Computing 2-terminal reliability for radio-broadcast networks

538 IEEE TRANSACTIONS ON RELIABILITY, VOL. 38, NO. 5, 1989 DECEMBER

Computing 2-Terminal Reliability for Radio-Broadcast Networks

Hosam M. AboEIFotoh, Member IEEE Kuwait University

Charles J. Colbourn, Affiliate IEEE University of Waterloo, Waterloo

Key Words - Network reliability, radio broadcast network, series-parallel graph, reliability bound, efficient algorithm.

Reader Aids - Purpose: Present a new method Special math needed for explanations: Graph theory Specal math needed to use results: Same Results useful to: Network designers

Summary & Conclusions - The 2-terminal reliability of a network is the probability that there exists an operating path from a source node to a sink node. Computing this measure is a difficult problem that has been studied extensively for wired point-to-point networks. However, little is known about the problem for radio broadcast networks. In this paper we present a probabilistic graph model for radio broadcast networks where nodes fail randomly and the edges are perfectly reliable. This model can represent the general case where both nodes and edges can fail. Using this model, we show that the 2-terminal reliability problem for radio broadcast networks is computationally difficult, in particular, #P-complete, even in two important restricted cases. We present efficient bounding techniques based on subgraph counts and vertex-packing methods. The subgraph counting and vertex-packing bounds are the counterpart of the subgraph counting and edge-packing bounds for wired point-to-point networks with reliable nodes and unreliable links. Further, we define series and parallel node reductions for arbitrary networks with unreliable nodes and reliable edges, and incorporate these reductions into a new polynomial time algorithm to improve the vertex-packing bounds via approximation by series-parallel reducible graphs.

1. INTRODUCTION

Radio packet broadcasting is an alternative to point-to- point wired networks; its importance is increasing as the size and number of networks increase and the cost of terrestrial links increases, while the cost of microwave communication technology goes down. The first computer system to employ radio instead of point-to-point wires for its communication facility was the ALOHA system at the University of Hawaii; it first went on the air in 1971 [29,3]. Another application for radio broadcast networks (RBN) is Mobile Radio Tele- phone Systems [ 12,291.

In RBN, communication is provided by equipping each site with radio transmitterlreceiver with a specified range.

A site can be a central computer, an end user terminal, a concentrator, a mobile station or merely a repeater. The repeaters help in establishing communication between sites not in range of each other.

Various schemes exist for RBN operation [29]; however in all of these schemes every site communicates with other sites by broadcasting the messages to all other sites within its transmitter range. This kind of operation gives rise to many design issues, eg, handling message collisions, re- transmission timing, acknowledgment, topology and reliability of the supporting repeater networks, and transmission time.

Reliability is an important issue in the design and evaluation of network topology. Various reliability measures can be used to assess the robustness of network topology against failures of its nodes. In this paper we deal with the RBN 2-terminal reliability problem. Two nodes are distinguished as source and terminal nodes. The rest of the nodes function as repeaters to provide a communication path between the source and the terminal. In an environment where repeaters fail, network operation is supported by introducing redun- dancy in repeaters, thus increasing the probability of having an operating path between the source and terminal nodes. This can yield various topologies from which the network designer has to choose the one with maximum reliability at reasonable cost. Methods for computing reliability are thus a valuable tool in network design and evaluation.

The 2-terminal reliability problem has been studied extensively for wired networks with unreliable links under the assumption that the nodes are perfect (we refer to this as the WN case). The network is modeled by a probabilistic graph G = ( Y E ) , where every communication site is represented by a node and every communication link is represented by an edge. Each edge e has operation probability p e (pe = 1 - qe) (see [13] for more details). For WN, the 2-terminal reliability problem is known to belong to the class of #P-complete problems. The class of #P-complete problems was introduced by Valiant [30]. The class #P contains those problems that involve counting the accepting computations for problems in NP; the class of #P-complete problems contains the hardest problems in #P [ 17,301. All known exact algorithms for these problems have exponential time complexity, and it is unlikely that efficient (polynomial) algorithms can be developed for this class of problems. Hence restricted classes of networks, or efficient methods for computing bounds, should be considered.

