An Extension of Menger s Theorem

Transcription

1 An Extension of Menger s Theorem April 2015

2 Merging

3 Merging

4 Merging merging

5 Merging

6 Merging merging

7 Merging merging merging

8 A Theorem Mergings Congestions a.s.!

9 A Proof

10 Motivation in Transportation Networks

11 Motivation in Transportation Networks

12 Motivation in Transportation Networks Minimizing Mergings Maximizing Throughput!

13 Motivation in Computer Networks a S b R 1 R 2

14 Motivation in Computer Networks a S b b a b b R 1 R 2

15 Motivation in Computer Networks a S b a a b a R 1 R 2

16 Motivation in Computer Networks a a S b b a b? a? b?? R 1 R 2

17 Motivation in Computer Networks a a S b b a a+b b a+b a+b R 1 R 2

18 Motivation in Computer Networks a a S b b a a+b network coding b a+b a+b R 1 R 2

19 Motivation in Computer Networks a a S b b a a+b network coding b a+b a+b R 1 R 2 Minimizing Mergings Minimizing Encoding Complexity!

20 Karl Menger

21 Menger s Theorem Theorem (Menger, 1927) Consider a directed graph G(V, E). For any two vertices A, B, the maximum number of pairwise edge-disjoint directed paths from A to B in G equals the min-cut between A and B, namely the minimum number of edges in E whose deletion destroys all directed paths from A to B.

22 Menger s Theorem Theorem (Menger, 1927) Consider a directed graph G(V, E). For any two vertices A, B, the maximum number of pairwise edge-disjoint directed paths from A to B in G equals the min-cut between A and B, namely the minimum number of edges in E whose deletion destroys all directed paths from A to B. Menger s paths

23 A Quick Example E C A F B D G H

24 A Quick Example E C A F B D G H

25 A Quick Example E C A F B D G H

26 We are interested in... Mergings among different groups of Menger s paths in networks with multiple sources and sinks.

27 Rerouting S 1 R 1 S 2 R 2

28 Rerouting S 1 R 1 S 2 R 2

29 Rerouting S 1 R 1 S 2 reroutable R 2

30 Rerouting S 1 R 1 S 2 R 2

31 Rerouting S 1 R 1 S 2 R 2

32 Rerouting (cont d)

33 Rerouting (cont d)

34 Rerouting (cont d)

35 Rerouting (cont d)

36 Rerouting (cont d)

37 Rerouting (cont d)

38 Network Model and Notations G: an acyclic directed graph.

39 Network Model and Notations G: an acyclic directed graph. S 1, S 2,, S n : sources.

40 Network Model and Notations G: an acyclic directed graph. S 1, S 2,, S n : sources. R 1, R 2,, R n : distinct sinks.

41 Network Model and Notations G: an acyclic directed graph. S 1, S 2,, S n : sources. R 1, R 2,, R n : distinct sinks. c i : the min-cut between S i and R i.

42 Network Model and Notations G: an acyclic directed graph. S 1, S 2,, S n : sources. R 1, R 2,, R n : distinct sinks. c i : the min-cut between S i and R i. α i = {α i,1, α i,2,, α i,ci }: a set of Menger s paths from S i to R i.

43 M (c 1, c 2,, c n ) Assume that all sources are identical. M (G):the minimum number of mergings over all possible Menger s path sets α i s, i = 1, 2,, n. M (c 1, c 2,, c n ): the supremum of M (G) over all possible choices of such G.

44 M (c 1, c 2,, c n ) Assume that all sources are identical. M (G):the minimum number of mergings over all possible Menger s path sets α i s, i = 1, 2,, n. M (c 1, c 2,, c n ): the supremum of M (G) over all possible choices of such G. R 1 S R 2

45 M (c 1, c 2,, c n ) Assume that all sources are identical. M (G):the minimum number of mergings over all possible Menger s path sets α i s, i = 1, 2,, n. M (c 1, c 2,, c n ): the supremum of M (G) over all possible choices of such G. R 1 S R 2

46 M (c 1, c 2,, c n ) Assume that all sources are identical. M (G):the minimum number of mergings over all possible Menger s path sets α i s, i = 1, 2,, n. M (c 1, c 2,, c n ): the supremum of M (G) over all possible choices of such G. R 1 S R 2

47 M (c 1, c 2,, c n ) Assume that all sources are identical. M (G):the minimum number of mergings over all possible Menger s path sets α i s, i = 1, 2,, n. M (c 1, c 2,, c n ): the supremum of M (G) over all possible choices of such G. R 1 S R 2

48 M (c 1, c 2,, c n ) Assume that all sources are identical. M (G):the minimum number of mergings over all possible Menger s path sets α i s, i = 1, 2,, n. M (c 1, c 2,, c n ): the supremum of M (G) over all possible choices of such G. R 1 S R 2

