1. 1 Source-to-all-terminal diameter constrained network reliability Louis Petingi Computer Science Dept. College of Staten Island City University of New York

2. 2 Historical background Edge Reliability Model (1960’s) Distinguished set of vertices K (terminals) Edges fail independently Each edge e i fails with probability q i =1-p i Reliability of a graph G R K (G)= Probability that every pair of terminal vertices of G remain connected by an operational path, after removal of the failing edges.

3. 3 Diameter constrained reliability K-diameter – max {distance(u,v): u,v  K} K-diameter = 3 Delay T at each node Delay at least 3.T between terminal nodes

4. 4 Diameter constrained reliability Petingi, and Rodriguez (2001) Suppose that we want to know what is the probability that the terminal nodes meet a delay constrained D.T, for some upper bound D. R K (G,D) = Pr{After random failures of the edges, there exists an operational path of length <=D between every pair of terminal nodes u,v}

5. 5 Applications This measure gives an indicator of the suitability of an existing network topology to support good quality voice over IP applications between a pair of terminals.Videoconference, we take K to be the set of the participating nodes, and the Diameter-constrained reliability gives the probability that we can find short enough paths between all of them. Another potential case of interest are a number of protocols which, in order to avoid congestion by looping data, assign a maximum number of hops to each data packet, to control information. In this case, the diameter constrained unreliability (the complement to one of the reliability) gives the probability that, due to failed links, there are some nodes of the network which are not reachable by using these protocols.

6. 6 Example – Let D = 3 G=(V,E) = Operating States

7. 7 Definition. Let spans a subgraph whose K-diameter  D}

8. 8 Diameter constrained reliability; a generalization of the classical reliability Let G=(V,E), K  V, n=|V|, and e=|E|. In the classical reliability the operational paths are of unconstrained length, and by noticing that the maximum path length in a graph on n nodes is n-1, then: Diameter constrained rel. = classical rel. when D=n-1, i.e. R K (G,n-1) = R K (G) Thus in general to compute R K (G,D) is NP-hard as R K (G) is NP-hard

9. 9 Computational complexity –diameter constrained two-terminal reliability Consider a graph G= (V,E), K = {s,t}, and D=2. Example G = st irrelevant edges Reduced graph e1e1 e2e2 e3e3 e4e4 e5e5

10. 10 What is the computational complexity for the two-terminal case for fixed D=3 Consider any bipartite G=(V,E), where V=X  Y, and let G’ be the following: Bipartite G st G’ p=1/2 p=1 Cancela, Petingi (2002) X Y

11. 11 What about for fixed D  3 and fixed |K|? The proof is similar to the previous case. Open cases – Fixed D and arbitrary K. – Fixed D and all-terminal case K=V.

12. 12 Source-to-K-terminal diameter- constrained reliability Consider a directed graph G=(V,E), without self-loops or parallel arcs, with terminal set K  V, root s  K, and diameter bound D. s R s,K (G,D)=Pr{there exists an operational dipath of at atmost D edges from s to any terminal-vertex u.} G ab c dipath dicycle s,K-diameter = 2 d

13. 13 Minpaths and hierarchical systems Def. An operating state P i of is called a minpath if for any e  P i,, P i - e is not an operating state. Def. A system (E, O) is called hierarchical iff A  O, and B  A, then B  O. - The system is hierarchical. - Every operating state of must contain a minpath, thus R s,K (G,D) = Pr {That at least one minpath is operating}

14. 14 Characterization of the minpaths Let G=(V,E) be a digraph, with terminal set K, distinguished node s, and bound D. A tree of digraph G is a connected subgraph with no cycles independently of the direction of arcs. At rooted tree T=(V’,E’), rooted at s, of G, is a tree of G where ind T (s)=0 (indegree), and ind T (u)=1, u  V’-{s}. A K-tree T=(V’,E’) of G, is a rooted tree, rooted at s, with K  V’, and any pendant vertex u (out T (u)=0) must belong to K. A D,K-tree of G is a K-tree whose s,K-diameter  D.

