827 research outputs found
The Norms of Graph Spanners
A -spanner of a graph is a subgraph in which all distances are
preserved up to a multiplicative factor. A classical result of Alth\"ofer
et al. is that for every integer and every graph , there is a
-spanner of with at most edges. But for some
settings the more interesting notion is not the number of edges, but the
degrees of the nodes. This spurred interest in and study of spanners with small
maximum degree. However, this is not necessarily a robust enough objective: we
would like spanners that not only have small maximum degree, but also have
"few" nodes of "large" degree. To interpolate between these two extremes, in
this paper we initiate the study of graph spanners with respect to the
-norm of their degree vector, thus simultaneously modeling the number
of edges (the -norm) and the maximum degree (the -norm).
We give precise upper bounds for all ranges of and stretch : we prove
that the greedy -spanner has norm of at most , and that this bound is tight (assuming the Erd\H{o}s girth
conjecture). We also study universal lower bounds, allowing us to give
"generic" guarantees on the approximation ratio of the greedy algorithm which
generalize and interpolate between the known approximations for the
and norm. Finally, we show that at least in some situations,
the norm behaves fundamentally differently from or
: there are regimes ( and stretch in particular) where
the greedy spanner has a provably superior approximation to the generic
guarantee
On Strong Diameter Padded Decompositions
Given a weighted graph G=(V,E,w), a partition of V is Delta-bounded if the diameter of each cluster is bounded by Delta. A distribution over Delta-bounded partitions is a beta-padded decomposition if every ball of radius gamma Delta is contained in a single cluster with probability at least e^{-beta * gamma}. The weak diameter of a cluster C is measured w.r.t. distances in G, while the strong diameter is measured w.r.t. distances in the induced graph G[C]. The decomposition is weak/strong according to the diameter guarantee.
Formerly, it was proven that K_r free graphs admit weak decompositions with padding parameter O(r), while for strong decompositions only O(r^2) padding parameter was known. Furthermore, for the case of a graph G, for which the induced shortest path metric d_G has doubling dimension ddim, a weak O(ddim)-padded decomposition was constructed, which is also known to be tight. For the case of strong diameter, nothing was known.
We construct strong O(r)-padded decompositions for K_r free graphs, matching the state of the art for weak decompositions. Similarly, for graphs with doubling dimension ddim we construct a strong O(ddim)-padded decomposition, which is also tight. We use this decomposition to construct (O(ddim),O~(ddim))-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
Near-Optimal Approximate Shortest Paths and Transshipment in Distributed and Streaming Models
We present a method for solving the transshipment problem - also known as
uncapacitated minimum cost flow - up to a multiplicative error of in undirected graphs with non-negative edge weights using a
tailored gradient descent algorithm. Using to hide
polylogarithmic factors in (the number of nodes in the graph), our gradient
descent algorithm takes iterations, and in each
iteration it solves an instance of the transshipment problem up to a
multiplicative error of . In particular, this allows
us to perform a single iteration by computing a solution on a sparse spanner of
logarithmic stretch. Using a randomized rounding scheme, we can further extend
the method to finding approximate solutions for the single-source shortest
paths (SSSP) problem. As a consequence, we improve upon prior work by obtaining
the following results: (1) Broadcast CONGEST model: -approximate SSSP using rounds, where is the (hop) diameter of the network.
(2) Broadcast congested clique model: -approximate
transshipment and SSSP using rounds. (3)
Multipass streaming model: -approximate transshipment and
SSSP using space and passes. The
previously fastest SSSP algorithms for these models leverage sparse hop sets.
We bypass the hop set construction; computing a spanner is sufficient with our
method. The above bounds assume non-negative edge weights that are polynomially
bounded in ; for general non-negative weights, running times scale with the
logarithm of the maximum ratio between non-zero weights.Comment: Accepted to SIAM Journal on Computing. Preliminary version in DISC
2017. Abstract shortened to fit arXiv's limitation to 1920 character
Light Spanners for High Dimensional Norms via Stochastic Decompositions
Spanners for low dimensional spaces (e.g. Euclidean space of constant dimension, or doubling metrics) are well understood. This lies in contrast to the situation in high dimensional spaces, where except for the work of Har-Peled, Indyk and Sidiropoulos (SODA 2013), who showed that any n-point Euclidean metric has an O(t)-spanner with O~(n^{1+1/t^2}) edges, little is known.
In this paper we study several aspects of spanners in high dimensional normed spaces. First, we build spanners for finite subsets of l_p with 1<p <=2. Second, our construction yields a spanner which is both sparse and also light, i.e., its total weight is not much larger than that of the minimum spanning tree. In particular, we show that any n-point subset of l_p for 1<p <=2 has an O(t)-spanner with n^{1+O~(1/t^p)} edges and lightness n^{O~(1/t^p)}.
In fact, our results are more general, and they apply to any metric space admitting a certain low diameter stochastic decomposition. It is known that arbitrary metric spaces have an O(t)-spanner with lightness O(n^{1/t}). We exhibit the following tradeoff: metrics with decomposability parameter nu=nu(t) admit an O(t)-spanner with lightness O~(nu^{1/t}). For example, n-point Euclidean metrics have nu <=n^{1/t}, metrics with doubling constant lambda have nu <=lambda, and graphs of genus g have nu <=g. While these families do admit a (1+epsilon)-spanner, its lightness depend exponentially on the dimension (resp. log g). Our construction alleviates this exponential dependency, at the cost of incurring larger stretch
Minimum Weight Euclidean -Spanners
Given a set of points in the plane and a parameter , a
Euclidean -spanner is a geometric graph that
contains, for all , a -path of weight at most
. We show that the minimum weight of a Euclidean
-spanner for points in the unit square is
, and this bound is the best possible. The
upper bound is based on a new spanner algorithm that sparsifies Yao-graphs. It
improves upon the baseline , obtained by combining
a tight bound for the weight of a Euclidean minimum spanning tree (MST) on
points in , and a tight bound for the lightness of Euclidean
-spanners, which is the ratio of the spanner weight to the
weight of the MST. The result generalizes to Euclidean -space for every
dimension : The minimum weight of a Euclidean
-spanner for points in the unit cube is
, and this bound is the best possible.
For the section of the integer lattice, we show that the minimum
weight of a Euclidean -spanner is between
and
. These bounds become
and
when scaled to a grid
of points in the unit square. In particular, this shows that the integer
grid is \emph{not} an extremal configuration for minimum weight Euclidean
-spanners.Comment: 27 pages, 9 figures. An extended abstract appears in the Proceedings
of WG 202
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