Reliability Analysis of Communication Networks

Transcription

1 Reliability Analysis of Communication Networks Mathematical models and algorithms Peter Tittmann Hochschule Mittweida May 2007 Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

2 Models for Communication Networks Communication network Network topology Server Switch Base station Fiber optical transmission line Transmission line (copper) Wireless channel Server failure probability Link availability Line length (costs) Transfer rate User terminal PoP Mathematical model (un-) directed graph Vertex Edge (arc) Weights of vertices Edge weights Terminal vertex Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

3 Problems What is the probability that a certain vertex is reachable from another given vertex? Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

4 Problems What is the probability that a certain vertex is reachable from another given vertex? What is the probability of network overload? Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

5 Problems What is the probability that a certain vertex is reachable from another given vertex? What is the probability of network overload? Which mean transmission rate can be realized? Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

6 Problems What is the probability that a certain vertex is reachable from another given vertex? What is the probability of network overload? Which mean transmission rate can be realized? Where are the weak points of the network (e.g. w.r.t. reliability or security)? Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

7 Problems What is the probability that a certain vertex is reachable from another given vertex? What is the probability of network overload? Which mean transmission rate can be realized? Where are the weak points of the network (e.g. w.r.t. reliability or security)? Is a spread of failure within the network possible? How? Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

8 Problems What is the probability that a certain vertex is reachable from another given vertex? What is the probability of network overload? Which mean transmission rate can be realized? Where are the weak points of the network (e.g. w.r.t. reliability or security)? Is a spread of failure within the network possible? How? What is the mean (average) packet delay? Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

9 Problems What is the probability that a certain vertex is reachable from another given vertex? What is the probability of network overload? Which mean transmission rate can be realized? Where are the weak points of the network (e.g. w.r.t. reliability or security)? Is a spread of failure within the network possible? How? What is the mean (average) packet delay? What is the probability of packet loss? Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

10 Problems What is the probability that a certain vertex is reachable from another given vertex? What is the probability of network overload? Which mean transmission rate can be realized? Where are the weak points of the network (e.g. w.r.t. reliability or security)? Is a spread of failure within the network possible? How? What is the mean (average) packet delay? What is the probability of packet loss? How is tra c load distributed over the network? Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

11 Constraints Network technology, used protocol (IP, ATM, SS7, GSM) Routing, redundancy properties Service under consideration Supply of data Type of failure Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

12 Mathematical Model Network Structure Graphs Directed or undirected graph G = (V, E ) p e... availability of edge e 2 E All edges are assumed to fail independently. W set of paths C set of cuts Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

13 Reliability Measures All-terminal reliability (connectedness probability) R(G ) = P (fg is connectedg) Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

14 Reliability Measures All-terminal reliability (connectedness probability) R(G ) = P (fg is connectedg) K-terminal probability R(G, K ), jk j 2 Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

15 Reliability Measures All-terminal reliability (connectedness probability) R(G ) = P (fg is connectedg) K-terminal probability R(G, K ), jk j 2 The residual connectedness probability: G is considered operating if and only if at least one vertex is in operating state and all operating vertices of G belong to one component. Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

16 Reliability Measures All-terminal reliability (connectedness probability) R(G ) = P (fg is connectedg) K-terminal probability R(G, K ), jk j 2 The residual connectedness probability: G is considered operating if and only if at least one vertex is in operating state and all operating vertices of G belong to one component. Probability of a given data transfer between two vertices. Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

17 Reliability Measures All-terminal reliability (connectedness probability) R(G ) = P (fg is connectedg) K-terminal probability R(G, K ), jk j 2 The residual connectedness probability: G is considered operating if and only if at least one vertex is in operating state and all operating vertices of G belong to one component. Probability of a given data transfer between two vertices. Resilience: Expectation for the number of vertex pairs that are connected by operating paths. Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

18 Reliability Measures All-terminal reliability (connectedness probability) R(G ) = P (fg is connectedg) K-terminal probability R(G, K ), jk j 2 The residual connectedness probability: G is considered operating if and only if at least one vertex is in operating state and all operating vertices of G belong to one component. Probability of a given data transfer between two vertices. Resilience: Expectation for the number of vertex pairs that are connected by operating paths. Importance measures Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

