1 IM.CJCU Hsin-Hung Chou The Node-Searching Problem on Special Graphs 周信宏長榮大學資訊管理學系 [email protected]

1IM.CJCU Hsin-Hung Chou

The Node-Searching Problem on Special Graphs

周信宏長榮大學資訊管理學系[email protected]


Outline

• Introduction• Properties of the problem• Previous results• Result on unicyclic graphs• Conclusions


Outline

• Introduction• graph searching problem• problem definition

• Properties of the problem• Previous results• Result on unicyclic graphs• Conclusions

4

Graph-searching problem was first proposed by Parsons, 1976.

Goal: To find a search strategy using the least number of searchers to capture the fugitive.

Graph-Searching Problem

IM.CJCU Hsin-Hung Chou

5

Problem Definition

Variations: Operations:

• Place a searcher• Remove a searcher• Move along an edge

Clearing rules:

• Move along an edge• Guard two endpoints of an edge

RulesOperations

Edge-

Node-

Mixed-

• Place• Remove• Move

• Place• Remove

• Place• Remove• Move

• Move

• Guard

• Move• Guard


6

Node-Searching Problem

a

b

c

d

e

1

2 # of searchers = 3

3

1

2

The node-searching problem was first proposed by Kirousis and Papadimitriou, 1986.



Outline

• Introduction• Properties of the problem

• progressive strategy• related problems

• Previous results• Result on unicyclic graphs• Conclusions

8

Recontamination


a

b

c

d

e

3

1

9

Kirousis and Papadimitriou showed that there exists an optimal search strategy without recontamination for any graph.

Progressive Strategy


We can only consider search strategies without recontamination.

There exists an optimal search strategy in which no vertex is visited twice by a searcher, and in which every searcher is deleted immediately after all the edges incident on it have been cleared.

10

Interval Model


e

2

b2

a

c

d1 1

1 2 876543 109

3

abcde

11

Guard Sets


1 2 876543 109

abcde

abc bc c cd c

de de ea

X1

ab

X2 X3 X4 X5 X6 X7 X8 X9

12

Path-decomposition


abc bc c cd c

de de ea

X1

ab

X2 X3 X4 X5 X6 X7 X8 X9

1. ).(1

GVri iX

2. For every edge (u,v)E(G), there exists an Xi

containing both u and v.

3. if i < j < k.jX

kXiX

13

Path-width Problem


The width of a path-decomposition is max { |Xi| | 1 i r} – 1.

The path-width of G is the minimum width over all path-decompositions of G.

The path-width of G is equal to the node-search number of G minus one.

14

Guard Sequence


1 2 876543 109

abcde

c d ea b

15

Linear Layout


c d ea b

A linear layout of a graph G is a one-to-one mapping L : V(G) {1,2, …, |V(G)|}.

1 2 3 4 5

16

Cut Numbers


The i-th cut number of L, denoted by cutL(i), is the number of vertices which are mapped to integers less than or equal to i and adjacent to a vertex mapped to an integer larger than i.

c d ea b

1 2 3 4 5

cutL(i) : 1 2 1 2 0

17

Vertex Separation


The vertex separation with respect to G and L: vsL(G) = max {cutL(i) | 1 i |V(G)|}.

The vertex separation of G: vs(G) = min {vsL(G) | L is a linear layout of G}.

The vertex separation of G is equal to the node-search number of G minus one.

18

Related Problems

• path-width problem• vertex separation problem• interval thickness problem• gate matrix layout problem• narrowness problem

The node-searching problem is equivalent to



Outline

• Introduction• Properties of the problem• Previous results• Result on unicyclic graphs• Conclusions

20

Previous Results


Classes of graphs Complexity

general with bounded degree 3 NP-complete

planar with bounded degree 3 NP-complete

chordal NP-complete

chordal bipartite NP-complete

bipartite NP-complete

bipartite distance-hereditary NP-complete

co-comparability NP-complete

starlike NP-complete

circle NP-complete

lattice NP-complete

21

Previous Results (cont.)


Classes of graphs Complexity

k-starlike (for a fixed k) O(mnk)

split O(mn)

co-chordal O(n+1),=2.37…

interval O(m+n)

permutation O(npw)

partial k-tree (for a fixed k) (n4k+3)

cograph, tree O(n)

circular-arc O(n2)

block

unicyclic

O(bc+c2+n)

O(n)


Outline

• Motivation• Avenue system on trees• Previous results• Result on unicyclic graphs

• motivation• avenue system on trees• linear-time algorithm

• Conclusions

236-tree


k-trees Recursive definition of k-trees:

• A k-clique is a k-tree.• If T = (V,E) is a k-tree and C is a k-clique of T and xV(T), then T’ = (V{x},E {cx | cC}) is a k-tree.

5-tree5-tree

24

5-tree


Partial k-trees A graph is a partial k-trees if it is a spanning subgraph of a k-tree.

partial 5-tree


Unicyclic Graphs A unicyclic graph is a graph composed of a tree with one extra edge.

A unicyclic graph is a partial 2-tree.


Results on Unicyclic Graphs• Hans L. Bodlaender and Ton Kloks, “Efficient and constructive algorithms for the pathwidth and treewidth of graphs”, Journal of Algorithms, 21(no.2):pp. 358–402, 1996.Time complexity: (n4k+3) for a partial k-tree with fixed k.

• J. A. Ellis and M. Markov, “Computing the vertex separation of unicyclic graphs”, Information and Computation, 192:pp. 123–161, 2004.Time complexity: O(n log n).

• Our result: O(n).


