clip tree applications

OutlineIntroduction

Algorithmic problems on intersection graphsThree Applications

Clique Tree Application

Chih Yi Huang

September 21, 2005

Chih Yi Huang Clique Tree Application

OutlineIntroduction


Introduction

Algorithmic problems on intersection graphsIntroductionAlgorithms: Pefect Elimination Scheme and Independent set.Pefect Elimination SchemeIndependent Set.

Three Applications


OutlineIntroduction


Discuss Paper

I Topic: Fast Parallel Algorithms for Chordal Graph

I Source: Algorithmic problems on intersection graphs

I Authors: Alejandro Alberto Schaffer.


OutlineIntroduction


Survey

I Database System

I Sparse Matrix

I Graph theory (i.e., PEO, clique cover, . . . )


OutlineIntroduction


Survey

I Database SystemI Power of Natural SemijoinsI On the Desirability of Acyclic Database Schemes, Catriel Beeri

and Ronald Fagin and David Maier and Mihalis Yannakakis (沒有直接使用到 clique tree 的技術。但是有提到 clique 相關的

hypergraph.)

I Sparse Matrix



OutlineIntroduction


Survey

I Database SystemI Sparse Matrix

I Compact clique tree data structures in sparse matrixfactorizations, A Pothen (Foucs on the two new datastructures derived from Clique Tree.)

I A clique tree algorithm for partitioning a chordal graph intotransitive subgraphs, BW Peyton, A Pothen, X Yuan (談如何將

clique tree 應用在解 sparse triangulated linear system.)I An introduction to chordal graphs and clique trees, Jean R.S.

Blair and Barry Peyton, Graph theory and sparse matrixcomputation.



OutlineIntroduction


Survey

I Database SystemI Sparse MatrixI Graph theory (i.e., PEO, clique cover, . . . )

I Separating clique tree and bipartition inequalities in polynomialtime, Robert D. Carr

I Clique tree generalization and new subclasses of chordalgraphs, PS Kumar, CEV Madhavan (引入一些的 treerepresentation 修正 clique tree not unique 的缺點, 並提出幾個新

的 subgraph 來討論。I Efficient Implementation of a Minimal Triangulation Algorithm,

Pinar Heggernes and Yngve Villanger (配合 Clique Tree 和 TreeDecomposition 來做 Minimal Triangulation in O(nm).)

I Chordal Graphs and Their Clique Graphs, P Galinier, M Habib,C Paul (拓展 Clique Tree 到 Clique Graph 然後配合自 LBFS 與

MCS 改良的 Greedy Strategy, 提出建構 Clique Tree 的 lineartime 演算法。)


OutlineIntroduction


Survey

I Database SystemI Sparse MatrixI Graph theory (i.e., PEO, clique cover, . . . )

I Fast parallel algorithms for the clique separator decomposition,Elias Dahlhaus and Marek Karpinski and Mark B. Novick (給了

有效率的找出 clique separator 的 algorithms 並更進一步的處理以

下問題。 These optimization problems include: finding amaximum-weight clique, a minimum coloring, amaximum-weight independent set, and a minimum fill-inelimination order. We also give the first parallel algorithms forsolving these problems by using the clique separatordecomposition. Our maximumweight independent setalgorithm applied to chordal graphs yields the most efficientknown parallel algorithm for finding a maximum-weightindependent set of a chordal graph.)


OutlineIntroduction


Survey

I Database System

I Sparse MatrixI Graph theory (i.e., PEO, clique cover, . . . )

I Fast parallel algorithms for chordal graphs, J. Naor and M.Naor and A. Schaffer (present NC algorithms for finding thefollowing objects on chordal graphs: all maximal cliques, anintersection graph represention, a11 optimal coloring, a perfectelimination scheme, a maximum independent set, a minimumclique cover, and the chromatic polynomial. 基本上沒有用到

clique tree, 而是使用 subtree graph, 但是做了一點延伸, 使他能應

用到解決上述問題。)I Dominating Sets in Chordal Graphs, KS Booth (使用 Clique

Tree 去解 Domination set 的問題。)I Algorithmic problems on intersection graphs, Alejandro Alberto

Schaffer. Ph.D. thesis


OutlineIntroduction


Survey — results

I new structures

I seperator

I parallel (use other infomation)


OutlineIntroduction


IntroductionPefect Elimination SchemeIndependent Set.

