345 research outputs found
Fast approximation of centrality and distances in hyperbolic graphs
We show that the eccentricities (and thus the centrality indices) of all
vertices of a -hyperbolic graph can be computed in linear
time with an additive one-sided error of at most , i.e., after a
linear time preprocessing, for every vertex of one can compute in
time an estimate of its eccentricity such that
for a small constant . We
prove that every -hyperbolic graph has a shortest path tree,
constructible in linear time, such that for every vertex of ,
. These results are based on an
interesting monotonicity property of the eccentricity function of hyperbolic
graphs: the closer a vertex is to the center of , the smaller its
eccentricity is. We also show that the distance matrix of with an additive
one-sided error of at most can be computed in
time, where is a small constant. Recent empirical studies show that
many real-world graphs (including Internet application networks, web networks,
collaboration networks, social networks, biological networks, and others) have
small hyperbolicity. So, we analyze the performance of our algorithms for
approximating centrality and distance matrix on a number of real-world
networks. Our experimental results show that the obtained estimates are even
better than the theoretical bounds.Comment: arXiv admin note: text overlap with arXiv:1506.01799 by other author
Distance Oracles for Time-Dependent Networks
We present the first approximate distance oracle for sparse directed networks
with time-dependent arc-travel-times determined by continuous, piecewise
linear, positive functions possessing the FIFO property.
Our approach precomputes approximate distance summaries from
selected landmark vertices to all other vertices in the network. Our oracle
uses subquadratic space and time preprocessing, and provides two sublinear-time
query algorithms that deliver constant and approximate
shortest-travel-times, respectively, for arbitrary origin-destination pairs in
the network, for any constant . Our oracle is based only on
the sparsity of the network, along with two quite natural assumptions about
travel-time functions which allow the smooth transition towards asymmetric and
time-dependent distance metrics.Comment: A preliminary version appeared as Technical Report ECOMPASS-TR-025 of
EU funded research project eCOMPASS (http://www.ecompass-project.eu/). An
extended abstract also appeared in the 41st International Colloquium on
Automata, Languages, and Programming (ICALP 2014, track-A
Fast 2-Approximate All-Pairs Shortest Paths
In this paper, we revisit the classic approximate All-Pairs Shortest Paths
(APSP) problem in undirected graphs. For unweighted graphs, we provide an
algorithm for -approximate APSP in time,
for any . This is time, using known bounds for
rectangular matrix multiplication~~[Le Gall, Urrutia, SODA
2018]. Our result improves on the bound of [Roddity, STOC
2023], and on the bound of [Baswana, Kavitha, SICOMP
2010] for graphs with edges.
For weighted graphs, we obtain -approximate APSP in time, for any . This is
time using known bounds for . It improves on the state of the art
bound of by [Kavitha, Algorithmica 2012]. Our techniques further
lead to improved bounds in a wide range of density for weighted graphs. In
particular, for the sparse regime we construct a distance oracle in time that supports -approximate queries in constant time. For
sparse graphs, the preprocessing time of the algorithm matches conditional
lower bounds [Patrascu, Roditty, Thorup, FOCS 2012; Abboud, Bringmann, Fischer,
STOC 2023]. To the best of our knowledge, this is the first 2-approximate
distance oracle that has subquadratic preprocessing time in sparse graphs.
We also obtain new bounds in the near additive regime for unweighted graphs.
We give faster algorithms for -approximate APSP, for
.
We obtain these results by incorporating fast rectangular matrix
multiplications into various combinatorial algorithms that carefully balance
out distance computation on layers of sparse graphs preserving certain distance
information
On the Space Usage of Approximate Distance Oracles with Sub-2 Stretch
For an undirected unweighted graph with vertices and edges,
let denote the distance from to in . An
-stretch approximate distance oracle (ADO) for is a data
structure that given returns in constant (or near constant) time a
value such that , for some reals . If , we say that the
ADO has stretch .
Thorup and Zwick~\cite{thorup2005approximate} showed that one cannot beat
stretch 3 with subquadratic space (in terms of ) for general graphs.
P\v{a}tra\c{s}cu and Roditty~\cite{patrascu2010distance} showed that one can
obtain stretch 2 using space, and so if is subquadratic
in then the space usage is also subquadratic. Moreover, P\v{a}tra\c{s}cu
and Roditty~\cite{patrascu2010distance} showed that one cannot beat stretch 2
with subquadratic space even for graphs where , based on the
set-intersection hypothesis.
In this paper we explore the conditions for which an ADO can be stored using
subquadratic space while supporting a sub-2 stretch. In particular, we show
that if the maximum degree in is for
some , then there exists an ADO for that uses
space and has a sub-2 stretch.
Moreover, we prove a conditional lower bound, based on the set intersection
hypothesis, which states that for any positive integer ,
obtaining a sub- stretch for graphs with maximum degree
requires quadratic space. Thus, for graphs with maximum
degree , obtaining a sub-2 stretch requires quadratic space
On Diameter Approximation in Directed Graphs
Computing the diameter of a graph, i.e. the largest distance, is a fundamental problem that is central in fine-grained complexity. In undirected graphs, the Strong Exponential Time Hypothesis (SETH) yields a lower bound on the time vs. approximation trade-off that is quite close to the upper bounds.
In directed graphs, however, where only some of the upper bounds apply, much larger gaps remain. Since d(u,v) may not be the same as d(v,u), there are multiple ways to define the problem, the two most natural being the (one-way) diameter (max_(u,v) d(u,v)) and the roundtrip diameter (max_{u,v} d(u,v)+d(v,u)). In this paper we make progress on the outstanding open question for each of them.
- We design the first algorithm for diameter in sparse directed graphs to achieve n^{1.5-?} time with an approximation factor better than 2. The new upper bound trade-off makes the directed case appear more similar to the undirected case. Notably, this is the first algorithm for diameter in sparse graphs that benefits from fast matrix multiplication.
- We design new hardness reductions separating roundtrip diameter from directed and undirected diameter. In particular, a 1.5-approximation in subquadratic time would refute the All-Nodes k-Cycle hypothesis, and any (2-?)-approximation would imply a breakthrough algorithm for approximate ?_?-Closest-Pair. Notably, these are the first conditional lower bounds for diameter that are not based on SETH
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