最大流問題是現今網路流理論中最根本的問題之一，近年來已有許多單一程序執行的最大流演算法被發表出來。然而，隨著分散式環境的日漸普及，分散式系統開始蓬勃發展。在這篇論文，我們將探討在一個同時有許多電腦平行運作的分散式環境中，如何快速地得到最大流。我們將實作並評比三個不同的最大流演算法，其中包括目前運算速度最快的Push Relabel演算法、在不同程序間的溝通行為上再進行優化的Push Relabel演算法、以及一個最近發表，專用於分散式環境上的最大流演算法。

The maximum flow problem is one of the most basic problems in network
flow theory. Many single-machine sequential algorithms are proposed over
the years. However, distributed environments are much more common nowadays.
In this thesis, our focus is to compute maximum flow on a distributed
environment, where each distributed group may contain multiple computers
running in parallel. We implement and evaluate three maximum-flow
algorithms, including the Push-Relabel algorithm (the best sequential algorithm),
a modified version of the Push-Relabel algorithm that is more
communication-aware, and a recently proposed algorithm by Chen et al. [1]
that is dedicated to run in a distributed environment.

1 Introduction 2
2 Model Assumptions 5
2.1 Distributed-Parallel Model . . . . . . . . . . . . . . . . . . . . 5
2.2 Distributed Graph . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Algorithms Review 8
3.1 Ford-Fulkerson and Edmonds-Karp . . . . . . . . . . . . . . . 8
3.2 Push-Relabel . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3 Maximum Flow via Graph Summary . . . . . . . . . . . . . . 10
3.3.1 The Preprocessing Step . . . . . . . . . . . . . . . . . . 10
3.3.2 The Evaluation Step . . . . . . . . . . . . . . . . . . . 13
4 Experiment 16
4.1 Input Graph Generation . . . . . . . . . . . . . . . . . . . . . 17
4.2 Algorithms to be Compared . . . . . . . . . . . . . . . . . . . 18
4.2.1 Push-Relabel . . . . . . . . . . . . . . . . . . . . . . . 19
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4.2.2 Modified Push-Relabel . . . . . . . . . . . . . . . . . . 19
4.2.3 Maximum Flow via Graph Summary . . . . . . . . . . 19
4.3 Experiment Results . . . . . . . . . . . . . . . . . . . . . . . . 20
4.3.1 Influence by Communication . . . . . . . . . . . . . . . 20
4.3.2 Influence by the Number of Nodes . . . . . . . . . . . . 21
4.3.3 Influence by the Number of Groups . . . . . . . . . . . 22
4.3.4 Influence by the Number of Connecting Vertices . . . . 23
5 Conclusion 25
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[1] Y. M. Chen, P. C. Wu, and W. K. Hon. Maximum Flow via Graph
Summaries. Manuscript in preparation, 2014.
[2] T. H. Cormen, C. E. Leicerson, R. L. Rivest, and C. Stein. Introduction
to Algorithms. MIT Press, 2009.
[3] J. Edmonds and R. M. Karp. Theoretical Improvements in Algorithmic
Efficiency for Network Flow Problems. Journal of the ACM, 19(2):248–
264, 1972.
[4] L. R. Ford and D. R. Fulkerson. Maximal Flow through a Network.
Canadian Journal of Mathematics, 8(3):399–404, 1956.
[5] A. V. Goldberg and R. E. Tarjan. A New Approach to the Maximum-
Flow Problem. Journal of the ACM, 35(4):921–940, 1988.
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