Tree Decompositio

nBenoit Vanalderweireldt

Phan Quoc TrungTram Minh TriVu Thi Phuong

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Summary

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• Introduction• Definition• Requirement• Data structure • Graph generators• Algorithm• Example• Benchmark and tests• Interpretation• Conclusion

Definition

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Tree decomposition

Definition

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Tree width

1.The width of a bag is the cardinality of the bag. 2.The width of a tree decomposition is the size of the

largest bag minus one 3.The treewidth of a graph G is the minimum width

among all possible tree decompositions of G minus 1, denoted as tw(G). Note that trees and forests are precisely the structures

with treewidth of 1.

Introduction

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Graph reachability answering using tree decomposition ?

Is it profitable to decompose a graphIn order to test reachability ?

Requirement

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1. Understand notion of tree decomposition2. Develop a data structure for graphs3. Develop different graph generators4. Implement the 4 parts of the algorithms5. Make benchmark6. Analyze result7. Make conclusion8. Write report9. Present our work

Data structure

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• Vertex• Tree • Node

Graph generators

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In order to test the algorithm, we generated various kind of graphs :1. Inclusive square graph2. Friendship graph3. Random graph

Friendship graph

Algorithm

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Algorithm 1 start with reduction of an undirected graph before construction the tree

Algorithm

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Explanation in following examples

Algorithm

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Example

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Random graph

Example

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Random graph

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Example

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1

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34

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34

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34

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Complete graph

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Example

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6 7

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6 7

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6 7

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Wheel Graph

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6 7

1 2

3 45

Example

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31 2

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46 5

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Bipartite Graph

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Example

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Cycle graph

Benchmark

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The main goal of benchmark is to reveal in what condition it is gainful to reduced graph in order to check reachability.We run multiple series of benchmark :• Various number of vertices and neighbors• …..

Every test series have been past at least 2 times (to operating system accident)We compare effective results with expected resultsOur series (around 20 measures) vary (increase or decrease) numbers of edges or vertices

What else ?

Benchmark

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In order to have workable measure, we have to establish a protocol to collect data. We quickly realize that to have meaningful results, we need to apply the algorithm of really large graph (over hundred of vertices).

Put some capture of result like in report

Interpretation

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From these tests we found that the execution time of this algorithm is influenced by number of edges not really by number of vertices. If the density of edges increases then the computing time for the graph reduction increase. In order to use this algorithm in a goodresponding time, the density of edges must be well know.

Advantage & Disadvantage

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Advantage of Tree Decomposition:Solving shortest path query answering over undirected large

graphs.Many intractable problems (Independent Set, the Hamiltonian

Circuits) can be solved in polynomial time or even in linear timeDisadvantage of Tree Decomposition:

Calculating the tree width of a graph is hard. (NP-complete)It’s impossible to guarantee that good performance will be

obtained even though the tree width can be determined.

To solve really hard problems efficiently by using the tree decomposition based approaches, we have to require that the underlying graphs have bounded tree width (less than 10)

Conclusion

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1. In this project, we implemented the algorithm of tree decomposition as the index structure for a large graphs to answer reachability queries efficiently.

2. This approach used to solve intractable problems (NP-complete)3. One of the limitation of tree decomposition based approach is the

underlying graphs have bounded treewidth, but in this algorithm the treewidth is adapted to be related to degree-l reduction, instead of the treewidth.

Acknowledgements

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This project was supported by a grant from the professor Bruno Courcelle and Anne Dicky

Reference1.Efficient Graph Reachability Query Answering using Tree Decomposition, by

Fang Wei (Computer Science Department, University of Freiburg, Germany).2.Discrete Math for Bioinformatics, by A. Bockmayr/K3. A partial k-arboretum of graph with bounded treewidth, by Hans Bodlaender