Parallel Minimum Cuts in Near-linear Work and Low Depth

We present the first near-linear work and poly-logarithmic depth algorithm for computing a minimum cut in an undirected graph. Previous parallel algorithms with poly-logarithmic depth required at least quadratic work in the number of vertices. In a graph with n vertices and m edges, our randomized algorithm computes the minimum cut with high probability in O ( m log 4 n ) work and O (log 3 n ) depth. This result is obtained by parallelizing a data structure that aggregates weights along paths in a tree, in addition exploiting the connection between minimum cuts and approximate maximum packings of spanning trees. In addition, our algorithm improves upon bounds on the number of cache misses incurred to compute a minimum cut.

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THREE TECHNIQUES FOR PARALLEL MAINTENANCE OF A MINIMUM SPANNING TREE UNDER BATCH OF UPDATES

Parallel Processing Letters ◽

10.1142/s0129626496000212 ◽

1996 ◽

Vol 06 (02) ◽

pp. 213-222 ◽

Cited By ~ 3

Author(s):

PAOLO FERRAGINA ◽

FABRIZIO LUCCIO

Keyword(s):

Data Structure ◽

Parallel Algorithms ◽

Spanning Tree ◽

Minimum Spanning Tree ◽

Undirected Graph ◽

Batch Size ◽

Insertions And Deletions ◽

Erew Pram ◽

Pram Model ◽

Proper Values

In this paper we provide three simple techniques to maintain in parallel the minimum spanning tree of an undirected graph under single or batch of edge updates (i.e., insertions and deletions). Our results extend the use of the sparsification data structure to the EREW PRAM model. For proper values of the batch size, our algorithms require less time and work than the best known dynamic parallel algorithms.

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Cut Star of an Undirected Graph

Journal of Interconnection Networks ◽

10.1142/s0219265921500067 ◽

2021 ◽

pp. 2150006

Author(s):

Saeid Hanifehnezhad ◽

Ardeshir Dolati

Keyword(s):

Data Structure ◽

Global Minimum ◽

Undirected Graph ◽

Minimum Cut ◽

New Type

Suppose that [Formula: see text] is an undirected graph. An ordered pair [Formula: see text] of the vertices of the graph [Formula: see text] is called a pendant pair for the graph if [Formula: see text] is a minimum cut separating [Formula: see text] and [Formula: see text] Stoer and Wagner obtained a global minimum cut of [Formula: see text] by using pendant pairs of [Formula: see text] and its contractions. A Gomory Hu tree of the graph [Formula: see text] is a very useful data structure which gives us all the minimum s-t cuts of [Formula: see text] for every pair of distinct vertices [Formula: see text] and [Formula: see text] In this paper, we construct a new type of tree for the graph [Formula: see text] called cut star, by using pendant pairs of [Formula: see text] and its contractions. A cut star of the graph [Formula: see text] is constructed more quickly than a Gomory Hu tree of [Formula: see text] We characterize a class of graphs for which a cut star of a graph of this class is also a Gomory Hu tree.

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AN EREW PARM ALGORITHM FOR UPDATING MINIMUM SPANNING TREES

Parallel Processing Letters ◽

10.1142/s012962649900013x ◽

1999 ◽

Vol 09 (01) ◽

pp. 111-122 ◽

Cited By ~ 2

Author(s):

SAJAL K. DAS ◽

PAOLO FERRAGINA

Keyword(s):

Data Structure ◽

Spanning Tree ◽

Spanning Trees ◽

Minimum Spanning Tree ◽

Undirected Graph ◽

Substantial Improvement ◽

Minimum Spanning Trees ◽

Single Edge ◽

Sequential Approach ◽

Work Complexity

We propose a parallel algorithm for the EREW PRAW model that maintains a minimum spanning tree (MST) of an undirected graph under single edge insertions and deletions. For a graph of n nodes and m edges, each update requires O( log n) time and O(m 2/3 log n) work. This is a substantial improvement over the known bounds on the work complexity. Our algorithm uses a partition of the MST, similar to the sequential approach due to Frederickson [6], and also employs a novel data structure for efficiently managing edge insertions in parallel.

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New BSP/CGM algorithms for spanning trees

The International Journal of High Performance Computing Applications ◽

10.1177/1094342018803672 ◽

2018 ◽

Vol 33 (3) ◽

pp. 444-461

Author(s):

Jucele França de Alencar Vasconcellos ◽

Edson Norberto Cáceres ◽

Henrique Mongelli ◽

Siang Wun Song ◽

Frank Dehne ◽

...