Results

For chordal graph, all following probelms can be sovled inpolylogarithmic time on a PRAM having polynominal manyprocessors:

I all cliques and chromatic number [Da86]

I clique repersentation [Da86]

I optimal coloring

I Pefect Elimination Scheme [Da86]

I Unweighted max. Independent set.

I Weighted max. Independent set.

I Min. clique cover

[Da86]: Elias Dahlaus, Marek Karpinski, The Matching Problemfor Strongly Chordal Graph is in NC.


OutlineIntroduction



Results

I PRAM stands for Parallel Random Access Machine, which isan abstract machine for designing the algorithms applicable toparallel computers.

I In complexity theory, the class NC (”Nick’s Class”) is the setof decision problems decidable in polylogarithmic time on aparallel computer with a polynomial number of processors. Inother words, a problem is in NC if there are constants c and ksuch that it can be solved in time O((log n)c) using O(nk)parallel processors. NC is a subset of P because parallelcomputers can be simulated by sequential ones. It is unknownwhether NC = P, but most researchers suspect this to befalse, meaning that there are some tractable problems whichare probably ”inherently sequential” and cannot significantlybe sped up by using parallelism.


OutlineIntroduction



Pefect Elimination Scheme

DefinitionA terminal branch is any path consisting of tree nodes of degreetwo or less and containing a leaf.

Figure: terminal branch


OutlineIntroduction



Figure: Example Graph G


OutlineIntroduction



Figure: Clique Tree T of G


OutlineIntroduction



Algorithm: Terminal Branch Identifing and numbering - 1

Figure: Find node N which reachs leaf only via nodes with degree 2.


OutlineIntroduction




Figure: compute every node which labeled.


OutlineIntroduction




Figure: Output


OutlineIntroduction



Notice

I Path Doubling(?)

I 每一階段的迭代至少會使 nodes 少一半, 所以會是 log n

I Observation: If G is a chordal graph and an interval graph,then the vertex corresponding to the interval with the leftmostright endpoints is simplical.


OutlineIntroduction



Algorithm: Construct an interval model for the vertices onterminal branch.

Definition

C (B): the set of cliques corresponding to nodes on B.

Definition

V (B): the set of vertices of G occurring in some member of C (B).


OutlineIntroduction




OutlineIntroduction



Figure: C (B1) = {{b, a, c}, {b, f , a}}, |C (B1)| = 2 = PB1,V (B1) = {a, b, c , f } The output: interval graph GV (B1)


OutlineIntroduction




OutlineIntroduction



Figure: Interval of G


OutlineIntroduction



Algorithm: Pefect Elimination Scheme

DefinitionFor any terminal branch B of T , let U(B) be the set of verticesthat are in no cliques represented by tree nodes besides B.

Figure: example of vertices in U(B)


OutlineIntroduction



Algorithm: Pefect Elimination Scheme - 1

Figure:


OutlineIntroduction



I U(B1) = {a, c}I U(B2) = {g}I U(B3) = {d}

I {a, c , g , d}I G − {a, c , g , d} = e, f , b

I T − B1 − B2 − B3 = A


OutlineIntroduction



I U(B1) = {a, c}I U(B2) = {g}I U(B3) = {d}I {a, c , g , d}

I G − {a, c , g , d} = e, f , b

I T − B1 − B2 − B3 = A


OutlineIntroduction



I U(B1) = {a, c}I U(B2) = {g}I U(B3) = {d}I {a, c , g , d}I G − {a, c , g , d} = e, f , b

I T − B1 − B2 − B3 = A


OutlineIntroduction



Algorithm: Pefect Elimination Scheme - 2

Figure:


OutlineIntroduction



I U(B(T′)) = {e, f , b}

I PEO = {a, c , g , d} + {e, f , b}


OutlineIntroduction



Correctness

Observation: If G is a chordal graph and an interval graph, thenthe vertex corresponding to the interval with the leftmost rightendpoints is simplical.

