approximating shortest paths in undirected graphs with O(m poly log n) work. We show that every weighted connected graph @math contains as a subgraph a spanning tree into which the edges of @math can be embedded with average stretch @math . Moreover, we show that this tree can be constructed in time @math in general, and in time @math if the input graph is unweighted. The main ingredient in our construction is a novel graph decomposition technique. Our new algorithm can be immediately used to improve the running time of the recent solver for symmetric diagonally dominant linear systems of Spielman and Teng from @math to @math , and to @math when the system is planar. Our result can also be used to improve several earlier approximation algorithms that use low-stretch spanning trees.