Clip Tree Applications

Outline
Introduction
Algorithmic problems on intersection graphs
Three Applications
Clique Tree Application
Chih Yi Huang
September 21, 2005
Chih Yi Huang Clique Tree Application

Outline
Introduction
Three Applications
Introduction
Introduction
Algorithms: Pefect Elimination Scheme and Independent set.
Pefect Elimination Scheme
Independent Set.
Three Applications

Outline
Introduction
Three Applications
Discuss Paper
Topic: Fast Parallel Algorithms for Chordal Graph
Source: Algorithmic problems on intersection graphs
Authors: Alejandro Alberto Schaﬀer.

Outline
Introduction
Three Applications
Survey
Database System
Sparse Matrix
Graph theory (i.e., PEO, clique cover, . . . )

Outline
Introduction
Three Applications
Survey
Database System
Power of Natural Semijoins
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.)
Sparse Matrix

Outline
Introduction
Three Applications
Survey
Database System
Sparse Matrix
Compact clique tree data structures in sparse matrix
factorizations, A Pothen (Foucs on the two new data
structures derived from Clique Tree.)
A clique tree algorithm for partitioning a chordal graph into
transitive subgraphs, BW Peyton, A Pothen, X Yuan (談如何將
clique tree 應用在解 sparse triangulated linear system.)
An introduction to chordal graphs and clique trees, Jean R.S.
Blair and Barry Peyton, Graph theory and sparse matrix
computation.

Outline
Introduction
Three Applications
Survey
Database System
Sparse Matrix
Separating clique tree and bipartition inequalities in polynomial
time, Robert D. Carr
Clique tree generalization and new subclasses of chordal
graphs, PS Kumar, CEV Madhavan (引入一些的 tree
representation 修正 clique tree not unique 的缺點, 並提出幾個新
的 subgraph 來討論。
Eﬃcient Implementation of a Minimal Triangulation Algorithm,
Pinar Heggernes and Yngve Villanger (配合 Clique Tree 和 Tree
Decomposition 來做 Minimal Triangulation in O(nm).)
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 linear
time 演算法。)

Outline
Introduction
Three Applications
Survey
Database System
Sparse Matrix
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 a
maximum-weight clique, a minimum coloring, a
maximum-weight independent set, and a minimum fill-in
elimination order. We also give the first parallel algorithms for
solving these problems by using the clique separator
decomposition. Our maximumweight independent set
algorithm applied to chordal graphs yields the most efficient
known parallel algorithm for finding a maximum-weight
independent set of a chordal graph.)

Outline
Introduction
Three Applications
Survey
Database System
Sparse Matrix
Fast parallel algorithms for chordal graphs, J. Naor and M.
Naor and A. Schaffer (present NC algorithms for finding the
following objects on chordal graphs: all maximal cliques, an
intersection graph represention, a11 optimal coloring, a perfect
elimination scheme, a maximum independent set, a minimum
clique 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。)
Dominating Sets in Chordal Graphs, KS Booth (使用 Clique
Tree 去解 Domination set 的問題。)
Algorithmic problems on intersection graphs, Alejandro Alberto
Schaffer. Ph.D. thesis

Outline
Introduction
Three Applications
Survey — results
new structures
seperator
parallel (use other infomation)

Outline
Introduction
Three Applications
Introduction
Independent Set.
Results
For chordal graph, all following probelms can be sovled in
polylogarithmic time on a PRAM having polynominal many
processors:
all cliques and chromatic number [Da86]
clique repersentation [Da86]
optimal coloring
Pefect Elimination Scheme [Da86]
Unweighted max. Independent set.
Weighted max. Independent set.
Min. clique cover
[Da86]: Elias Dahlaus, Marek Karpinski, The Matching Problem
for Strongly Chordal Graph is in NC.

Outline
Introduction
Three Applications
Introduction
Independent Set.
Results
PRAM stands for Parallel Random Access Machine, which is
an abstract machine for designing the algorithms applicable to
parallel computers.
In complexity theory, the class NC (”Nick’s Class”) is the set
of decision problems decidable in polylogarithmic time on a
parallel computer with a polynomial number of processors. In
other words, a problem is in NC if there are constants c and k
such that it can be solved in time O((log n)c) using O(nk)
parallel processors. NC is a subset of P because parallel
computers can be simulated by sequential ones. It is unknown
whether NC = P, but most researchers suspect this to be
false, meaning that there are some tractable problems which
are probably ”inherently sequential” and cannot signiﬁcantly
be sped up by using parallelism.