The reliability of RBN has not been as extensively studied. Ball et al. [6] present algorithms for computing the reliability of RBNs for three routing strategies. Ball [7] has presented an algorithm for both node and edge failures, and a specialization for only node failure. Van Slyke, Frank, Kershenbaum [ 331 used simulation techniques to estimate

00 18-9529/89/ 1200-0538$0 l.OOO1990 IEEE

ABOELFOTOHKOLBOURN: COMPUTING 2-TERMINAL RELIABILITY FOR RADIO-BROADCAST NETWORKS 539

some reliability measures for networks with unreliable nodes. Pullen [25] has presented a complete graph model for radio networks where all sites can communicate directly to each other under perfect transmission conditions. In her model, both nodes and links are assumed to be unreliable. The assumption of unreliable links is based on natural conditions and possibility of jamming in wartime scenario. However, the links are still assumed to fail statistically in- dependently. Pullen used this model to evaluate some measures including the probability that all the nodes are connected. To the best of our knowledge, the complexity and bounding techniques have not been investigated when links are perfect and nodes are unreliable, and in particular, not for RBNs.

We present a probabilistic graph model for RBN in which only nodes can fail. However, we show that the RBN model can be used to represent the general case where both nodes and edges can fail. This implies that the RBN reliability problems are at least as hard as the general problems. We also present the unit-disk graph (UDG) and the grid- graph models for some restricted, but practical, situations. We show that the 2-terminal reliability problem is #P-complete even when restricted to those models. We present the counterpart of subgraph counting and edge-packing bounds for RBN (Kruskal-Katona and vertex-packing bounds). We present a new bounding techniques based on approximation by series-parallel graphs. We prove that the new technique improves upon vertex-packing bounds. A similar technique has been applied for WN [ 1,2], and computational results showed an appreciable improvement upon edge-packing bounds.

The rest of this paper is organized as follows. Section 2 presents RBN models. Section 3 discusses their relation to the WN model and the general case. Section 4 surveys some of the relevant work that has been done on the WN model, in particular, the Kruskal-Katona and edge-packing bounds. Section 5 shows that the RBN 2-terminal reliability problem is #P-complete even for the grid graph model. Sec- tion 6 presents efficient methods for computing lower and upper bounds based on subgraph counting and vertex-packing. These methods extend the methods discussed in section 4. Section 7 presents new techniques for improving vertex- packing bounds via approximation by series-parallel graphs. Caution: The terms series and parallel do not have their usual hardware meanings, but are specially defined logic terms in section 7.

2. PROBABILISTIC GRAPH MODELS FOR RBN

2 .I. Assumptions

1. An RBN network model is an undirected graph G = ( K E ) , where every site is represented by a node and an edge exists between two nodes if and only if the corresponding sites can directly communicate with each other. Nodes are either operating or failed. Every node has a stated

probability of operating. Operation or failure of nodes are mutually statistically independent.

2. The surrounding medium is prefect for radio transmission within the range under consideration. Therefore, the existence of the edge between two nodes depends only on the following:

a. the distance between the corresponding sites, b. the orientation of their antennas, c. the power of their transmitter/receivers, and d . the absence of physical obstacles (eg, moun-

3 . Sites are static during the communication period. Therefore, an edge between two nodes is either existent or nonexistent, and any existing edge is perfectly reliable.

4. Any node that is failed (operable) remains failed (operable) throughout the entire communication period. Therefore, the model represents a relatively short time com- pared with the mean time between node failures.

An RBN model under these assumptions is an arbitrary graph.

tains, high buildings).

Notation

G n m P e

P"

graph whose reliability is to be bounded. number of nodes of G . number of edges of G . 1 - qe, operation probability for edge e . 1 - q", operation probability for node v.

2 . 2 . The Unit-Disk Graph Model

The RBN model can be restricted further while retain- ing a model of practical value.

More Assumptions

5. The earth is flat from a geometrical point of view. 6. All sites are equipped with similar transmitters and

receivers in terms of their antenna power and height, and all the antennas are omnidirectional. Hence each site can communicate to all the sites within a circle centered at that site having radius equal to the range of transmission. More formally, let d( i , j ) = the distance between two sites i and j , and r = the range of the transmittedreceiver. Site i can communicate directly with site j if and only if d ( i , j ) 6 r .

