9 research outputs found
Parallel Greedy Spanners
A -spanner of a graph is a subgraph that -approximates pairwise
distances. The greedy algorithm is one of the simplest and most well-studied
algorithms for constructing a sparse spanner: it computes a -spanner with
edges by repeatedly choosing any edge which does not close a
cycle of chosen edges with or fewer edges.
We demonstrate that the greedy algorithm computes a -spanner with edges even when a matching of such edges are added in parallel. In
particular, it suffices to repeatedly add any matching where each individual
edge does not close a cycle with or fewer edges but where adding the
entire matching might. Our analysis makes use of and illustrates the power of
new advances in length-constrained expander decompositions
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
Online Directed Spanners and Steiner Forests
We present online algorithms for directed spanners and Steiner forests. These
problems fall under the unifying framework of online covering linear
programming formulations, developed by Buchbinder and Naor (MOR, 34, 2009),
based on primal-dual techniques. Our results include the following:
For the pairwise spanner problem, in which the pairs of vertices to be
spanned arrive online, we present an efficient randomized
-competitive algorithm for graphs with general lengths,
where is the number of vertices. With uniform lengths, we give an efficient
randomized -competitive algorithm, and an
efficient deterministic -competitive algorithm,
where is the number of terminal pairs. These are the first online
algorithms for directed spanners. In the offline setting, the current best
approximation ratio with uniform lengths is ,
due to Chlamtac, Dinitz, Kortsarz, and Laekhanukit (TALG 2020).
For the directed Steiner forest problem with uniform costs, in which the
pairs of vertices to be connected arrive online, we present an efficient
randomized -competitive algorithm. The
state-of-the-art online algorithm for general costs is due to Chakrabarty, Ene,
Krishnaswamy, and Panigrahi (SICOMP 2018) and is -competitive. In the offline version, the current best approximation
ratio with uniform costs is , due to Abboud
and Bodwin (SODA 2018).
A small modification of the online covering framework by Buchbinder and Naor
implies a polynomial-time primal-dual approach with separation oracles, which a
priori might perform exponentially many calls. We convert the online spanner
problem and the online Steiner forest problem into online covering problems and
round in a problem-specific fashion
Approximation Algorithms for Directed Weighted Spanners
In the pairwise weighted spanner problem, the input consists of an
-vertex-directed graph, where each edge is assigned a cost and a length.
Given vertex pairs and a distance constraint for each pair, the goal is to
find a minimum-cost subgraph in which the distance constraints are satisfied.
This formulation captures many well-studied connectivity problems, including
spanners, distance preservers, and Steiner forests.
In the offline setting, we show:
1. An -approximation algorithm for pairwise
weighted spanners. When the edges have unit costs and lengths, the best
previous algorithm gives an -approximation, due
to Chlamt\'a\v{c}, Dinitz, Kortsarz, and Laekhanukit (TALG, 2020).
2. An -approximation algorithm for all-pair
weighted distance preservers. When the edges have unit costs and arbitrary
lengths, the best previous algorithm gives an
-approximation for all-pair spanners, due to Berman,
Bhattacharyya, Makarychev, Raskhodnikova, and Yaroslavtsev (Information and
Computation, 2013).
In the online setting, we show:
1. An -competitive algorithm for pairwise
weighted spanners. The state-of-the-art results are
-competitive when edges have unit costs and arbitrary
lengths, and -competitive when edges have unit costs and lengths, due to
Grigorescu, Lin, and Quanrud (APPROX, 2021).
2. An -competitive algorithm for single-source
weighted spanners. Without distance constraints, this problem is equivalent to
the directed Steiner tree problem. The best previous algorithm for online
directed Steiner trees is -competitive, due to
Chakrabarty, Ene, Krishnaswamy, and Panigrahi (SICOMP, 2018)
A Unified and Fine-Grained Approach for Light Spanners
Seminal works on light spanners from recent years provide near-optimal
tradeoffs between the stretch and lightness of spanners in general graphs,
minor-free graphs, and doubling metrics. In FOCS'19 the authors provided a
"truly optimal" tradeoff for Euclidean low-dimensional spaces. Some of these
papers employ inherently different techniques than others. Moreover, the
runtime of these constructions is rather high.
In this work, we present a unified and fine-grained approach for light
spanners. Besides the obvious theoretical importance of unification, we
demonstrate the power of our approach in obtaining (1) stronger lightness
bounds, and (2) faster construction times. Our results include:
_ -minor-free graphs: A truly optimal spanner construction and a fast
construction.
_ General graphs: A truly optimal spanner -- almost and a linear-time
construction with near-optimal lightness.
_ Low dimensional Euclidean spaces: We demonstrate that Steiner points help
in reducing the lightness of Euclidean -spanners almost
quadratically for .Comment: We split this paper into two papers: arXiv:2106.15596 and
arXiv:2111.1374