Graph Searching and Related Problems

2005/11/04 中山大學資訊工程研究所

Graph Searching and Graph Searching and Related ProblemsRelated Problems

中央研究院中央研究院資訊科學研究所資訊科學研究所

高明達高明達

Graph Searching ProblemsGraph Searching Problems

Input:Input:

A graph is looked as a system of A graph is looked as a system of contaminatedcontaminated tunnels. tunnels.

Goal:Goal:

To find a To find a strategystrategy using the least using the least number of number of searcherssearchers to clear the to clear the entire graph. entire graph.

Graph Searching ProblemsGraph Searching Problems

• Operations• Place Remove Move

• Clearing rules• move along an edge

• guard the two endpoints of an edge

• Recontamination

a cleared edge connects a contaminated edge by a path with no searcher

• Versions:

OperationsRule

Edge placemove

remove

move

Node placeguard

remove

Mixed placeboth

remove

move

Searching StrategySearching Strategy

Node search number =3

Edge search number = 2

There exists an optimal search strategy without recontamination.

Equivalent ProblemsEquivalent Problems

Node searching problem is Node searching problem is equivalent to equivalent to • Interval thickness problemInterval thickness problem• Gate matrix layout problem Gate matrix layout problem • Vertex separation problem Vertex separation problem • Narrowness problemNarrowness problem• Path-width problemPath-width problem

Interval Thickness ProblemInterval Thickness Problem

Given a graph G=(V, E), find an Given a graph G=(V, E), find an interval graph G’= (V, E’), E interval graph G’= (V, E’), E E’, E’, such that the maximum clique of G’ such that the maximum clique of G’ is minimum. is minimum.

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Gate Matrix Layout ProblemGate Matrix Layout Problem

Given a gate matrix, each column Given a gate matrix, each column represents a gate and each row represents a gate and each row represents a net, the problem is to represents a net, the problem is to permute the columns such that the permute the columns such that the nets can be arranged in minimal nets can be arranged in minimal number of tracks. number of tracks.

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incompatible graph

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G2 G5

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G1G2 G3G5 G6G4 G7G8

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Vertex Separation ProblemVertex Separation Problem

GG=(=(VV, , EE)) Linear layout: Linear layout: LL: : VV {1, 2, …, | {1, 2, …, |VV|}|} VVLL((ii) = ) = {{uu||LL((uu) ) ii, , ww, , LL((ww)>)>ii, (, (uu, , ww))EE}.}. vsvsLL((GG)) = = maxmaxii {|{|VVLL((ii)|})|}

vsvs((GG) = min) = minLL vsvsLL((GG))

b

a

e

d

h

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f g

a b e d c f g h

V(1) = {a}V(2) = {a, b}V(3) = {a, b, e}V(4) = {a, e, d}V(5) = {d, e}V(6) = {e, f}V(7) = {e, f}V(8) = {}

c a d b e f h g

V(1) = {c}V(2) = {a}V(3) = {d}V(4) = {d}V(5) = {d, e}V(6) = {e, f}V(7) = {e, f}V(8) = {}

a b e d c f g h

c a d b e f h g

Narrowness ProblemNarrowness Problem

Inner

Memory

Outer

MemoryShack

Limited

in-sequence out-sequence

Short term memory constraint: A vertex can be moved from the shack to OM only if all of the vertices connected to it by an arc are also in the shack or already in OM.


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a

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In-sequenceb a d e c f h g

b a e d ch fgOut-sequence

In-sequencea b c e d f h g

b ae d ch fgOut-sequence

Path-width ProblemPath-width ProblemPath decompositionPath decomposition of of GG=(=(VV, , EE):): XX={={XX11, , XX22, …, , …, XXnn}, }, XXi i VV, , 1 1 i i n.n. ((aa,, b b))E, E, i i such that {such that {aa,, b b} } XXii.. i < j < k, Xi < j < k, Xii XXk k XXjj..Path-widthPath-width of of G = G = minminXX maxmaxii {{|X|Xii|-|-1}1}

A

C

B

D

E F

X ={{A, B}, {B, C, E}, {C, D, E}, {E, F}}

Tree-width ProblemTree-width ProblemTree decompositionTree decomposition of of GG=(=(VV, , EE):): XX={={XX11, , XX22, …, , …, XXnn}, }, XXi i VV, , 1 1 i i n.n. ((aa,, b b))E, E, i i such that {such that {aa,, b b} } XXii.. XXii, , XXjj,, XXkk,, XXjj is on the path from is on the path from XXii to to XXkk,, XXii

XXk k XXjj..

