10 research outputs found
Additive Spanners and Distance Oracles in Quadratic Time
Let G be an unweighted, undirected graph. An additive k-spanner of G is a subgraph H that approximates all distances between pairs of nodes up to an additive error of +k, that is, it satisfies d_H(u,v) <= d_G(u,v)+k for all nodes u,v, where d is the shortest path distance. We give a deterministic algorithm that constructs an additive O(1)-spanner with O(n^(4/3)) edges in O(n^2) time. This should be compared with the randomized Monte Carlo algorithm by Woodruff [ICALP 2010] giving an additive 6-spanner with O(n^(4/3)log^3 n) edges in expected time O(n^2 log^2 n).
An (alpha,beta)-approximate distance oracle for G is a data structure that supports the following distance queries between pairs of nodes in G. Given two nodes u, v it can in constant time compute a distance estimate d\u27 that satisfies d <= d\u27 <= alpha d + beta where d is the distance between u and v in G. Sommer [ICALP 2016] gave a randomized Monte Carlo (2,1)-distance oracle of size O(n^(5/3) polylog n) in expected time O(n^2 polylog n). As an application of the additive O(1)-spanner we improve the construction by Sommer [ICALP 2016] and give a Las Vegas (2,1)-distance oracle of size O(n^(5/3)) in time O(n^2). This also implies an algorithm that in O(n^2) time gives approximate distance for all pairs of nodes in G improving on the O(n^2 log n) algorithm by Baswana and Kavitha [SICOMP 2010]
Improved Approximate Distance Oracles: Bypassing the Thorup-Zwick Bound in Dense Graphs
Despite extensive research on distance oracles, there are still large gaps
between the best constructions for spanners and distance oracles. Notably,
there exist sparse spanners with a multiplicative stretch of
plus some additive stretch. A fundamental open problem is whether such a bound
is achievable for distance oracles as well. Specifically, can we construct a
distance oracle with multiplicative stretch better than 2, along with some
additive stretch, while maintaining subquadratic space complexity? This
question remains a crucial area of investigation, and finding a positive answer
would be a significant step forward for distance oracles. Indeed, such oracles
have been constructed for sparse graphs. However, in the more general case of
dense graphs, it is currently unknown whether such oracles exist.
In this paper, we contribute to the field by presenting the first distance
oracles that achieve a multiplicative stretch of along with a
small additive stretch while maintaining subquadratic space complexity. Our
results represent an advancement particularly for constructing efficient
distance oracles for dense graphs. In addition, we present a whole family of
oracles that, for any positive integer , achieve a multiplicative stretch of
using space
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
Stronger 3-SUM Lower Bounds for Approximate Distance Oracles via Additive Combinatorics
The "short cycle removal" technique was recently introduced by Abboud,
Bringmann, Khoury and Zamir (STOC '22) to prove fine-grained hardness of
approximation. Its main technical result is that listing all triangles in an
-regular graph is -hard under the 3-SUM conjecture even
when the number of short cycles is small; namely, when the number of -cycles
is for .
Abboud et al. achieve by applying structure vs. randomness
arguments on graphs. In this paper, we take a step back and apply conceptually
similar arguments on the numbers of the 3-SUM problem. Consequently, we achieve
the best possible and the following lower bounds under the 3-SUM
conjecture:
* Approximate distance oracles: The seminal Thorup-Zwick distance oracles
achieve stretch after preprocessing a graph in
time. For the same stretch, and assuming the query time is Abboud et
al. proved an lower bound on the
preprocessing time; we improve it to which is only a
factor 2 away from the upper bound. We also obtain tight bounds for stretch
and and higher lower bounds for dynamic shortest paths.
* Listing 4-cycles: Abboud et al. proved the first super-linear lower bound
for listing all 4-cycles in a graph, ruling out time
algorithms where is the number of 4-cycles. We settle the complexity of
this basic problem by showing that the
upper bound is tight up to factors.
Our results exploit a rich tool set from additive combinatorics, most notably
the Balog-Szemer\'edi-Gowers theorem and Rusza's covering lemma. A key
ingredient that may be of independent interest is a subquadratic algorithm for
3-SUM if one of the sets has small doubling.Comment: Abstract shortened to fit arXiv requirement
Bridge Girth: A Unifying Notion in Network Design
A classic 1993 paper by Alth\H{o}fer et al. proved a tight reduction from
spanners, emulators, and distance oracles to the extremal function of
high-girth graphs. This paper initiated a large body of work in network design,
in which problems are attacked by reduction to or the analogous
extremal function for other girth concepts. In this paper, we introduce and
study a new girth concept that we call the bridge girth of path systems, and we
show that it can be used to significantly expand and improve this web of
connections between girth problems and network design. We prove two kinds of
results:
1) We write the maximum possible size of an -node, -path system with
bridge girth as , and we write a certain variant for
"ordered" path systems as . We identify several arguments in
the literature that implicitly show upper or lower bounds on ,
and we provide some polynomially improvements to these bounds. In particular,
we construct a tight lower bound for , and we polynomially
improve the upper bounds for and .
2) We show that many state-of-the-art results in network design can be
recovered or improved via black-box reductions to or .
Examples include bounds for distance/reachability preservers, exact hopsets,
shortcut sets, the flow-cut gaps for directed multicut and sparsest cut, an
integrality gap for directed Steiner forest.
We believe that the concept of bridge girth can lead to a stronger and more
organized map of the research area. Towards this, we leave many open problems,
related to both bridge girth reductions and extremal bounds on the size of path
systems with high bridge girth