Outline
Introduction
Three Applications
Introduction
Independent Set.
Deﬁnition
A terminal branch is any path consisting of tree nodes of degree
two or less and containing a leaf.

Outline
Introduction
Three Applications
Introduction
Independent Set.
Figure: Example Graph G

Outline
Introduction
Three Applications
Introduction
Independent Set.
Figure: Clique Tree T of G

Outline
Introduction
Three Applications
Introduction
Independent Set.
Algorithm: Terminal Branch Identiﬁng and numbering - 1

Outline
Introduction
Three Applications
Introduction
Independent Set.
Algorithm: Terminal Branch Identiﬁng and numbering - 2

Outline
Introduction
Three Applications
Introduction
Independent Set.
Notice
Path Doubling(?)
每一階段的迭代至少會使 nodes 少一半, 所以會是 log n
Observation: If G is a chordal graph and an interval graph,
then the vertex corresponding to the interval with the leftmost
right endpoints is simplical.

Outline
Introduction
Three Applications
Introduction
Independent Set.
Algorithm: Construct an interval model for the vertices on
terminal 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).

Outline
Introduction
Three Applications
Introduction
Independent Set.

Outline
Introduction
Three Applications
Introduction
Independent Set.
Figure: C(B1) = {{b, a, c}, {b, f , a}}, |C(B1)| = 2 = PB 1,
V (B1) = {a, b, c, f } The output: interval graph GV (B1)

Outline
Introduction
Three Applications
Introduction
Independent Set.
Figure: Interval of G

Outline
Introduction
Three Applications
Introduction
Independent Set.
Algorithm: Pefect Elimination Scheme
Deﬁnition
For any terminal branch B of T, let U(B) be the set of vertices
that are in no cliques represented by tree nodes besides B.

Outline
Introduction
Three Applications
Introduction
Independent Set.
Algorithm: Pefect Elimination Scheme - 1

Outline
Introduction
Three Applications
Introduction
Independent Set.
U(B1) = {a, c}
U(B2) = {g}
U(B3) = {d}
T − B1 − B2 − B3 = A

Outline
Introduction
Three Applications
Introduction
Independent Set.
U(B1) = {a, c}
U(B2) = {g}
U(B3) = {d}
{a, c, g, d}
T − B1 − B2 − B3 = A

Outline
Introduction
Three Applications
Introduction
Independent Set.
U(B1) = {a, c}
U(B2) = {g}
U(B3) = {d}
{a, c, g, d}
G − {a, c, g, d} = e, f , b
T − B1 − B2 − B3 = A

Outline
Introduction
Three Applications
Introduction
Independent Set.
Algorithm: Pefect Elimination Scheme - 2

Outline
Introduction
Three Applications
Introduction
Independent Set.
U(B(T )) = {e, f , b}
PEO = {a, c, g, d} + {e, f , b}

Outline
Introduction
Three Applications
Introduction
Independent Set.
Correctness
Observation: If G is a chordal graph and an interval graph, then
the vertex corresponding to the interval with the leftmost right
endpoints is simplical.
Observation: A vertex in U(Bi ) can never be adjance to a vertex in
U(Bj ) if Bi = Bj

Outline
Introduction
Three Applications
Introduction
Independent Set.
Independent Set: Sequential algorithms
He86 David P. Helmbold, Ernst W. Mayr: Perfect Graphs and Parallel
Algorithms. International Conference on Parallel Processing (ICPP
’86) page 853 – 860, University Park, PA, USA, August 1986.
IEEE Computer Society Press, 1986
Fr75 Andr´as Frank, Some Polynominal Algorithmss for Certain Graphs
and Hypergraph, Proceedings of the Fifth British Combinatorial
Conference, Congressus Numerantium, No. XV. pp. 211–226

Outline
Introduction
Three Applications
Introduction
Independent Set.
Unweighted Max. Independent Set.
Use Lexicographic frist order([He86]) based on PEO.
This paper is unavialiable.