19 All-Terminal Reliability Example Sets of minimum paths spanning trees W = ffa, b, dg, fa, b, eg, fa, c, dg, fa, c, eg, fa, d, eg, fb, c, dg, fb, c, eg, Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

20 All-Terminal Reliability Example R (G ) = p a p b p d + p a p b p e + p a p c p d + p a p c p e + p a p d p e + p b p c p d +p b p c p e + p b p d p e 2p a p b p c p d 2p a p b p c p e 3p a p b p d p e 2p a p c p d p e 2p b p c p d p e + 4p a p b p c p d p e Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

21 Decomposition R (G ) = (1 p c ) R (G c) + p c R (G /c) Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

22 All-Terminal Reliability Reductions 1 Parallel reduction e f g 2 Degree-2-reduction p g = p e + p f p e p f e f g Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

23 All-Terminal Reliability Reductions 1 Parallel reduction e f g 2 Degree-2-reduction p g = p e + p f p e p f e f g ω = p e + p f p e p f, p g = p e p f p e + p f p e p f. R (G ) = ωr G 0 Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

24 All-Terminal Reliability Reductions G w (1 p)(1 p) e f u (1 p)p e f + p(1 e p) f G 1 u e w v p p f e f u G 2 uw v w R (G ) = (p e + p f 2p e p f ) R (G 1 ) +p e p f R (G 2 ) Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

25 All-Terminal Reliability Reductions G w (1 p)(1 p) e f u (1 p)p e f + p(1 e p) f G 1 u e w v p p f e f u G 2 uw G' u g w 1 p g p g u w G 1 G 2 uw v w R G 0 = (1 p g ) R (G 1 ) +p g R (G 2 ) R (G ) = (p e + p f 2p e p f ) R (G 1 ) +p e p f R (G 2 ) Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

26 All-Terminal Reliability Reductions Reduction principle R (G ) = ωr G 0 R (G ) = (p e + p f 2p e p f ) R (G 1 ) + p e p f R (G 2 ) R G 0 = (1 p g ) R (G 1 ) + p g R (G 2 ) Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

27 All-Terminal Reliability Reductions Reduction principle R (G ) = ωr G 0 R (G ) = (p e + p f 2p e p f ) R (G 1 ) + p e p f R (G 2 ) R G 0 = (1 p g ) R (G 1 ) + p g R (G 2 ) Method of coe cient comparison p e + p f 2p e p f = ω (1 p g ) p e p f = ωp g Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

28 All-Terminal Reliability Reductions Reduction principle R (G ) = ωr G 0 R (G ) = (p e + p f 2p e p f ) R (G 1 ) + p e p f R (G 2 ) R G 0 = (1 p g ) R (G 1 ) + p g R (G 2 ) Method of coe cient comparison Solution (reduction parameter) p e + p f 2p e p f = ω (1 p g ) p e p f = ωp g ω = p e + p f p e p f p e p p g = f p e + p f p e p f Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

29 All-Terminal Reliability Reductions Bridge reduction R(G ) = p e R G 0 Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

30 All-Terminal Reliability Vertex Separators Cut vertex articulation G 1 [ G 2 = G G 1 \ G 2 = (fvg, ) R(G ) = R G 1 R G 2 Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

31 All-Terminal Reliability Vertex Separators Separating vertex pair G 1 [ G 2 = G G 1 \ G 2 = (fu, vg, ) R(G ) = R G 1 R Guv 2 + R G 1 uv R G 2 R G 1 R G 2 Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

32 All-Terminal Reliability Vertex Separators U P (U) P G 1, π G 2 π separating vertex set partition lattice of U probability that G 1 induces π obtained from G 2 merging the blocks of π R(G ) = P G 1, π R G 2 π π2p(u ) Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

33 Reliability polynomials De nition The reliability polynomial R(G, p) is the probability that the undirected graph G = (V, E ) is connected, assuming all edges of G fail independently with probability 1 p. Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