Outline



• Conclusions


Results on Trees• J. A. Ellis, I. H. Sudborough, and J. S. Turner, “The vertex separation and search number of a graph”, Information and Computation, 113(no. 1):pp. 50–79, 1994.Search number: O(n); Optimal search strategy: O(n log n).

• S. L. Peng, C. W. Ho, T. S. Hsu, M. T. Ko, and C. Y. Tang, “A linear-time algorithm for constructing an optimal node-search strategy of a tree”, LNCS 1449:pp. 279–288, 1998.• K. Skodinis, “Construction of linear tree-layouts which are optimal with respect to vertex separation in linear time”, Journal of Algorithms, 47:pp. 40–59, 2003.

Optimal search strategy: O(n).


k k k

k+1

[PAR76] For any tree T and an integer k 2, ns(T) k+1 if and only if there exists a vertex v with at least three branches having search numbers at least k.

Parsons’ Lemma

branch


Example

2

1

2

3

2

3

3

ns(T) = 3


Hub

ns(T1) = 2

ns(T) = 3

ns(T2) = 2 ns(T3) = 2


Example

2

1

2

3

2

3

2

1

3

1

3

3

1

1

2

1

2

3 2

ns(T) = 3


ns(T2) = 3ns(T1) = 3

ns(T) = 3

Critical Vertices


Outlet Vertices

ns(T’) = 3

ns(T) = 3


Avenue on Trees

[MEG88] For any tree T, a path P = [v1, v2, . . . , vr] is an avenue of T, if the following conditions hold:

• If r = 1, then v1 is a hub.• If r > 1, then each of v1 and vr is an outlet vertex,

and for every j, 2 j r-1, vj is a critical vertex.

[MEG88] For any tree T, T has an avenue. If the length of the avenue is at least two, then the avenue is unique.


Dynamic Programming• Every tree under construction has a specified vertex called the root.

• The algorithm computes the search number based on the tree decomposition using dynamic programming.


Types of Rooted Trees

type H

u u

u

type E

u

type Iu

type M


Label of Trees

Label:

pppp aaaaaaaa 121121 ...),,,...,,(

)(,,...,, 121 uuuuu pp

up= u

u1

Vertices:

Types: M,M, …, M, H(E,I)

a1

u2

a2

u3a3

39

kk


Merge Rules – Case 1

= (k+1’)

k

• If there exist at least three labels containing k which is the maximum element in all labels, then

…

40

kk



= (k+1’)

< k

• If there exist exactly two labels containing k and one of them contains a k-critical vertex , then

… < k

k

41

kk



= (k)

< k

• If there exist exactly two labels containing k and neither of them contains a k-critical vertex , then

… < k

42

< kk



= (k+1’)

< k

• If there exist exactly one label containing k and it contains a k-critical vertex x, and ns(Tu[x]) = k, then

… < k

k

x

43

< kk



= (k)&(Tu[x])

< k

• If there exist exactly one label containing k and it contains a k-critical vertex x, and ns(Tu[x]) < k, then

… < k

k

x

44

< kk



= (k’)

< k

• If there exists exactly one label containing k and it contains no k-critical vertex , then

… < k


Outline



• Conclusions


Oriented Search Strategy A search strategy in which u is the start vertex and v is the end vertex is called an oriented search strategy for G from u to v.

c4

v2

u

a

b

11

3

start

end

# of searchers = 4

os(G,u,v) = os(G,v,u). os(G,u) = min { os(G,u,v) | vV(G) }. ns(G) = min { os(G,u,v) | u,vV(G) }.


Main Algorithm (Phase 1)

• Compute the labels of rooted constituent trees.



• Compute the label of rooted U-e.

ns(U-e) ns(U) ns(U-e) + 1.



• Construct the label array of U. Example: ={1=(9,8,6,4’), 2=(8’), 3=(7,6)}.

]1[A

],,6[ 32vI

],,4[ 14vH

],,6[ 13vM

]9[A

]8[A

]7[A

]6[A

]5[A

]4[A

]3[A

]2[A

],,7[ 31vM

],,8[ 21vE],,8[ 1

2vM

],,9[ 11vM

iALLn i

Hn iEn i

In iMn iptr

1

1

1

2

2

8

7

6

4

01

11

1

1 1

1

0 0 0

0 0

0

0

0

0

0

0

0

0

H E I



• Decide ns(U) = k or k+1, where k = ns(U-e), based on the labels of U-e and constituent trees.


Phase 2 (Case 1)

• One k-critical constituent tree (containing k-critical vertex).

k

k

T’

u

u

c

If ns(T’) = k , then ns(U) = k+1; else ns(U) = k.

U’If ns(U’-e) < k-1, then ns(U) = k; else ns(U) = ns(U’)+1.

e


Phase 2 (Case 2)

• Three or more k-non-critical constituent trees.

k

ns(U) = k+1.

k k


Phase 2 (Case 3)

• Exactly two k-non-critical constituent trees.

k ku v

If os(U’,u,v) k , then ns(U) = k; else ns(U) = k+1.

U’


Phase 2 (Case 4)

• Exactly one k-non-critical constituent tree.


k k-1u v

U’


Phase 2 (Case 5)

• No k-non-critical constituent tree.


k-1 k-1u v

U’


Conclusion

• unicyclic cactus partial k-tree (O(n4k+3)).

Unicyclic graph: O(n).

Cactus graph:


Thank you for your attention!


Q & A

Documents

1 IM.CJCU Hsin-Hung Chou The Node-Searching Problem on Special Graphs 周信宏 長榮大學 資訊管理學系 [email protected]

1 IM.CJCU Hsin-Hung Chou The Node-Searching Problem on Special Graphs 周信宏長榮大學資訊管理學系 [email protected]