Tree-widthTree-width of of G = G = minminXX maxmaxii {{|X|Xii|-|-1}1}Tree decomposition is equivalent to the emTree decomposition is equivalent to the em

bedding into a chordal graph. bedding into a chordal graph.

A computing paradigm based on tree deA computing paradigm based on tree decomposition using dynamic programmicomposition using dynamic programming. ng. • Maximum independent set problemMaximum independent set problem• Maximum clique problemMaximum clique problem• Pathwidth problemPathwidth problem• Minimum fill-in problemMinimum fill-in problem• Minimum dominating set problemMinimum dominating set problem• Hamiltonian cycle problemHamiltonian cycle problem• Minimum coloring problemMinimum coloring problem• Minimum edge coloring problemMinimum edge coloring problem• …………………………………………

Maximum Independent SetMaximum Independent Set

X1 X2 X3

X5

X7

X6

X4

I: an independent set of [X7]

{Xi}= subgraph induced by Xi and its descendents.

Ii = I [Xi], i = 5. 6.

{Ii} the MIS containing Ii in {Xi}.

{I} = I {I5} {I6}

Bounded treewidth graphs have Bounded treewidth graphs have linear tilinear timeme algorithm for the decision problem o algorithm for the decision problem of pathwidth problem. f pathwidth problem.

Tree Tree serial-parallel graphsserial-parallel graphs cactus graphscactus graphs partial 2-treepartial 2-tree

Previous ResultsClasses of graphsClasses of graphs ComplexityComplexity

GeneralGeneral NP-completeNP-complete

General, degree General, degree 3 3 NP-completeNP-complete

Planar, degree Planar, degree 3 3 NP-completeNP-complete

chordalchordal NP-completeNP-complete

Chordal dominoChordal domino NP-completeNP-complete

Chordal bipartiteChordal bipartite NP-completeNP-complete

starlikestarlike NP-completeNP-complete

bipartitebipartite NP-completeNP-complete

Bipartite distance-hereditaryBipartite distance-hereditary NP-completeNP-complete

Circle Circle NP-completeNP-complete

Previous Results

Classes of graphsClasses of graphs ComplexityComplexity

k-starlikek-starlike P P OO((mnmnkk))Split Split P P OO((mnmn))Co-chordalCo-chordal PP

Trapezoid Trapezoid PP

Interval Interval P P OO((mm++nn))Permutation Permutation P P OO((nknk))Partial k-treePartial k-tree P P OO((nn))UnicyclicUnicyclic P P O(O(nn))CographCograph PP

TreeTree P P O(O(nn))Block Block P O(bc+c2+n)

Previous Results on TreesPrevious Results on Trees

Search number: O(Search number: O(nn)) Search strategy:O(Search strategy:O(nn log lognn) ) O(O(nn)) Node search: Node search:

• R. H. Moehring 1990, R. H. Moehring 1990, P. Scheffler 1990, P. Scheffler 1990, J. Ellis, I. Sudborough, J. Turner 1994J. Ellis, I. Sudborough, J. Turner 1994

BranchBranch

t v TTtvtv

Three-Branch LemmaThree-Branch Lemma [Pa76,Sc90, EST94, TUK95][Pa76,Sc90, EST94, TUK95] For any tree For any tree TT, , ((TT) ) kk+1+1, k ≧ 2 if and , k ≧ 2 if and

only if there exists a vertex only if there exists a vertex tt with at le with at least three branches ast three branches TTtutu, , TTtvtv,, T Ttwtw such tha such that t ((TTtutu)≧)≧kk, , ((TTtvtv)≧)≧kk, , ((TTtwtw)≧)≧kk. .