34 Reliability polynomials De nition The reliability polynomial R(G, p) is the probability that the undirected graph G = (V, E ) is connected, assuming all edges of G fail independently with probability 1 p. R (G, p) = = = 1 m i=n 1 m i=n 1 a i p i c i... number of cut sets of cardinality i n i... number of spanning subgraphs of G n i p i (1 p) m i m c i p m i (1 p) i i=λ Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

35 Reliability polynomials Recursive De nition 8 < p n 1, if G is a tree with n vertices, R (G, p) = 0, if G is disconnected, : pr (G /e) + (1 p) R (G e), else. Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

36 Reliability polynomials Recursive De nition 8 < p n 1, if G is a tree with n vertices, R (G, p) = 0, if G is disconnected, : pr (G /e) + (1 p) R (G e), else. Fact This de nition does not require any meaning of the variable p. Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

37 Reliability polynomials Properties 1 If G is connected then and R (G, 0) = 0, R (G, 1) = 1 p m R (G, p) 1 (1 p) m. Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

38 Reliability polynomials Properties 1 If G is connected then and 2 Monotonicity: R (G, 0) = 0, R (G, 1) = 1 p m R (G, p) 1 (1 p) m. p 1 < p 2 ) R (G, p 1 ) < R (G, p 2 ) Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

39 Reliability polynomials Properties 1 If G is connected then and 2 Monotonicity: R (G, 0) = 0, R (G, 1) = 1 p m R (G, p) 1 (1 p) m. p 1 < p 2 ) R (G, p 1 ) < R (G, p 2 ) 3 If G is a graph with at least three vertices then dr (G, p) dp = 0. p=0 Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

40 Reliability polynomials Properties 1 If G is connected then and 2 Monotonicity: R (G, 0) = 0, R (G, 1) = 1 p m R (G, p) 1 (1 p) m. p 1 < p 2 ) R (G, p 1 ) < R (G, p 2 ) 3 If G is a graph with at least three vertices then dr (G, p) dp = 0. p=0 4 If G is biconnected then dr (G, p) dp = 0. p=1 Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

41 Reliability polynomials Reliability Function R(G) p Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

42 Reliability polynomials Special Graphs Trees R (T n, p) = p n 1 Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

43 Reliability polynomials Special Graphs Cycles R (C n, p) = np n 1 (n 1) p n Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

44 Reliability polynomials Special Graphs Ladder R (L n, p) = p2n 1 2 n α [(4 3p + α)n (4 3p α) n ] with α = p 12 20p + 9p 2 Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

45 Reliability polynomials Special Graphs Complete graphs Recurrence equation r n := R (K n, q), q := 1 n 1 r n = 1 k 1 r 1 = 1 n 1 k=1 q k(n p k) r k Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

46 Reliability polynomials Special Graphs Explicit representation with r n = ( 1) jλj+1 n λ λ`n λ = (λ 1,..., λ k ) = jλj k 1 jλj qa 2(λ) 1 k 1 2 k 2 n k n and a 2 (λ 1,..., λ s ) = 1 2 n 2! s λ 2 i i=1 Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

47 Reliability polynomials Special Graphs Explicit representation with r n = ( 1) jλj+1 n λ λ`n λ = (λ 1,..., λ k ) = jλj k 1 jλj qa 2(λ) 1 k 1 2 k 2 n k n and a 2 (λ 1,..., λ s ) = 1 2 Exponential generating function r n = q n2 z n 2 ln n! n 2! s λ 2 i i=1! q n2 zn 2 n0 n! Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

48 The K-terminal reliability Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

49 The K-terminal reliability De nition G is K-connected if all vertices of K V belong to one component of G. Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

50 The K-terminal reliability 8 < 0 if G is not K-connected, R (G, K ) = 1 if G = (fvg, ), v 2 K, : p e R (G /e, K 0 ) + (1 p e ) R (G e, K ) else, where K 0 = (K n fu, vg) [ X, e = fu, vg, w is the vertex obtained merging vertices u and v and fwg, if K \ fu, vg 6= X =, else.. Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

