The Node-Searching Problem on Special Graphs

1. The Node-Searching Problem on Special Graphs 周信宏 長榮大學 資訊管理學系 chouhh@mail.cjcu.edu.tw IM.CJCU Hsin-Hung Chou

2. Outline • Introduction • Properties of the problem • Previous results • Result on unicyclic graphs • Conclusions IM.CJCU Hsin-Hung Chou

3. Outline • Introduction • graph searching problem • problem definition • Properties of the problem • Previous results • Result on unicyclic graphs • Conclusions IM.CJCU Hsin-Hung Chou

4. Graph-Searching Problem • 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. IM.CJCU Hsin-Hung Chou

5. Problem Definition • Variations: • Operations: Operations Rules • Place a searcher • Remove a searcher • Move along an edge • Place • Remove • Move Edge- • Move • Clearing rules: • Place • Remove Node- • Guard • Move along an edge • Guard two endpoints • of an edge • Place • Remove • Move • Move • Guard Mixed- IM.CJCU Hsin-Hung Chou

6. Node-Searching Problem • The node-searching problem was first proposed • by Kirousis and Papadimitriou, 1986. a d c b e # of searchers = 3 IM.CJCU Hsin-Hung Chou

7. Outline • Introduction • Properties of the problem • progressive strategy • related problems • Previous results • Result on unicyclic graphs • Conclusions IM.CJCU Hsin-Hung Chou

8. Recontamination a d c b e IM.CJCU Hsin-Hung Chou

9. Progressive Strategy • Kirousis and Papadimitriou showed that there exists an • optimal search strategy without recontamination for any • graph. • 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. • We can only consider search strategies without • recontamination. IM.CJCU Hsin-Hung Chou

10. 1 2 3 4 5 6 7 8 9 10 Interval Model a d c b e a b c d e IM.CJCU Hsin-Hung Chou

11. a c a bc c de e ab cd bc de 1 2 3 4 5 6 7 8 9 10 X1 X2 X3 X4 X5 X6 X7 X8 X9 Guard Sets a b c d e IM.CJCU Hsin-Hung Chou

12. 1. a c a bc c de e ab cd bc de X1 X2 X3 X4 X5 X6 X7 X8 X9 3. if i < j < k. Path-decomposition 2. For every edge (u,v)E(G), there exists an Xi containing both u and v. IM.CJCU Hsin-Hung Chou

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. IM.CJCU Hsin-Hung Chou

14. 1 2 3 4 5 6 7 8 9 10 Guard Sequence a b c d e a d e c b IM.CJCU Hsin-Hung Chou

15. Linear Layout a e c b d 1 2 3 4 5 A linear layout of a graph G is a one-to-one mapping L : V(G)  {1,2, …, |V(G)|}. IM.CJCU Hsin-Hung Chou

16. Cut Numbers a e c b d 1 2 3 4 5 cutL(i) : 1 2 1 2 0 The i-thcut 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. IM.CJCU Hsin-Hung Chou

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. IM.CJCU Hsin-Hung Chou

18. Related Problems • The node-searching problem is equivalent to • path-width problem • vertex separation problem • interval thickness problem • gate matrix layout problem • narrowness problem IM.CJCU Hsin-Hung Chou

19. Outline • Introduction • Properties of the problem • Previous results • Result on unicyclic graphs • Conclusions IM.CJCU Hsin-Hung Chou

20. Previous Results IM.CJCU Hsin-Hung Chou

21. Previous Results (cont.) block O(bc+c2+n) unicyclic O(n) IM.CJCU Hsin-Hung Chou

22. Outline • Motivation • Avenue system on trees • Previous results • Result on unicyclic graphs • motivation • avenue system on trees • linear-time algorithm • Conclusions IM.CJCU Hsin-Hung Chou

23. 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 • xV(T), then T’ = (V{x},E {cx | cC}) is a k-tree. 5-tree 6-tree 5-tree IM.CJCU Hsin-Hung Chou

24. Partial k-trees • A graph is a partial k-trees if it is a spanning subgraph of a k-tree. partial 5-tree 5-tree IM.CJCU Hsin-Hung Chou

25. Unicyclic Graphs • Aunicyclic graphis a graph composed of a • tree with one extra edge. • Aunicyclic graphis a partial 2-tree. IM.CJCU Hsin-Hung Chou

26. 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). IM.CJCU Hsin-Hung Chou

27. Outline • Motivation • Avenue system on trees • Previous results • Result on unicyclic graphs • motivation • avenue system on trees • linear-time algorithm • Conclusions IM.CJCU Hsin-Hung Chou