15. 15 Domination Let G=(V,E) be a digraph, with terminal set K, distinguished node s, and bound D. Lemma 1. M is a minpath of G iff M is a D,K-tree. G=(V,E) is a D,K-digraph if every arc of G belongs to some D,K-tree. Let C (G) be the set of all D,K-trees of G. A formation F of G is a collection of D,K-trees whose union constitutes the set of arcs E. A formation is odd or even dependent whether F contains an odd or even number of D,K-trees. The sign domination of G=(V,E), denoted as d(E, C (G)), is the number of odd formations – the number even formations of G.

16. 16 Let P 1, P 2,…, P s be the set of all minpaths of R s,K (G,D) = Pr {That at least one minpath is operating} Using Inclusion-Exclusion Where the event E 1 E 2 … E m is the event that all the arcs of the subgraph obtained by the union of P 1,…,P m are operating.

17. 17 Domination From inclusion-exclusion and the fact that R s,K (G,D) = Pr {That at least one minpath is operating} we obtain Where H is the set of all D,K-digraphs of G, and Pr(H) is the probability that the arcs of H are operative. The concept of domination was originally introduced by Satyanarayana and Prabhakar for source-to-terminal case (1978).

18. 18 Example G D=3 s minpaths t P1P1 P2P2 P3P3 P4P4 Formations= {(P 1, P 2, P 3 ) (P 1, P 2, P 3, P 4 ) } d(E,C(G))=# odd-formations - #even-formations =1-1=0

19. 19 General systems Domination generalized for general systems. (Barlow, Huseby). A system (E, C ), where C  P(E) is called a clutter of E if for any two elements C 1, C 2  C, whenever C 1  C 2,, C 1 = C 2., A system (E, C ) is coherent if each element of E is contained in some element of C. Let x  E. C – x = {C – x: C  C }, C -x = {C  C : x  C}. C – x may not be a clutter, C +x = collection of elements of C – x that are not a proper supersets of other elements. Lemma 2- (Huseby) For any system (E, C ), and x  E

20. 20 Source-to-all-terminal DC reliability For source-to-K-terminal DC rel., for digraph G=(V,E). d D,K (G) = d(E, C ), and for any x d D,K (G) = d(E-{x}, C +x ) - d D,K (G-x). For the case K=V, source s  K, and bound D a) A D,V-tree is a s-rooted spanning tree with s,V-diameter  D. b) G is s,V-connected if there exists an s,u-dipath, for all u  V. c) Parallel arcs e 1, e 2, … e k, replaced by an arc e with rel. 1- q 1 q 2 …q k. d) If ind G (s) > 0, then d D,K (G)=0 since any arc directed into s can not be in any D,V-tree. Thus if ind G (s) = 0 we call this digraph s-rooted, and for this point on we consider only s-rooted digraphs.

21. 21 Source-to-all-terminal DC reliability Operation SP (G) – If there exists vertex u in V-{s} with ind G (u) > 1, and dist G (u)  dist G (v) for v in V-{s}, with ind G (v) > 1, this operation returns a s,u-dipath P s,u = of length dist G (u), otherwise returns . Properties : ind G (s) = 0; ind G (u j ) = 1, for 2  i  i-1. ab P s,u = s

22. 22 Source-to-all-terminal DC reliability Operation LP (G) – If G is s,V-connected, this operation returns the length of the longest dipath from s to any vertex u in G, otherwise it returns . ab LP (G)=5 s

23. 23 Source-to-all-terminal DC reliability Lemma 3. Let G=(V,E) be a s-rooted digraph, and bound D. Suppose that P s,u = is returned by SP (G) and let x=(u i- 1,u), then d D,V (G) = - d D,V (G-x). Sketch. d D,K (G) = d(E-{x}, C +x ) - d D,V (G-x). We want to show that system (E-{x}, C +x ) is not coherent. Let x’  x be another edge into u, and let T’ be a D,V-tree containing x’. But ind G (u j ) = 1, for 2  j  i-1, thus every V-tree must contain the path. Thus T=T’-x’+x is a V-tree, but also a D,K-tree. T-x = T’ – x’  C - x, but T’  C - x, therefore T’  C +x  No clutter element in C +x contains x’.

24. 24 Source-to-all-terminal DC reliability Lemma 4. Let G=(V,E) be a s-rooted digraph, and e > n-1. If SP (G) returns  then G is not s,V-connected. By the contraposite of Lemma 4 we obtain Lemma 5. Let G=(V,E) be a s-rooted digraph, and e > n-1. If G is s,V-connected then SP (G) returns a dipath P s,u =. Claim 1. If G is not s,V-connected then d D,V (G) = 0.