51 The K-terminal reliability Complete state enumeration: with R (G, K ) = p e F E e2f (1 p) ξ (G [F ], K ) e2e nf 1, if G [F ] is K-connected, ξ (G [F ], K ) = 0, else. For K = fs, tg, the probability R(G, K ) is called two-terminal reliability (or st-reliability).. Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

52 The K-terminal reliability Properties 1 Let p = (p 1,..., p m ) be the vector of edge availabilities of G. Then p p 0 ) (G, K, p) R G, K, p 0. Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

53 The K-terminal reliability Properties 1 Let p = (p 1,..., p m ) be the vector of edge availabilities of G. Then p p 0 ) (G, K, p) R G, K, p 0. 2 Monotonicity with respect to terminal vertex sets: K L ) R (G, K ) R (G, L) Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

54 The K-terminal reliability Properties 1 Let p = (p 1,..., p m ) be the vector of edge availabilities of G. Then p p 0 ) (G, K, p) R G, K, p 0. 2 Monotonicity with respect to terminal vertex sets: 3 If K \ L 6= then K L ) R (G, K ) R (G, L) Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

55 The K-terminal reliability Properties 1 Let p = (p 1,..., p m ) be the vector of edge availabilities of G. Then p p 0 ) (G, K, p) R G, K, p 0. 2 Monotonicity with respect to terminal vertex sets: 3 If K \ L 6= then K L ) R (G, K ) R (G, L) 1 R (G, K [ L) R (G, K ) + R (G, L) 1, Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

56 The K-terminal reliability Properties 1 Let p = (p 1,..., p m ) be the vector of edge availabilities of G. Then p p 0 ) (G, K, p) R G, K, p 0. 2 Monotonicity with respect to terminal vertex sets: 3 If K \ L 6= then K L ) R (G, K ) R (G, L) 1 R (G, K [ L) R (G, K ) + R (G, L) 1, 2 R (G, K [ L) R (G, K ) R (G, L). Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

57 The K-terminal reliability Properties 1 Let p = (p 1,..., p m ) be the vector of edge availabilities of G. Then p p 0 ) (G, K, p) R G, K, p 0. 2 Monotonicity with respect to terminal vertex sets: 3 If K \ L 6= then K L ) R (G, K ) R (G, L) 1 R (G, K [ L) R (G, K ) + R (G, L) 1, 2 R (G, K [ L) R (G, K ) R (G, L). 4 The K-terminal reliability of the complete graph K n : n k j(n j) R (K n, k) = r j q j k n j=k Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

58 Reliability Calculations Reductions Series reduction e f g p g = p e p f Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

59 Reliability Calculations Reductions Polygon-to chain reductions R (G ) = ωr (G 0 ) a b c f g h d e α = ab (1 c) d (1 e), β = (1 a) bc (1 d) e γ = a(1 b)c(d + e 2de) + (a + c 2ac)bd(1 e) δ = abcde a + 1 b + 1 c + 1 d + 1 e f = ω = δ β + δ, g = δ γ + δ, h = (α + δ) (β + δ) (γ + δ) δ 2 δ α + δ, Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

60 Reliability Calculations Reductions Delta-Star reduction a b c g e f x Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

61 Reliability Calculations Reductions Delta-Star reduction a b c g e f x α = a + b + c ab ac cc + abc β = a + bc abc γ = b + ac abc δ = c + ab abc e = α β, f = α γ, g = α δ, x = βγδ α 2 Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

62 Partitions of Vertex Sets Partition Lattice M = fa, b, c, dg, P (M).. set of all partitions of M π σ if and only if π is a re nement of σ abcd abc/d ab/cd abd/c ac/bd acd/b ad/bc bcd/a ab/c/d ac/b/d ad/b/c bc/a/d bd/a/c cd/a/b a/b/c/d Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

63 Partitions of Vertex Sets Incidence Algebra Möbius function i=1 jσj jπj jσj µ (π, σ) = ( 1) (p i 1)! Supremum function Inverse a (x, y) = x _ y = ˆ1 a 1 (x, z) = y µ (y, z) µ (y, x) µ y, ˆ1 Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

