13,428 research outputs found
A Linear-Size Logarithmic Stretch Path-Reporting Distance Oracle for General Graphs
In 2001 Thorup and Zwick devised a distance oracle, which given an -vertex
undirected graph and a parameter , has size . Upon a query
their oracle constructs a -approximate path between
and . The query time of the Thorup-Zwick's oracle is , and it was
subsequently improved to by Chechik. A major drawback of the oracle of
Thorup and Zwick is that its space is . Mendel and Naor
devised an oracle with space and stretch , but their
oracle can only report distance estimates and not actual paths. In this paper
we devise a path-reporting distance oracle with size , stretch
and query time , for an arbitrarily small .
In particular, our oracle can provide logarithmic stretch using linear size.
Another variant of our oracle has size , polylogarithmic
stretch, and query time .
For unweighted graphs we devise a distance oracle with multiplicative stretch
, additive stretch , for a function , space
, and query time , for an arbitrarily
small constant . The tradeoff between multiplicative stretch and
size in these oracles is far below girth conjecture threshold (which is stretch
and size ). Breaking the girth conjecture tradeoff is
achieved by exhibiting a tradeoff of different nature between additive stretch
and size . A similar type of tradeoff was exhibited by
a construction of -spanners due to Elkin and Peleg.
However, so far -spanners had no counterpart in the
distance oracles' world.
An important novel tool that we develop on the way to these results is a
{distance-preserving path-reporting oracle}
Compact routing on the Internet AS-graph
Compact routing algorithms have been presented as candidates for scalable routing in the future Internet, achieving near-shortest path routing with considerably less forwarding state than the Border Gateway Protocol. Prior analyses have shown strong performance on power-law random graphs, but to better understand the applicability of compact routing algorithms in the context of the Internet, they must be evaluated against real- world data. To this end, we present the first systematic analysis of the behaviour of the Thorup-Zwick (TZ) and Brady-Cowen (BC) compact routing algorithms on snapshots of the Internet Autonomous System graph spanning a 14 year period. Both algorithms are shown to offer consistently strong performance on the AS graph, producing small forwarding tables with low stretch for all snapshots tested. We find that the average stretch for the TZ algorithm increases slightly as the AS graph has grown, while previous results on synthetic data suggested the opposite would be true. We also present new results to show which features of the algorithms contribute to their strong performance on these graphs
Improved Parallel Algorithms for Spanners and Hopsets
We use exponential start time clustering to design faster and more
work-efficient 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 stretch and size in unweighted
graphs, and size in weighted graphs. * Hopsets that lead
to the first parallel algorithm for approximating shortest paths in undirected
graphs with work
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
Compact Routing on Internet-Like Graphs
The Thorup-Zwick (TZ) routing scheme is the first generic stretch-3 routing
scheme delivering a nearly optimal local memory upper bound. Using both direct
analysis and simulation, we calculate the stretch distribution of this routing
scheme on random graphs with power-law node degree distributions, . We find that the average stretch is very low and virtually
independent of . In particular, for the Internet interdomain graph,
, the average stretch is around 1.1, with up to 70% of paths
being shortest. As the network grows, the average stretch slowly decreases. The
routing table is very small, too. It is well below its upper bounds, and its
size is around 50 records for -node networks. Furthermore, we find that
both the average shortest path length (i.e. distance) and width of
the distance distribution observed in the real Internet inter-AS graph
have values that are very close to the minimums of the average stretch in the
- and -directions. This leads us to the discovery of a unique
critical quasi-stationary point of the average TZ stretch as a function of
and . The Internet distance distribution is located in a
close neighborhood of this point. This observation suggests the analytical
structure of the average stretch function may be an indirect indicator of some
hidden optimization criteria influencing the Internet's interdomain topology
evolution.Comment: 29 pages, 16 figure
Pruning based Distance Sketches with Provable Guarantees on Random Graphs
Measuring the distances between vertices on graphs is one of the most
fundamental components in network analysis. Since finding shortest paths
requires traversing the graph, it is challenging to obtain distance information
on large graphs very quickly. In this work, we present a preprocessing
algorithm that is able to create landmark based distance sketches efficiently,
with strong theoretical guarantees. When evaluated on a diverse set of social
and information networks, our algorithm significantly improves over existing
approaches by reducing the number of landmarks stored, preprocessing time, or
stretch of the estimated distances.
On Erd\"{o}s-R\'{e}nyi graphs and random power law graphs with degree
distribution exponent , our algorithm outputs an exact distance
data structure with space between and
depending on the value of , where is the number of vertices. We
complement the algorithm with tight lower bounds for Erdos-Renyi graphs and the
case when is close to two.Comment: Full version for the conference paper to appear in The Web
Conference'1
Towards Bypassing Lower Bounds for Graph Shortcuts
For a given (possibly directed) graph G, a hopset (a.k.a. shortcut set) is a (small) set of edges whose addition reduces the graph diameter while preserving desired properties from the given graph G, such as, reachability and shortest-path distances. The key objective is in optimizing the tradeoff between the achieved diameter and the size of the shortcut set (possibly also, the distance distortion). Despite the centrality of these objects and their thorough study over the years, there are still significant gaps between the known upper and lower bound results.
A common property shared by almost all known shortcut lower bounds is that they hold for the seemingly simpler task of reducing the diameter of the given graph, D_G, by a constant additive term, in fact, even by just one! We denote such restricted structures by (D_G-1)-diameter hopsets. In this paper we show that this relaxation can be leveraged to narrow the current gaps, and in certain cases to also bypass the known lower bound results, when restricting to sparse graphs (with O(n) edges):
- {Hopsets for Directed Weighted Sparse Graphs.} For every n-vertex directed and weighted sparse graph G with D_G ? n^{1/4}, one can compute an exact (D_G-1)-diameter hopset of linear size. Combining this with known lower bound results for dense graphs, we get a separation between dense and sparse graphs, hence shortcutting sparse graphs is provably easier. For reachability hopsets, we can provide (D_G-1)-diameter hopsets of linear size, for sparse DAGs, already for D_G ? n^{1/5}. This should be compared with the diameter bound of O?(n^{1/3}) [Kogan and Parter, SODA 2022], and the lower bound of D_G = n^{1/6} by [Huang and Pettie, {SIAM} J. Discret. Math. 2018].
- {Additive Hopsets for Undirected and Unweighted Graphs.} We show a construction of +24 additive (D_G-1)-diameter hopsets with linear number of edges for D_G ? n^{1/12} for sparse graphs. This bypasses the current lower bound of D_G = n^{1/6} obtained for exact (D_G-1)-diameter hopset by [HP\u2718]. For general graphs, the bound becomes D_G ? n^{1/6} which matches the lower bound of exact (D_G-1) hopsets implied by [HP\u2718]. We also provide new additive D-diameter hopsets with linear size, for any given diameter D.
Altogether, we show that the current lower bounds can be bypassed by restricting to sparse graphs (with O(n) edges). Moreover, the gaps are narrowed significantly for any graph by allowing for a constant additive stretch
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