42 research outputs found
New Pairwise Spanners
Let G = (V,E) be an undirected unweighted graph on n vertices. A subgraph H of G is called an (all-pairs) purely additive spanner with stretch beta if for every (u,v) in V times V, mathsf{dist}_H(u,v) le mathsf{dist}_G(u,v) + beta. The problem of computing sparse spanners with small stretch beta is well-studied. Here we consider the following relaxation: we are given psubseteq V times V and we seek a sparse subgraph H where mathsf{dist}_H(u,v)le mathsf{dist}_G(u,v) + beta for each (u,v) in p. Such a subgraph is called a pairwise spanner with additive stretch beta and our goal is to construct
such subgraphs that are sparser than all-pairs spanners with the same stretch. We show sparse pairwise spanners with additive stretch 4 and with additive stretch 6. We also consider the following special cases: p = S times V and p = S times T, where Ssubseteq V and Tsubseteq V, and show sparser pairwise spanners for these cases
Preserving Distances in Very Faulty Graphs
Preservers and additive spanners are sparse (hence cheap to store) subgraphs that preserve the distances between given pairs of nodes exactly or with some small additive error, respectively. Since real-world networks are prone to failures, it makes sense to study fault-tolerant versions of the above structures. This turns out to be a surprisingly difficult task. For every small but arbitrary set of edge or vertex failures, the preservers and spanners need to contain replacement paths around the faulted set. Unfortunately, the complexity of the interaction between replacement paths blows up significantly, even from 1 to 2 faults, and the structure of optimal preservers and spanners is poorly understood. In particular, no nontrivial bounds for preservers and additive spanners are known when the number of faults is bigger than 2.
Even the answer to the following innocent question is completely unknown: what is the worst-case size of a preserver for a single pair of nodes in the presence of f edge faults? There are no super-linear lower bounds, nor subquadratic upper bounds for f>2. In this paper we make substantial progress on this and other fundamental questions:
- We present the first truly sub-quadratic size fault-tolerant single-pair preserver in unweighted (possibly directed) graphs: for any n node graph and any fixed number f of faults, O~(fn^{2-1/2^f}) size suffices. Our result also generalizes to the single-source (all targets) case, and can be used to build new fault-tolerant additive spanners (for all pairs).
- The size of the above single-pair preserver grows to O(n^2) for increasing f. We show that this is necessary even in undirected unweighted graphs, and even if you allow for a small additive error: If you aim at size O(n^{2-eps}) for eps>0, then the additive error has to be Omega(eps f). This surprisingly matches known upper bounds in the literature.
- For weighted graphs, we provide matching upper and lower bounds for the single pair case. Namely, the size of the preserver is Theta(n^2) for f > 1 in both directed and undirected graphs, while for f=1 the size is Theta(n) in undirected graphs. For directed graphs, we have a superlinear upper bound and a matching lower bound.
Most of our lower bounds extend to the distance oracle setting, where rather than a subgraph we ask for any compact data structure
Improved Purely Additive Fault-Tolerant Spanners
Let be an unweighted -node undirected graph. A \emph{-additive
spanner} of is a spanning subgraph of such that distances in
are stretched at most by an additive term w.r.t. the corresponding
distances in . A natural research goal related with spanners is that of
designing \emph{sparse} spanners with \emph{low} stretch.
In this paper, we focus on \emph{fault-tolerant} additive spanners, namely
additive spanners which are able to preserve their additive stretch even when
one edge fails. We are able to improve all known such spanners, in terms of
either sparsity or stretch. In particular, we consider the sparsest known
spanners with stretch , , and , and reduce the stretch to , ,
and , respectively (while keeping the same sparsity).
Our results are based on two different constructions. On one hand, we show
how to augment (by adding a \emph{small} number of edges) a fault-tolerant
additive \emph{sourcewise spanner} (that approximately preserves distances only
from a given set of source nodes) into one such spanner that preserves all
pairwise distances. On the other hand, we show how to augment some known
fault-tolerant additive spanners, based on clustering techniques. This way we
decrease the additive stretch without any asymptotic increase in their size. We
also obtain improved fault-tolerant additive spanners for the case of one
vertex failure, and for the case of edge failures.Comment: 17 pages, 4 figures, ESA 201
Simpler and Higher Lower Bounds for Shortcut Sets
We provide a variety of lower bounds for the well-known shortcut set problem:
how much can one decrease the diameter of a directed graph on vertices and
edges by adding or of shortcuts from the transitive closure
of the graph. Our results are based on a vast simplification of the recent
construction of Bodwin and Hoppenworth [FOCS 2023] which was used to show an
lower bound for the -sized shortcut set
problem. We highlight that our simplification completely removes the use of the
convex sets by B\'ar\'any and Larman [Math. Ann. 1998] used in all previous
lower bound constructions. Our simplification also removes the need for
randomness and further removes some log factors. This allows us to generalize
the construction to higher dimensions, which in turn can be used to show the
following results. For -sized shortcut sets, we show an
lower bound, improving on the previous best lower bound. For
all , we show that there exists a such that there
are -vertex -edge graphs where adding any shortcut set of size
keeps the diameter of at . This
improves the sparsity of the constructed graph compared to a known similar
result by Hesse [SODA 2003].
We also consider the sourcewise setting for shortcut sets: given a graph
, a set , how much can we decrease the sourcewise
diameter of , by adding a set of edges from the transitive closure of
? We show that for any integer , there exists a graph on
vertices and with ,
such that when adding or shortcuts, the sourcewise diameter is
.Comment: To appear in SODA 2024. Abstract shortened to fit arXiv requirement