Outline
Introduction
Three Applications
Introduction
Independent Set.
Weighted Max. Independent Set: Sequential
PEO= a, c, g, d, e, f , b

Outline
Introduction
Three Applications
Introduction
Independent Set.
Figure:

Outline
Introduction
Three Applications
Introduction
Independent Set.
Weighted Max. Independent Set: Parallel
Proof: refer [He86]
Ref.: Observation: A vertex in U(Bi ) can never be adjance to a
vertex in U(Bj ) if Bi = Bj

Outline
Introduction
Three Applications
Introduction
Independent Set.
Min. Clique Cover
Theorem
Given a PEO for a chordal graph G and the corresponding
lexicographically frist maximal independent set, it is possible to
compute a min. clique cover for G in O(log n) time using O(m)
processors.

Outline
Introduction
Three Applications
Introduction
Independent Set.
Min. Clique Cover
Theorem
processors.
PEO: v1, v2, v3, , vn

Outline
Introduction
Three Applications
Introduction
Independent Set.
Min. Clique Cover
Theorem
processors.
PEO: v1, v2, v3, , vn
X(v) be the nieghbor of v listed follow v in PEO.

Outline
Introduction
Three Applications
Introduction
Independent Set.
Min. Clique Cover
Theorem
processors.
PEO: v1, v2, v3, , vn
X(v) be the nieghbor of v listed follow v in PEO.
Gravil: w1, w2, · · · , wk lexicographically frist maximal independent
set, then w1 ∪ X(w1), w2 ∪ X(w2), . . . form a min. clique cover for
G.

Outline
Introduction
Three Applications
Introduction
Chromatic polynominal of chordal graph
Testing whether a database scheme is acyclic
Large k-colorable subgraph of a chordal graph.

Outline
Introduction
Three Applications
Introduction
chromatic polynominal of chordal graph
Assume we have large x colors to color.
testing whether a database scheme is acyclic
large k-colorable subgraph of a chordal graph.

Outline
Introduction
Three Applications
Introduction
If v is simplical and deg(v) = d, then
f (G, x) = (x − d)f (G − {v}, x)

Outline
Introduction
Three Applications
Introduction
f (G, x) = (x − d)f (G − {v}, x)
By PEO, there is
f (G, x) = (x − deg(v1))(x − deg(v2)) · · · (x − deg(vn))x

Outline
Introduction
Three Applications
Introduction
f (G, x) = (x − d)f (G − {v}, x)
By PEO, there is
f (G, x) = (x − deg(v1))(x − deg(v2)) · · · (x − deg(vn))x
(x − deg(vi )) can be computed in parallel.

Outline
Introduction
Three Applications
Database Scheme: Hypergraph H with vertex set A and
hypergraph set R.

Outline
Introduction
Three Applications
hypergraph set R.
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.

Outline
Introduction
Three Applications
hypergraph set R.
The database scheme represented by H is acyclic if
G(H) is chordal.
every clique of G(H) is containrd in some hyperedge of H.

Outline
Introduction
Three Applications
hypergraph set R.
G(H) is chordal.
Easy construct in parallel.

Outline
Introduction
Three Applications
hypergraph set R.
G(H) is chordal.
Easy construct in parallel.
Based on PEO.

Outline
Introduction
Three Applications
The Dynamic Tree Expression Problem, Ernst W. Mayr1
Jeﬀrey D. Ullman, Allen Van Gelder: Parallel Complexity of Logical
Query Programs FOCS 1986: 438-454
. . . . . .
1
http://www.stormingmedia.us/73/7393/A739323.html

Outline
Introduction
Three Applications
. . . . . .
Dynamic programming
1

Outline
Introduction
Three Applications
. . . . . .
Dynamic programming
Terminal Branch
1

Outline
Introduction
Three Applications
. . . . . .
Dynamic programming
Terminal Branch
Interval Graph Model.
1

Clip Tree Applications

More Related Content

What's hot

Viewers also liked

Similar to Clip Tree Applications

More from Abner Chih Yi Huang

Recently uploaded

Clip Tree Applications