49 M (c 1, c 2,, c n ) Assume that all sources are identical. M (G):the minimum number of mergings over all possible Menger s path sets α i s, i = 1, 2,, n. M (c 1, c 2,, c n ): the supremum of M (G) over all possible choices of such G. R 1 S R 2

50 M (c 1, c 2,, c n ) Assume that all sources are identical. best decision M (G):the minimum number of mergings over all possible Menger s path sets α i s, i = 1, 2,, n. M (c 1, c 2,, c n ): the supremum of M (G) over all possible choices of such G. R 1 S R 2

51 M (c 1, c 2,, c n ) Assume that all sources are identical. best decision worst performance M (G):the minimum number of mergings over all possible Menger s path sets α i s, i = 1, 2,, n. M (c 1, c 2,, c n ): the supremum of M (G) over all possible choices of such G. R 1 S R 2

52 M(c 1, c 2,, c n ) Assume that all sources are distinct. M(G): the minimum number of mergings over all possible Menger s path sets α i s, i = 1, 2,, n. M(c 1, c 2,, c n ): the supremum of M(G) over all possible choices of such G. R 1 S 1 S 2 R 2

53 Previous Work It was first conjectured by Tavory, Feder and Ron that M(c 1, c 2,, c n ) is finite. More specifically the authors proved that if M(c 1, c 2 ) is finite for all c 1, c 2, then M(c 1, c 2,, c n ) is finite as well. As for M, Fragouli and Soljanin use the idea of subtree decomposition to first prove that M (2, 2,, 2) = n 1. }{{} n It was first shown Langberg, Sprintson and Bruck that M (c 1, c 2 ) is finite for all c 1, c 2, and subsequently M (c 1, c 2,, c n ) is finite all c 1, c 2,, c n.

54 Main Results Main Results For fixed c 1, c 2,, c n, M(c 1, c 2,, c n ) and M (c 1, c 2,, c n ) are always finite. And as functions of c 1, c 2,, c n, they have interesting properties. We give exact values of and tighter bounds on M and M with certain parameters.

55 Some Remarks When n = 1, Ford-Fulkerson algorithm can find the min-cut and a set of Menger s path between S 1 and R 1 in polynomial time.

56 Some Remarks When n = 1, Ford-Fulkerson algorithm can find the min-cut and a set of Menger s path between S 1 and R 1 in polynomial time. The LDP (Link Disjoint Problem) asks if there are two edge-disjoint paths from S 1, S 2 to R 1, R 2, respectively. The fact that the LDP problem is NP-complete suggests the intricacy of the problem when n 2.

57 Outline of the Proof Lemma For any c 1, c 2, M(c 1, c 2 ) c 1 c 2 (c 1 + c 2 )/2. Theorem For any c 1, c 2,, c n, we have M(c 1, c 2,, c n ) i<j M(c i, c j ). Observation Note that for any c 1, c 2,, c n, we always have M (c 1, c 2,, c n ) M(c 1, c 2,, c n )

58 A Key Idea If too many mergings:

59 A Key Idea If too many mergings:

60 A Key Idea If too many mergings:

61 A Key Idea If too many mergings:

62 A Key Idea If too many mergings:

63 A Key Idea If too many mergings:

64 A Key Idea If too many mergings:

65 A Key Idea If too many mergings:

66 A Key Idea If too many mergings:

67 Properties of M Proposition For c 1 c 2 c n, if c 1 + c c n 1 c n, then M (c 1, c 2,, c n ) = M (c 1, c 2,, c n 1, c 1 + c c n 1 ). Proposition For c 1 = 1 c 2 c n, we have M (c 1, c 2,, c n ) = M (c 2,, c n 1, c n ). Proposition For n 1 n 2 n k, k 1 M (n 1, n 2,..., n k ) M (n i, n i ). i=1

68 Properties of M Proposition For any c 1,0, c 1,1, c 2, we have M(c 1,0 + c 1,1, c 2 ) M(c 1,0, c 2 ) + M(c 1,1, c 2 ). Proposition For any c 1, c 2,, c n and any fixed k with 1 k n, we have M(c 1, c 2,, c n ) M(c i, c j ). i k,j k+1

69 Properties of M Proposition For any c 1,0, c 1,1, c 2, we have M(c 1,0 + c 1,1, c 2 ) M(c 1,0, c 2 ) + M(c 1,1, c 2 ). Proposition For any c 1, c 2,, c n and any fixed k with 1 k n, we have M(c i, c j ) M(c 1, c 2,, c n ) M(c i, c j ). i<j i k,j k+1

70 Properties of M (cont d) Proposition For any m n, we have where U(m, n) = and m 1 j=1 M(m, n) U(m, n) + V (m, n) + m 2, (M(j, m 1) M(m j, n))+m(m, m 1)+1, m 1 V (m, n) = M(m, n 1)+ (M(j, n) M(m j, n)) M(1, n). j=1

71 Properties of M (cont d) Proposition For any fixed k, there exists a positive constant C k such that for all n, M(k, n) C k n. Proof.