25. 25 Source-to-all-terminal DC reliability A digraph G is cyclic if it contains a dicycle, otherwise is acyclic. Lemma 6. Let G=(V,E) be a s-rooted cyclic digraph. If SP (G) returns a dipath P s,u =, and let x=(u i-1,u), then G-x is also cyclic.. Sketch. Let U ={s=u 1, u 2, …., u i-1 }, ind G (s)=0, and ind G (u)=1, u  U -{s}. Moreover u i-1 can only be reached from vertices in U, thus if x=(u i-1,u) belongs to a cycle, then ind G (s)> 0, or ind G (u)>1, u  U -{s}, a contradiction. s s s s x x x

26. 26 Source-to-all-terminal DC reliability Theorem 1. Let G=(V,E) be a s-rooted cyclic digraph with n > 2 vertices, and D be a diameter bound, then d D,V (G) = 0. Sketch. We will consider all s-rooted cyclic digraphs with n>2 vertices. Induction on e=|E|. Basis e=n-1. With this number of arcs the only s,V-connected s-rooted digraph is a rooted spanning tree, thus G is not s,V-connected thus d D,K (G) = 0. Ind. Step. Suppose that hyp. Is true for all s-rooted cyclic digraphs with e = m  n-1 arcs, and n > 2 vertices. Suppose that e=m+1 > n-1 arcs and n vertices. If G is not s,V-connected, then d D,V (G) = 0. If G is s,V-connected then SP (G) returns a dipath P s,u =, thus from Lemma 3, d D,V (G) = - d D,V (G-x). But by the Ind. Step d D,V (G-x)=0.

27. 27 Source-to-all-terminal DC reliability Lemma 7. Let G=(V,E) be a s-rooted, s,V-connected acyclic digraph. Suppose that SP (G) returns a dipath P s,u =, and let x=(u i-1,u), then a) G – x is also s-rooted, acyclic, and s,V-connected. b) LP (G) = LP (G-x). Example : D=4 s x x x x x SP LP

28. 28 Source-to-all-terminal DC reliability Theorem 2. Let G=(V,E) be a s-rooted, s,V-connected acyclic digraph, with terminal set K=V, e arcs and n nodes, and diameter bound D, then Sketch. By induction on e. Basis. e=n-1. The only s-rooted acyclic is a s-rooted spanning tree. If LP (G) > D, then G is not a D,V-tree thus d D,V (G) = 0. If LP (G)  D, then G is a D,V-tree thus d D,V (G) = 1 = (-1) e-n+1. Giving that d D,V (G) = - d D,V (G-x), we proceed according Lemma 5, and Lemma 7.

29. 29 Algorithm to determine reliability. We assume that G is s-rooted (if not delete arcs into s), without self-loops or parallel arcs (replace a bank of parallel arcs with an arc with corresponding reliability ). Rooted Directed Tree Generation. Starting from the root vertex (k=0, G k =G), grow tree progressively by generating children, if any, of every vertex. States duplications are avoided by a simple check. G k,j = G k -e j, is a new state with arc label e j, provided e j is not the label of an arc incident into any elder brother (generated previously) or elder brother of an ancestor of k. GkGk ejej

30. 30 Algorithm to determine reliability. Four rules to generate children: 1)If G k is not s,V-connected (DFS), do not generate any children. 2)If G k is s,V-connected and cyclic (DFS), let C={e 1, e 2, …., e c } be a dicycle, then G k,j = G k -e j, j=1,2,…,c, provided that e j is not duplicated. 3)If G k is s,V-connected and acyclic, determine LP (G k ) (longest path). For acyclic digraphs we can use PERT algorithm (linear complexity). 3a) If G k has LP (G k ) > D, let P={e 1, e 2, …., e p } be a longest s,u- dipath, then G k,j = G k -e j, j=1,2,…,p, provided that e j is not duplicated. 3b) If G k has LP (G k )  D, let G k,j = G k -e j, e j is an arc of G k...

31. 31 Open problems Determine d D,k (G) for arbitrary K. –Empirical results lead to conjecture that d D,k (G) = (-1) e-n+1, provided G is a D,K-digraph whose longest (s,u)-dipath is of length at most D. Otherwise d D,k (G) = 0.