64 Partitions of Vertex Sets Network Reliability Theorem R (G, K ) = σπ(k ) τ2p(v ) a 1 (σ, τ) R (G τ ) Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

65 Partitions of Vertex Sets Network Reliability Theorem R (G, K ) = σπ(k ) τ2p(v ) a 1 (σ, τ) R (G τ ) Theorem R (G ) = ( 1) jσj+1 (jσj 1)! qje (G,σ)j σ2p(v ) Tittmann, P.: Partitions and network reliability, Discrete Applied Mathematics 95 (1999), Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

66 Separating Vertex Sets De nition Let G 1 = V 1, E 1 and G 2 = V 2, E 2 be two subgraphs of G = (V, E ) such that V 1 [ V 2 = V, V 1 \ V 2 = U, E 1 [ E 2 = E, E 1 \ E 2 =. Then U is called a separating vertex set of G. a U 1 G b 2 G c d Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

67 Separating Vertex Sets a b c d a b c d a b c d induced partition: π = a/bc/d Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

68 Separating Vertex Sets a b d c ab cd G! G ab/cd Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

69 Separating Vertex Sets The Splitting Formula Number of terms: R (G ) = Pπ 1 R G 2 π π2p(u ) B (u) 1 p u e u(r +1/r 1) 1 with r e r = u Exponential generating function: e e z 1 = B(n) zn n0 n! Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

70 Separating Vertex Sets Symmetric graphs Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

71 Separating Vertex Sets Symmetric graphs a a 1 2 G G G 1 a G2 a b b b b c c c c d d d d R (G ) = λ`u u λ jλj k 1 jλj P1 λ R G 2 λ Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

72 Separating Vertex Sets Symmetric graphs Number of terms: n B (n) p (n) p n t n Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

73 Separating Vertex Sets Symmetric graphs A form of the splitting formula that requires only all-terminal reliability calculations for G 1 and G 2 : R (G ) = π2p(u ) σ2p(u ) R Gπ 1 a 1 (π, σ) R Gσ 2 Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

74 Separating Vertex Sets Symmetric graphs Special cases juj = 1 : R (G ) = R G 1 R G 2 juj = 2 : R (G ) = R G 1 R Guv 2 + R G 1 uv R G 2 juj = 3 : A 1 3 = C A abc, ab/c, ac/b, bc/a, a/b/c R G 1 R G 2 Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

75 Separating Vertex Sets Multiple splitting R (G 0 G 1... G n ) = q T 0 A1 1 Q 1A2 1 Q 2 An 1 1 Q n 1An 1 q n Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

76 Separating Vertex Sets Multiple splitting Gorlov, V.; Tittmann, P.: A uni ed approach to the reliability of recurrent structures, in Ellart von Collani et al. (editors): Advances in stochastic models for reliability, quality and safety, Birkhäuser, Boston, 1998 Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

77 Tree decompositions De nition A tree decomposition of a graph G = (V, E ) is a pair (T, φ) with a tree T = (W, F ) and a map φ : W! 2 V which assigns a vertex subset of G to each vertex of T such that S 1 w 2W φ (w) = V. 2 For each edge fu, vg 2 E there exists w in T such that fu, vg φ (w). 3 Let v 2 W be on the path between u and w in T. Then we have φ (u) \ φ (w) φ (v). Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

78 Tree decompositions De nition The width of a tree decomposition is maxfφ (w) : w 2 W g 1. The treewidth tw(g ) of G is the minimum width of a tree decomposition of G. Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

79 Tree decompositions Example Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

80 Partial k-trees De nition A graph G = (V, E ) is a k-tree if it is a complete graph with k + 1 vertices or if G contains a vertex v 2 V whose neighborhood induces a k-clique of G and G v is again a k-tree. A partial k-tree is a subgraph of a k-tree. Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

81 Partial k-trees Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

82 Path Decompositions De nition A path decomposition is a sequence X = (X 1,..., X r ) of vertex subsets of G such that the following conditions are satis ed r[ 1 X i = V. i=1 2 Each edge of G is contained in at least one of X 1,..., X r. 3 For all i, j, k with 1 i < j < k r the relation X i \ X k X j is valid. Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