72 Properties of M (cont d) Proposition For any fixed k, there exists a positive constant C k such that for all n, M(k, n) C k n. Proof. C k mergings

73 Properties of M (cont d) Proposition For any fixed k, there exists a positive constant C k such that for all n, M(k, n) C k n. Proof. C k mergings C k mergings

74 Properties of M (cont d) Proposition For any fixed k, there exists a positive constant C k such that for all n, M(k, n) C k n. Proof. C k mergings C k mergings C k mergings

75 Tighter Bounds It has been established [Langberg et al.] that n(n 1)/2 M (n, n) n 3. Next, we give tighter bounds on M (n, n) and M(m, n).

76 Bounds on M Proposition (n 1) 2 M (n, n) n 2 (n 2 4n + 5). Proof. S A graph showing M (3, 3) 4. R 1 R 2

77 Bounds on M Proposition m + n 2 2mn m n+1 M(m, n) (m+n 1)+(mn 2). 2 Proof. S 1 S 2 A graph showing M(2, 2) 5. R 1 R 2

78 Exact Values M (1, 1) = 0. M (2, 2) = 1. M (3, 3) = 4. M (4, 4) = 9. M (5, 5) = 16.

79 Exact Values M (1, 1) = 0. M (m, m) = (m 1) 2? M (2, 2) = 1. M (3, 3) = 4. M (4, 4) = 9. M (5, 5) = 16.

80 Exact Values M (1, 1) = 0. M (m, m) = (m 1) 2? M (2, 2) = 1. M (3, 3) = 4. M (6, 6) = 27! M (4, 4) = 9. M (5, 5) = 16.

81 Exact Values M(1, n) = n. M(2, n) = 3n 1. M(3, 3) = 13. M(3, 4) = 18. M(3, 5) = 23. M(3, 6) = 28. M(4, 4) = 27.

82 Exact Values M (1, 1,, 1) = 0. }{{} n M (2, 2,, 2) = n 1. }{{} n M (2, m, m) = 1 + (m 1) 2, m = 2, 3, 4, 5. M (m, m, m) = 2(m 1) 2, m = 1, 2, 3, 4. M (3, 4, 4) = 13. For m n p and (m, n) (3, 4) or (2, 5) M (m, n, p) = M (m, n, n).

83 Exact Values M(1,..., 1, 2) = }{{} k M(1, 2, n) = k 2 M(1, 1,, 1) =. }{{} 4 k { 3k 1 if k 6, k2 4 + k + 2 if k > 6. M(1, 1, n) = 2n + 1. { 4n if n = 2, 3, 4n + 1 if n = 1 or n 4. M(2, 2, 2) = 11, M(1, 3, 3) = 17, M(2, 2, 3) = 18.

84 The Finiteness Result Does Not Hold for Cyclic Graphs S R 1 R 2

85 The Finiteness Result Does Not Hold for Cyclic Graphs S R 1 R 2

86 The Finiteness Result Does Not Hold for Cyclic Graphs S R 1 R 2

87 The Finiteness Result Does Not Hold for Cyclic Graphs S R 1 R 2

88 The Finiteness Result Does Not Hold for Cyclic Graphs S R 1 R 2

89 The Finiteness Result Does Not Hold for Cyclic Graphs S R 1 R 2

90 The Finiteness Result Does Not Hold for Cyclic Graphs S R 1 R 2

91 The Finiteness Result Does Not Hold for Cyclic Graphs S R 1 R 2

92 Open Problems Tighter Upper Bounds There exists C > 0 such that for all m, n, M(m, n) Cmn.

93 Open Problems Tighter Upper Bounds There exists C > 0 such that for all m, n, M(m, n) Cmn. Mergings On Other Classes of Graphs undirected graphs, planar graphs, and so on.

94 References L. R. Ford and D. R. Fulkerson Maximal flow through a network. Canadian Journal of Mathematics 8: , C. Fragouli and E. Soljanin. Information Flow Decomposition for Network Coding. IEEE Transactions on Information Theory, Volume 52, Issue 3, March 2006, Page(s): M. Langberg, A. Sprintson, A. and J. Bruck. The encoding complexity of network coding IEEE Transactions on Information Theory, Volume 52, Issue 6, June 2006, Page(s): K. Menger. Zur allgemeinen Kurventhoerie. Fund. Math., 10: A. Tavory, M. Feder and D. Ron. Bounds on Linear Codes for Network Multicast. Electronic Colloquium on Computational Complexity (ECCC), 10(033), Li Xu, Weiping2003. Shang, Guangyue Han

95 Thank You!