17 research outputs found
Fast Distributed Approximation for TAP and 2-Edge-Connectivity
The tree augmentation problem (TAP) is a fundamental network design problem,
in which the input is a graph and a spanning tree for it, and the goal
is to augment with a minimum set of edges from , such that is 2-edge-connected.
TAP has been widely studied in the sequential setting. The best known
approximation ratio of 2 for the weighted case dates back to the work of
Frederickson and J\'{a}J\'{a}, SICOMP 1981. Recently, a 3/2-approximation was
given for unweighted TAP by Kortsarz and Nutov, TALG 2016. Recent breakthroughs
give an approximation of 1.458 for unweighted TAP [Grandoni et al., STOC 2018],
and approximations better than 2 for bounded weights [Adjiashvili, SODA 2017;
Fiorini et al., SODA 2018].
In this paper, we provide the first fast distributed approximations for TAP.
We present a distributed -approximation for weighted TAP which completes in
rounds, where is the height of . When is large, we show a
much faster 4-approximation algorithm for the unweighted case, completing in
rounds, where is the number of vertices and is
the diameter of .
Immediate consequences of our results are an -round 2-approximation
algorithm for the minimum size 2-edge-connected spanning subgraph, which
significantly improves upon the running time of previous approximation
algorithms, and an -round 3-approximation
algorithm for the weighted case, where is the height of the MST of
the graph. Additional applications are algorithms for verifying
2-edge-connectivity and for augmenting the connectivity of any connected
spanning subgraph to 2.
Finally, we complement our study with proving lower bounds for distributed
approximations of TAP
Massively Parallel Algorithms for Distance Approximation and Spanners
Over the past decade, there has been increasing interest in
distributed/parallel algorithms for processing large-scale graphs. By now, we
have quite fast algorithms -- usually sublogarithmic-time and often
-time, or even faster -- for a number of fundamental graph
problems in the massively parallel computation (MPC) model. This model is a
widely-adopted theoretical abstraction of MapReduce style settings, where a
number of machines communicate in an all-to-all manner to process large-scale
data. Contributing to this line of work on MPC graph algorithms, we present
round MPC algorithms for computing
-spanners in the strongly sublinear regime of local memory. To
the best of our knowledge, these are the first sublogarithmic-time MPC
algorithms for spanner construction. As primary applications of our spanners,
we get two important implications, as follows:
-For the MPC setting, we get an -round algorithm for
approximation of all pairs shortest paths (APSP) in the
near-linear regime of local memory. To the best of our knowledge, this is the
first sublogarithmic-time MPC algorithm for distance approximations.
-Our result above also extends to the Congested Clique model of distributed
computing, with the same round complexity and approximation guarantee. This
gives the first sub-logarithmic algorithm for approximating APSP in weighted
graphs in the Congested Clique model
New Tradeoffs for Decremental Approximate All-Pairs Shortest Paths
We provide new tradeoffs between approximation and running time for the
decremental all-pairs shortest paths (APSP) problem. For undirected graphs with
edges and nodes undergoing edge deletions, we provide two new
approximate decremental APSP algorithms, one for weighted and one for
unweighted graphs. Our first result is an algorithm that supports -approximate all-pairs constant-time distance queries with total
update time when (and
for any constant ), or when .
Prior to our work the fastest algorithm for weighted graphs with approximation
at most had total update time providing a
-approximation [Bernstein, SICOMP 2016]. Our technique also
yields a decremental algorithm with total update time
supporting -approximate queries where the second term is
an additional additive term and is the maximum weight on the shortest
path from to .
Our second result is a decremental algorithm that given an unweighted graph
and a constant integer , supports -approximate
queries and has total update time (when
for any constant ). For comparison, in the special case of
-approximation, this improves over the state-of-the-art
algorithm by [Henzinger, Krinninger, Nanongkai, SICOMP 2016] with total update
time of . All of our results are randomized and work
against an oblivious adversary
Distributed Weighted Min-Cut in Nearly-Optimal Time
Minimum-weight cut (min-cut) is a basic measure of a network's connectivity
strength. While the min-cut can be computed efficiently in the sequential
setting [Karger STOC'96], there was no efficient way for a distributed network
to compute its own min-cut without limiting the input structure or dropping the
output quality: In the standard CONGEST model, existing algorithms with
nearly-optimal time (e.g. [Ghaffari, Kuhn, DISC'13; Nanongkai, Su, DISC'14])
can guarantee a solution that is -approximation at best while the
exact -time algorithm [Ghaffari, Nowicki,
Thorup, SODA'20] works only on *simple* networks (no weights and no parallel
edges). Here and denote the network's number of vertices and
hop-diameter, respectively. For the weighted case, the best bound was [Daga, Henzinger, Nanongkai, Saranurak, STOC'19].
In this paper, we provide an *exact* -time algorithm
for computing min-cut on *weighted* networks. Our result improves even the
previous algorithm that works only on simple networks. Its time complexity
matches the known lower bound up to polylogarithmic factors. At the heart of
our algorithm are a clever routing trick and two structural lemmas regarding
the structure of a minimum cut of a graph. These two structural lemmas
considerably strengthen and generalize the framework of Mukhopadhyay-Nanongkai
[STOC'20] and can be of independent interest.Comment: Major changes: (i) The fragment decomposition technique is
simplified, (ii) Introduction and technical overview have been redone, and
(iii) The technical sections have been made simpler for better readabilit
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
Near-Optimal Distributed Dominating Set in Bounded Arboricity Graphs
We describe a simple deterministic O(-1 log ") round distributed algorithm for (2α+ 1) (1 + ) approximation of minimum weighted dominating set on graphs with arboricity at most α. Here Δdenotes the maximum degree. We also show a lower bound proving that this round complexity is nearly optimal even for the unweighted case, via a reduction from the celebrated KMW lower bound on distributed vertex cover approximation [Kuhn, Moscibroda, and Wattenhofer JACM'16]. Our algorithm improves on all the previous results (that work only for unweighted graphs) including a randomized O(α2) approximation inO(logn) rounds [Lenzen andWattenhofer DISC'10], a deterministic O(α log ") approximation in O(log ") rounds [Lenzen and Wattenhofer DISC'10], a deterministic O(α) approximation in O(log2 ") rounds [implicit in Bansal and Umboh IPL'17 and Kuhn, Moscibroda, and Wattenhofer SODA'06], and a randomized O(α) approximation in O(α logn) rounds [Morgan, Solomon and Wein DISC'21]. We also provide a randomized O(α log ") round distributed algorithm that sharpens the approximation factor to α (1 + o (1)). If each node is restricted to do polynomial-time computations, our approximation factor is tight in the first order as it is NP-hard to achieve α - 1 - approximation [Bansal and Umboh IPL'17]