kkk

k +1

σ = 3σ = 3

Node Search ExampleNode Search Example

Node search number = 3

σ = 2

AvenueAvenue

[MHGJP88] A path [MHGJP88] A path PP is an is an avenueavenue of of TT if if the following conditions hold:the following conditions hold:• For every path branch For every path branch T T ’ at ’ at PP, , ((T T ’) = ’) = ((TT),),• For every non-path branch For every non-path branch T T ’ at ’ at PP, , ((T T ’) < ’) <

((TT).).

[MHGJP88] [MHGJP88] For any tree For any tree TT, , TT has an has an avenue.avenue. If the length of the avenue is at If the length of the avenue is at least two, then the avenue is least two, then the avenue is uniqueunique..

AvenueAvenue

(T) = k

<k<k

<k

=kk=

Example (Node)Example (Node)

Avenue branchNon-avenue branch

Extended AvenueExtended Avenue

A path A path PP is an is an extended avenueextended avenue of of TT if the following conditions hold:if the following conditions hold:• For every path branch For every path branch T T ’ at ’ at PP, , ((T T ’) ’)

((TT),),• For every non-path branch For every non-path branch T T ’ at ’ at PP, , ((T T ’) ’)

< < ((TT).). An extended avenue must contain An extended avenue must contain

the avenue of the tree. the avenue of the tree.

Extended AvenueExtended Avenue

Avenue SystemAvenue System

If If TT is a basis, then is a basis, then

If If TT is is notnot a basis: a basis:

where where PP is the set of nonpath branch is the set of nonpath branches of P.es of P.

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Avenue System (Node)Avenue System (Node)

Searching Strategy (Node)Searching Strategy (Node)

Avenue Tree Avenue Tree

Avenue TreeAvenue Tree

Let Let P P AA((TT) and ) and TTPP denote the denote the branch in branch in FF((AA((TT)) with P as its main )) with P as its main avenue. avenue.

AA((TT) is the set of nodes of the logical ) is the set of nodes of the logical avenue tree. avenue tree.

PP is the parent of is the parent of QQ if and only if if and only if TTPP is is a non-path branch of a non-path branch of QQ. .

Avenue Tree ConstructionAvenue Tree Construction

Look tree Look tree TT as a rooted tree. as a rooted tree. Labeling algorithmLabeling algorithm Avenue tree constructionAvenue tree construction

Labeling AlgorithmLabeling Algorithm

A vertex A vertex uu is is kk-critical-critical if the subtree rooted if the subtree rooted at at uu is of is of ((TTuu)=)=kk and there exists and there exists exactlyexactly twtwo children o children vv and and ww of of uu such that such that ((TTvv)=)=kk an and d ((TTww)=)=kk..

k k<k <k

vw

u

Labeling AlgorithmLabeling Algorithm

Label Label

pp aaaaaa ...,),,...,,( 2121

puuu ,...,, 21 Critical verticesCritical vertices

up= u

u2 u1 u3

Labeling Algorithm (case 1)Labeling Algorithm (case 1)

If more than

2 children of = k

Then

= (k+1)’

(k+1)’

k k k


If exactly

2 children of = k and one of them contains a k-critical vertex

Then

= (k+1)’

k’k

(k+1)’

kk

k

< k


If exactly

2 children of = k and none of them contain a k-critical vertex

Then

= k

k’ k’

k

< k< k


If exactly

1 children of = k and it contains a k-critical vertex x and (Tu[x]) =k

Then

= (k+1)’

k

(k+1)’

x: k-critical

: Tu[x]

x

kk

k

< k < k


If exactly

1 children of = k and it contains a k-critical vertex x and (Tu[x]) < k

Then

= k

k

k

x: k-critical

: Tu[x]

x

kk

k

< k < k


If exactly

1 children of = k and it contains no k-critical vertex Then

= k’

k’

k’

< k < k

Running ExampleRunning Example

(8) (4)