Observation: A vertex in U(Bi ) can never be adjance to a vertex inU(Bj) if Bi 6= Bj


OutlineIntroduction



Independent Set: Sequential algorithms

He86 David P. Helmbold, Ernst W. Mayr: Perfect Graphs and ParallelAlgorithms. International Conference on Parallel Processing (ICPP’86) page 853 – 860, University Park, PA, USA, August 1986.IEEE Computer Society Press, 1986

Fr75 Andras Frank, Some Polynominal Algorithmss for Certain Graphsand Hypergraph, Proceedings of the Fifth British CombinatorialConference, Congressus Numerantium, No. XV. pp. 211–226


OutlineIntroduction



Unweighted Max. Independent Set.

I Use Lexicographic frist order([He86]) based on PEO.

I This paper is unavialiable.


OutlineIntroduction



Weighted Max. Independent Set: Sequential

PEO= a, c , g , d , e, f , b

Figure:


OutlineIntroduction



Figure:


OutlineIntroduction



Weighted Max. Independent Set: Parallel

I Proof: refer [He86]

I Ref.: Observation: A vertex in U(Bi ) can never be adjance to avertex in U(Bj) if Bi 6= Bj


OutlineIntroduction



Min. Clique Cover

TheoremGiven a PEO for a chordal graph G and the correspondinglexicographically frist maximal independent set, it is possible tocompute a min. clique cover for G in O(log n) time using O(m)processors.

I PEO: v1, v2, v3, , vn

I X (v) be the nieghbor of v listed follow v in PEO.

I Gravil: w1,w2, · · · ,wk lexicographically frist maximal independentset, then w1 ∪ X (w1), w2 ∪ X (w2), . . . form a min. clique cover forG .


OutlineIntroduction


Introduction

I Chromatic polynominal of chordal graph

I Testing whether a database scheme is acyclic

I Large k-colorable subgraph of a chordal graph.


OutlineIntroduction


Introduction

I chromatic polynominal of chordal graph

I Assume we have large x colors to color.

I If v is simplical and deg(v) = d , thenf (G , x) = (x − d)f (G − {v}, x)

I By PEO, there isf (G , x) = (x − deg(v1))(x − deg(v2)) · · · (x − deg(vn))x

I (x − deg(vi )) can be computed in parallel.

I testing whether a database scheme is acyclic

I large k-colorable subgraph of a chordal graph.


OutlineIntroduction


testing whether a database scheme is acyclic

I Database Scheme: Hypergraph H with vertex set A andhypergraph set R.

I G (H) is a graph with vertex set A that contain the edge a1—a2 iff.there is a relation schema r ∈ R containing both a1 and a2.

I The database scheme represented by H is acyclic if

I G (H) is chordal.

I every clique of G (H) is containrd in some hyperedge of H.

I Easy construct in parallel.

I Based on PEO.


OutlineIntroduction


Large k-colorable subgraph of a chordal graph.

I The Dynamic Tree Expression Problem, Ernst W. Mayr1

I Jeffrey D. Ullman, Allen Van Gelder: Parallel Complexity of LogicalQuery Programs FOCS 1986: 438-454

I . . . . . .

I Dynamic programming

I Terminal Branch

I Interval Graph Model.

1http://www.stormingmedia.us/73/7393/A739323.html


http://www.stormingmedia.us/73/7393/A739323.html

clip tree applications

Software

clique graphs

clique tree algorithm

clique cover

separating clique tree

clique trees

clique separator decomposition

maximumweight clique

intersection graphs