Keyword(s):

Parallel Algorithms ◽

Parallel Machines ◽

Spanning Trees ◽

Graphics Processing Unit ◽

Computation Time ◽

General Purpose ◽

Coarse Grained ◽

Processing Unit ◽

List Ranking ◽

Bulk Synchronous Parallel

Computing a spanning tree (ST) and a minimum ST (MST) of a graph are fundamental problems in graph theory and arise as a subproblem in many applications. In this article, we propose parallel algorithms to these problems. One of the steps of previous parallel MST algorithms relies on the heavy use of parallel list ranking which, though efficient in theory, is very time-consuming in practice. Using a different approach with a graph decomposition, we devised new parallel algorithms that do not make use of the list ranking procedure. We proved that our algorithms are correct, and for a graph [Formula: see text], [Formula: see text], and [Formula: see text], the algorithms can be executed on a Bulk Synchronous Parallel/Coarse Grained Multicomputer (BSP/CGM) model using [Formula: see text] communications rounds with [Formula: see text] computation time for each round. To show that our algorithms have good performance on real parallel machines, we have implemented them on graphics processing unit. The obtained speedups are competitive and showed that the BSP/CGM model is suitable for designing general purpose parallel algorithms.

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Simple randomized parallel algorithms for finding a maximal matching in an undirected graph

IEEE Proceedings of the SOUTHEASTCON '91 ◽

10.1109/secon.1991.147821 ◽

2002 ◽

Author(s):

S.B. Yang ◽

S.K. Dhall ◽

S. Lakshmivarahan

Keyword(s):

Parallel Algorithms ◽

Undirected Graph ◽

Maximal Matching

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Board Game Supporting Learning Prim’s Algorithm and Dijkstra’s Algorithm

Methods and Innovations for Multimedia Database Content Management ◽

10.4018/978-1-4666-1791-9.ch015 ◽

2012 ◽

pp. 256-270

Author(s):

Wen-Chih Chang ◽

Te-Hua Wang ◽

Yan-Da Chiu

Keyword(s):

Data Structure ◽

Spanning Tree ◽

Spanning Trees ◽

Minimum Spanning Tree ◽

Experimental Results ◽

Dijkstra’S Algorithm ◽

Board Game ◽

Tree Algorithm ◽

Dijkstra's Algorithm ◽

Tree Algorithms

The concept of minimum spanning tree algorithms in data structure is difficult for students to learn and to imagine without practice. Usually, learners need to diagram the spanning trees with pen to realize how the minimum spanning tree algorithm works. In this paper, the authors introduce a competitive board game to motivate students to learn the concept of minimum spanning tree algorithms. They discuss the reasons why it is beneficial to combine graph theories and board game for the Dijkstra and Prim minimum spanning tree theories. In the experimental results, this paper demonstrates the board game and examines the learning feedback for the mentioned two graph theories. Advantages summarizing the benefits of combining the graph theories with board game are discussed.

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DISTANCE PRESERVING SUBTREES IN MINIMUM AVERAGE DISTANCE SPANNING TREES

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830913500109 ◽

2013 ◽

Vol 05 (03) ◽

pp. 1350010

Author(s):

LAURENT LYAUDET ◽

PAULIN MELATAGIA YONTA ◽

MAURICE TCHUENTE ◽

RENÉ NDOUNDAM

Keyword(s):

Approximate Solution ◽

Spanning Tree ◽

Path Length ◽

Spanning Trees ◽

Undirected Graph ◽

Average Distance ◽

Edge Length ◽

The Other ◽

First Results ◽

Positive Length

Given an undirected graph G = (V, E) with n vertices and a positive length w(e) on each edge e ∈ E, we consider Minimum Average Distance (MAD) spanning trees i.e., trees that minimize the path length summed over all pairs of vertices. One of the first results on this problem is due to Wong who showed in 1980 that a Distance Preserving (DP) spanning tree rooted at the median of G is a 2-approximate solution. On the other hand, Dankelmann has exhibited in 2000 a class of graphs where no MAD spanning tree is distance preserving from a vertex. We establish here a new relation between MAD and DP trees in the particular case where the lengths are integers. We show that in a MAD spanning tree of G, each subtree H′ = (V′, E′) consisting of a vertex [Formula: see text] and the union of branches of [Formula: see text] that are each of size less than or equal to [Formula: see text], where w+ is the maximum edge-length in G, is a distance preserving spanning tree of the subgraph of G induced by V′.

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