(4,2‘)

(8,7,6‘)

(7)

(8,2‘)(7,5,3,2‘)

(4)

(4‘) (4‘)

critical vertex

(3) (5)

Avenue Tree Construction Avenue Tree Construction

Pointer Assignment (before)

(8) (4)

(4,2‘)

(8,7,6‘)

(7)

(8,2‘)

(5)

( 7,5,3,2‘)

(3)

Avenue Tree Construction Avenue Tree Construction

Pointer Assignment (after)

(8) (4)

(4,2‘)

(8,7,6‘)

(7)

(8,2‘)

(5)

( 7,5,3,2‘)

(3)

Notation Notation Let Let uu be the root with children be the root with children vv11, …, , …, vvkk.. = (= (aa11, …, , …, aapp) : the label of ) : the label of uu. . ii : the label of : the label of vvii for for ii = 1, …, = 1, …,kk..ii’: the elements of ’: the elements of ii which is greater than which is greater than aa

pp..bbii : : bbii is the maximal element in is the maximal element in ii no greater t no greater t

han han aapp..uuii: the vertex corresponding to : the vertex corresponding to bbii..ThenThen

.)( '1 pi

ki a

Pointer AssignmentPointer Assignment

aa11 a a22 …… a …… ap-1 p-1 aapp

ww11 w w2 2 …… w…… wp-1 p-1 wwpp

u1 b1

u2 b2

: :

: :

uk bk

Main TheoremMain Theorem

An optimal searching strategy on trees An optimal searching strategy on trees can be constructed in linear time. can be constructed in linear time.

Block Graphs A block graph is a graph in which every

block is a clique.

Block-cut-vertex tree

Recursive CharacterizationFor any block graph G and an integer k 2,

ns(G) k+1 if and only if G has either (1) a vertex x at which there exists at least three branches having search numbers at least k; or (2) a block K on which ns({G}u,v) k for any (u,v) E(K).

k k k

k+1

u

v

G :{G}u,v : Ku v

enclosure subgraph

Recursive Char. (cont.)

For any block graph G and an integer k 2, ns(G) = k if and only if G has

(1) for any vertex x at which there exists at most two branches having maximum search numbers k; and

(2) for any block K there exists (u,v) E(K) such that ns({G}u,v) < k.

Avenue of Length 1A vertex having all branches with search number smaller than ns(G) is called a hub of G.

A block graph may have more than one hub. ns(G) = 4

critical edges

v3

Avenue of Length > 1

v1 v4v2

critical vertices outlet vertices

For any critical edge (u,v) of a block graph G, ns({G}u,v) < ns(G).

ns(G) = 4

G may has more than one avenue, but the set of critical blocks is unique.


critical blocks

v3v1 v4v2

ns(G) = 4


v3v1 v4v2

ns(G) = 4

Block-cut-vertex tree

Vertex Merge

Block merge

MergeMergeVertex merge is similar to the case in trees.Vertex merge is similar to the case in trees.

Block merge is new. Block merge is new.

Greedy Strategy

K

u

v1 v2

v3 v4

vr

G1G2

Gr

G3G4

L R

3

1

4

2

r

Patterns of Greedy Strategies1. Centralized : L and > R

2. Balanced : L = R

3. Skewed : L > and L > R

L R

L R

L R

ns(G) = .

ns(G) = L.

ns(G) = L or L-1.

Unicyclic GraphsUnicyclic Graphs

Concluding RemarkConcluding Remark Based on concept of Based on concept of avenueavenue we get linea we get linea

r time algorithms and polynomial time ar time algorithms and polynomial time algorithm for node search problem on lgorithm for node search problem on tretrees, unicyclices, unicyclic graphsgraphs and and block graphsblock graphs, re, respectively.spectively.

How to extend the concept of avenue How to extend the concept of avenue to the general case to obtain a linear to the general case to obtain a linear time algorithm. time algorithm.

Cactus graphs Cactus graphs Block graphs Block graphs

2005/11/04 中山大學資訊工程研究所

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Graph Searching and Related Problems