1,461 research outputs found
Distributed Approximation of Minimum Routing Cost Trees
We study the NP-hard problem of approximating a Minimum Routing Cost Spanning
Tree in the message passing model with limited bandwidth (CONGEST model). In
this problem one tries to find a spanning tree of a graph over nodes
that minimizes the sum of distances between all pairs of nodes. In the
considered model every node can transmit a different (but short) message to
each of its neighbors in each synchronous round. We provide a randomized
-approximation with runtime for
unweighted graphs. Here, is the diameter of . This improves over both,
the (expected) approximation factor and the runtime
of the best previously known algorithm.
Due to stating our results in a very general way, we also derive an (optimal)
runtime of when considering -approximations as done by the
best previously known algorithm. In addition we derive a deterministic
-approximation
Balancing Minimum Spanning and Shortest Path Trees
This paper give a simple linear-time algorithm that, given a weighted
digraph, finds a spanning tree that simultaneously approximates a shortest-path
tree and a minimum spanning tree. The algorithm provides a continuous
trade-off: given the two trees and epsilon > 0, the algorithm returns a
spanning tree in which the distance between any vertex and the root of the
shortest-path tree is at most 1+epsilon times the shortest-path distance, and
yet the total weight of the tree is at most 1+2/epsilon times the weight of a
minimum spanning tree. This is the best tradeoff possible. The paper also
describes a fast parallel implementation.Comment: conference version: ACM-SIAM Symposium on Discrete Algorithms (1993
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
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