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Source-to-all-terminal diameter constrained network reliability

Louis Petingi

Computer Science Dept.

College of Staten Island

City University of New York

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Historical backgroundEdge Reliability Model (1960’s)

• Distinguished set of vertices K (terminals)

• Edges fail independently

• Each edge ei fails with probability qi=1-pi

• Reliability of a graph G  RK(G)= Probability that every pair of terminal vertices of

G remain connected by an operational path, after removal of the failing edges.

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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

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Diameter constrained reliabilityPetingi, 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.

• RK(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}

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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.

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• Example – Let D = 3

G=(V,E) =

Operating States

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Operating States

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

HEHEOKD :{)(

KD

KD i iOH OH He He

iiK qpHDGR

}Pr{),(

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Diameter constrained reliability; a generalization of the classical reliability

• Let G=(V,E), KV, 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. RK(G,n-1) = RK(G)

• Thus in general to compute RK(G,D) is NP-hard as RK(G) is NP-hard

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Computational complexity –diameter constrained two-terminal reliability

• Consider a graph G= (V,E), K = {s,t}, and D=2.

• Example G =

s t

irrelevant edges

Reduced graph

e1 e2

e3 e4

e5

}},{},{}{Pr{1)2,( 43215},{ eeeeeGR ts )1)(1)(1(1 43215 ppppp

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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

s t

G’ p=1/2

p=1

))(()2/1(1)3,'( ||||},{ GNVrtxcvGR YX

ts

Cancela, Petingi (2002)

X Y

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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.

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Source-to-K-terminal diameter-constrained reliability

• Consider a directed graph G=(V,E), without self-loops or parallel arcs, with terminal set KV, 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

a b

c

dipathdicycle

<s,a,b,c> <d,b,c,d>

s,K-diameter = 2

d

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Minpaths and hierarchical systems

• Def. An operating state Pi of is called a minpath if for any e Pi , , Pi - 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

• Rs,K(G,D) = Pr {That at least one minpath is operating}

)(, EO KsD

))(,( , EOE KsD

))(,( , EOE KsD

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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 indT(s)=0 (indegree), and indT(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 (outT(u)=0) must belong to K.

• A D,K-tree of G is a K-tree whose s,K-diameter D.

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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.

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Let P1, P2,…, Ps be the set of all minpaths of

Rs,K(G,D) = Pr {That at least one minpath is operating}

Using Inclusion-Exclusion

• Where the event E1E2… Em is the event that all the arcs of the subgraph

obtained by the union of

P1,…,Pm are operating.

}...Pr{)1(....

}Pr{}Pr{}Pr{),(

211

ss

kjikji

i jijiiK

EEE

EEEEEEDGR

)(, EO KsD

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Domination

• From inclusion-exclusion and the fact that

Rs,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).

HH

Ks HHHEdDGR }Pr{))(),((),(, C

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Example

• G• D=3 s

minpaths

t

P1 P2 P3 P4

Formations= {(P1, P2, P3) (P1, P2, P3, P4) }

d(E,C(G))=# odd-formations - #even-formations

=1-1=0

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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 C1, C2 C, whenever C1 C2, , C1 = C2. ,

• 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)},{()},{(),( xx xEdxEdEd CCC

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Source-to-all-terminal DC reliability• For source-to-K-terminal DC rel., for digraph G=(V,E).

dD,K(G) = d(E, C), and for any x

dD,K(G) = d(E-{x}, C+x) - dD,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 e1, e2, … ek, replaced by an arc e with rel. 1- q1 q2…qk.

d) If indG(s) > 0, then dD,K(G)=0 since any arc directed into s can not be in any D,V-tree. Thus if indG(s) = 0 we call this digraph s-rooted, and for this point on we consider only s-rooted digraphs.

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Source-to-all-terminal DC reliability• Operation SP(G) – If there exists vertex u in V-{s} with

indG(u) > 1, and distG(u) distG(v) for v in V-{s}, with

indG(v) > 1, this operation returns a s,u-dipath

Ps,u= <s=u1, u2, …., ui =u >of length distG(u), otherwise returns .

Properties : indG(s) = 0; indG(uj) = 1, for 2 i i-1.

a b

Ps,u=<s,a,b>s

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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 .

a b

LP(G)=5s

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Source-to-all-terminal DC reliability

• Lemma 3. Let G=(V,E) be a s-rooted digraph, and bound D. Suppose that Ps,u= <s=u1, u2, …., ui =u > is returned by SP(G) and let x=(ui-

1,u) , then dD,V(G) = - dD,V(G-x) .

Sketch. dD,K(G) = d(E-{x}, C+x) - dD,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 indG(uj) = 1, for 2 j i-1, thus every V-tree must contain the path <s=u1, u2, …., ui-1 >. 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’.

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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 Ps,u= <s=u1, u2, …., ui =u >.

• Claim 1. If G is not s,V-connected then dD,V(G) = 0.

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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 Ps,u= <s=u1, u2, …., ui =u > , and let x=(ui-1,u), then G-x is also cyclic.

. Sketch. Let U={s=u1, u2, …., ui-1 }, indG(s)=0, and indG(u)=1, u U-{s}. Moreover ui-1 can only be reached from vertices in U, thus if x=(ui-

1,u) belongs to a cycle, then indG(s)> 0, or indG(u)>1, u U-{s}, a contradiction.

ss s s

x

x

x

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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 dD,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 dD,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 dD,V(G) = 0.If G is s,V-connected then SP(G) returns a dipath Ps,u= <s=u1, u2, …., ui

=u >, thus from Lemma 3, dD,V(G) = - dD,V(G-x). But by the Ind. Step dD,V(G-x)=0.

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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 Ps,u= <s=u1, u2, …., ui =u > , and let x=(ui-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

xx

SP

LP

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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 dD,V(G) = 0.

If LP(G) D, then G is a D,V-tree thus dD,V(G) = 1 = (-1)e-n+1.

Giving that dD,V(G) = - dD,V(G-x), we proceed according Lemma 5, and Lemma 7.

D LP(G) :(-1)

D LP(G):0

(G)d1n-e

VD,

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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, Gk=G), grow tree progressively by

generating children, if any, of every vertex.

States duplications are avoided by a simple check. Gk,j= Gk-ej , is a new state with arc label ej, provided ej is not the label of an arc incident into any elder brother (generated previously) or elder brother of an ancestor of k.

Gk

ej

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Algorithm to determine reliability.

Four rules to generate children:

1) If Gk is not s,V-connected (DFS), do not generate any children.

2) If Gk is s,V-connected and cyclic (DFS), let C={e1, e2, …., ec } be a dicycle, then Gk,j= Gk-ej, j=1,2,…,c, provided that ej is not duplicated.

3) If Gk is s,V-connected and acyclic, determine LP(Gk) (longest path).

For acyclic digraphs we can use PERT algorithm (linear complexity).

3a) If Gk has LP(Gk) > D, let P={e1, e2, …., ep } be a longest s,u-dipath, then Gk,j= Gk-ej, j=1,2,…,p, provided that ej is not duplicated.

3b) If Gk has LP(Gk) D,

let Gk,j= Gk-ej, ej is an arc of Gk..

.

ki

k

Eei

ne pRR 1)1(

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Open problems

• Determine dD,k(G) for arbitrary K.

– Empirical results lead to conjecture that dD,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 dD,k(G) = 0.