83 Path Decompositions De nition A canonical path decomposition satis es in addition: 4. All subsets of the sequence X contain at most pw (G ) + 1 vertices. 5. jx i+1 X i j = 1 for i = 1,..., r 1. Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

84 Path Decompositions f1g, f1, 2g, f1, 2, 3g, f2, 3g, f2, 3, 5g, f2, 5g, f2, 4, 5g, f4, 5g, f4, 5, 8g, f4, 5g, f4g, f4, 6g, f4g, f4, 7g, f7g 1, 2, 3, 1, 5, 3, 4, 2, 8, 8, 5, 6, 6, 7, 7 Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

85 Path Decompositions De nition A composition order of G = (V, E ) is a sequence s = (s 1,..., s r ) of vertices and edges such that 1 The removal of all edges of s yields a canonical path decomposition of G. 2 Each edge is exactly once in s. 3 If s k = (u, v) 2 E then there are indices i, j, p, q with i, j < k and p, q > k such that u = s i = s p and v = s j = s q. Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

86 Path Decompositions Composition order: (1, 2, f1, 2g, 3, f1, 3g, 1, f2, 3g, 4, f2, 4g, 2, f3, 4g, 5, f3, 5g, 3, f4, 5g, 6, f4, 6g, 4, f5, 6g, 5, 6). Assigned path decomposition: (, f1g, f1, 2g, f1, 2, 3g, f2, 3g, f2, 3, 4g, f3, 4g, f3, 4, 5g, f4, 5g, f4, 5, 6g, f5, 6g, f6g, ) Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

87 Path Decompositions Composition Algorithms State = (index, value) Index: Partition of the set of active vertices Value: Polynomial Transformation of states 1. Vertex activation: (π, P π ) 7! (π/fvg, P π ) Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

88 Path Decompositions 2. Vertex deactivation: If fvg is a singleton of π then (π, P π ) 7!. Else: 0 1 (π, P π ) 7! v, v 2Y 2π jy j>1 3. The insertion of an edge e generates two successor states: P π (I) (π, P π ) 7! (π, (1 p) P π ) C A (II) (π, P π ) 7! (π _ e, pp π + P π_e ) Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

89 Path Decompositions Example s i X i+1 Z i+1 1 f1g f(1, 1)g 2 f1, 2g f(1/2, 1)g f1, 2g f1, 2g f(1/2, 1 p), (12, p)g 3 f1, 2, 3g f(1/2/3, 1 p), (12/3, p)g f1, 3g f1, 2, 3g f 1/2/3, 1 2p + p 2, 12/3, p p 2, 13/2, p p 2, 123, p 2 g 1 f2, 3g f 2/3, 2p 2p 2, 23, p 2 g f2, 3g f2, 3g f 2/3, 2p 4p 2 + 2p 3, 23, 3p 2 2p 3 g Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

90 Path Decompositions Numeric Example n = 53, m = 203 R(G ) = , (53 s) R st (G ) = , (221 s) Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

91 Path Decompositions Algorithmic Principles André Pönitz: Über eine Methode zur Konstruktion von Algorithmen für die Berechnung von Invarianten in endlichen ungerichteten Hypergraphen, (About a generation method for algorithms for the calculation of invariants of nite graphs and hypergraphs), PhDThesis, Technische Universität Freiberg, Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

92 Directed Graphs s e 2 e 1 v e 3 e u 4 e 5 e w 6 e 7 t Algebraic Methods R st (G ) = M O p e W 2W e2w Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

93 Directed Graphs Iteration procedure 0 0 p 1 p p 5 p 6 P = B 0 p 3 0 p 4 0 1, if i = s p 7 A, s i = 0 else x 0 = (0,..., 0), s = (s 1,..., s n ) x n = x n 1 P s for n > 0 Peter Tittmann (Hochschule Mittweida) Network Reliability / 64

94 Directed Graphs Splitting X s G 1 Y G 2 t R st (G ) = X U ( 1) jx j jy j+1 R s,u ny (G 1 U ny )R X,t(GX 2 ) Y X Peter Tittmann (Hochschule Mittweida) Network Reliability / 64