8 research outputs found
Light Spanners
A -spanner of a weighted undirected graph , is a subgraph
such that for all . The sparseness of
the spanner can be measured by its size (the number of edges) and weight (the
sum of all edge weights), both being important measures of the spanner's
quality -- in this work we focus on the latter.
Specifically, it is shown that for any parameters and ,
any weighted graph on vertices admits a
-stretch spanner of weight at most , where is the weight of a minimum
spanning tree of . Our result is obtained via a novel analysis of the
classic greedy algorithm, and improves previous work by a factor of .Comment: 10 pages, 1 figure, to appear in ICALP 201
Spanner Approximations in Practice
A multiplicative -spanner is a subgraph of with the
same vertices and fewer edges that preserves distances up to the factor
, i.e., for all vertices , .
While many algorithms have been developed to find good spanners in terms of
approximation guarantees, no experimental studies comparing different
approaches exist. We implemented a rich selection of those algorithms and
evaluate them on a variety of instances regarding, e.g., their running time,
sparseness, lightness, and effective stretch
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