28. 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). IM.CJCU Hsin-Hung Chou

29. Parsons’ Lemma • [PAR76] For any tree T and an integerk  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. k+1 branch k k k IM.CJCU Hsin-Hung Chou

30. Example ns(T) = 3 IM.CJCU Hsin-Hung Chou

31. Hub ns(T1) = 2 ns(T) = 3 ns(T3) = 2 ns(T2) = 2 IM.CJCU Hsin-Hung Chou

32. Example ns(T) = 3 IM.CJCU Hsin-Hung Chou

33. Critical Vertices ns(T) = 3 ns(T2) = 3 ns(T1) = 3 IM.CJCU Hsin-Hung Chou

34. Outlet Vertices ns(T) = 3 ns(T’) = 3 IM.CJCU Hsin-Hung Chou

35. 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 vris 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. IM.CJCU Hsin-Hung Chou

36. 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. IM.CJCU Hsin-Hung Chou

37. Types of Rooted Trees IM.CJCU Hsin-Hung Chou

38. Label of Trees • Label: • Vertices: Types: M,M, …, M, H(E,I) up= u u1 u3 u2 a3 a2 a1 IM.CJCU Hsin-Hung Chou

39. Merge Rules – Case 1 • If there exist at least three labels containing k which is the maximum element in all labels, then  = (k+1’) … k k k IM.CJCU Hsin-Hung Chou

40. Merge Rules – Case 2 • If there exist exactly two labels containing k and one of them contains a k-critical vertex , then  = (k+1’) … k k < k < k k IM.CJCU Hsin-Hung Chou

41. Merge Rules – Case 3 • If there exist exactly two labels containing k and neither of them contains a k-critical vertex , then  = (k) … k k < k < k IM.CJCU Hsin-Hung Chou

42. Merge Rules – Case 4 • If there exist exactly one label containing k and it contains a k-critical vertex x, and ns(Tu[x]) = k, then  = (k+1’) … < k k < k < k x k IM.CJCU Hsin-Hung Chou

43. Merge Rules – Case 5 • If there exist exactly one label containing k and it contains a k-critical vertex x, and ns(Tu[x]) < k, then  = (k)&(Tu[x]) … < k k < k < k x k IM.CJCU Hsin-Hung Chou

44. Merge Rules – Case 6 • If there exists exactly one label containing k and it contains no k-critical vertex , then  = (k’) … < k k < k < k IM.CJCU Hsin-Hung Chou

45. Outline • Motivation • Avenue system on trees • Previous results • Result on unicyclic graphs • motivation • avenue system on trees • linear-time algorithm • Conclusions IM.CJCU Hsin-Hung Chou

46. Oriented Search Strategy • A search strategy in which u is the start vertex and v is the end vertexis called an oriented search strategy for G from u to v. • os(G,u,v) = os(G,v,u). os(G,u) = min { os(G,u,v) | vV(G) }. ns(G) = min { os(G,u,v) | u,vV(G) }. b start u a end c v # of searchers = 4 IM.CJCU Hsin-Hung Chou

47. Main Algorithm (Phase 1) • Compute the labels of rooted constituent trees. IM.CJCU Hsin-Hung Chou

48. Main Algorithm (Phase 1) • Compute the label of rooted U-e. ns(U-e)  ns(U)  ns(U-e) + 1. IM.CJCU Hsin-Hung Chou

49. Main Algorithm (Phase 1) • Construct the label array of U. • Example: ={1=(9,8,6,4’), 2=(8’), 3=(7,6)}. H E I A [ 1 ] L A [ 2 ] L i i i i A [ 3 ] i i n n n n n ptr L H E I M ALL 1 v A [ 4 ] [ 4 , H , ] 1 0 0 0 1 0 L 4 A [ 5 ] L 1 3 v v A [ 6 ] [ 6 , M , ] [ 6 , I , ] 0 0 1 1 2 4 L 3 2 3 v A [ 7 ] [ 7 , M , ] 0 0 0 1 1 6 L 1 1 2 v v A [ 8 ] [ 8 , M , ] [ 8 , E , ] 0 1 0 1 2 7 L 2 1 1 v A [ 9 ] [ 9 , M , ] 0 0 0 1 1 8 L 1 IM.CJCU Hsin-Hung Chou

50. Main Algorithm (Phase 2) • Decide ns(U) = k or k+1, where k = ns(U-e), • based on the labels of U-e and constituent trees. IM.CJCU Hsin-Hung Chou