Under assumptions 5 and 6, an RBN network model is an undirected graph G = ( Y E ) , where every site is represented by a node v E V, and an edge e , E E if and only if d( i , j ) C r . This graph is a unit-disk graph (UDG); UDGs are the intersection graphs of unit-radius circles in the plane (taking the unit-distance as half of r ) . An example of RBN and the corresponding UDG are shown in figure 2-1. The circles in figure 2-1-a outline the range of the transmitted receiver of each station. The circles in figure 2-1-b are the unit-radius circles of the intersection graph representation. Figure 2-1-c is the UDG model.


2.3. Grid-Graph Model

UDG model to the grid-graph model.

More Assumptions

Two additional assumptions yield a specialization of the

7. Using usual Cartesian coordinates, each RBN site is located at an integer point in the plane.

8. The range r satisfies 1 S r < fl. (See figure 2- 2.)

A grid graph is a finite induced subgraph of an infinite grid. An infinite grid is a graph with a vertex at every point ( i J ) , where i j are integer coordinates in Euclidean space, and an edge between every two vertices that are one unit apart. Grid graphs are a proper subset of UDG. Restricting the RBN sites to grid points has been done in some practical networks (eg, [4]). Various restricted graph models can be obtained by changing the restrictions on the site locations, the power of their transmitter/receivers and the orientation and directionality of their antennas.

In the following section we show that the general case where both edges and nodes can fail, can be reduced to an

(2- 2- a)

m (2- 2- L )

Fig. 2-2. An RBN and Its Grid-Graph Model. (r = 1.2 x unit distance of the grid.)

RBN. This implies that, in general, the RBN case is at least as hard as the general case, and hence also at least as hard as the WN case.

3. RELATION BETWEEN RBN, WN, AND THE GENERAL CASE

Nomenclature

K-terminal and 2-terminal reliability: Given a graph G and a distinguished set of target nodes K , the K-ter- mina1 reliability is the probability that there exists communication paths between every pair of nodes in K , or equivalently, the probability that there exists, at least, an operating Steiner tree containing K . The 2-terminal problem is the special case where K = {s,t}. s is the source node and t is the terminal node. A pathset N in RBN (WN) is a set of nodes

(edges), such that if every node (edge) in N operates, the network operates according to the reliability measure under consideration.

A cutset C in RBN (WN) is a set of nodes (edges), such that if every node (edge) is C is failed, the network is failed according to the reliability measure. We use the term vertex-cutset, whenever there is an ambiguity, to distinguish it from an edge-cutset. A cutset in the 2-terminal problem is

w (2-1-a)

Pathset:

Minpath: A minimal pathset. Cutset:

("1-c)

@o +a c +

+s t+

+b d+

(2- 1 - L )

Fig. 2-1. An RBN and Its UDG Model.

ABOELFOTOH/COLBOURN: COMPUTING 2-TERMINAL RELIABILITY FOR RADIO-BROADCAST NETWORKS 54 1

an s,t-curset, and a cutset in the K-terminal problem is a Steiner cutset.

Mincut: A minimal cutset. Edge graphs [ 101: The edge graph of an undirected graph

G (Y E ) is the graph with vertex set E in which two vertices are adjacent if and only if they are adjacent edges in G.

Edge subdivision [lo]: An edge e is subdivided when it is deleted and replaced by a path of length two connecting its ends, the internal vertex of this path being a new vertex.

3.1. Reducing a WN Model to an RBN Model

Proposition 3.1. For any WN model G, there exists an RBN model H, such that there is one-to-one correspondence between pathsets in G and pathsets in H , and therefore Rel,(G) = Rel,(H), and for equal operation probabilities, the two reliability polynomials are identical.

Proof: Given an undirected graph G = ( V , E ) (WN model), a set of target nodes K C V , and associated with each edge e E E an operation probability p e , we construct the RBN model as follows: 1 ) obtain the edge graph H of G, and associate with every vertex e the operation probability pe . 2) add to H the set of target nodes K and an edge between every vertex v E K and a vertex e @ K if the edge e is incident with v in G.

To show that there is one-to-one correspondence between pathsets in G and pathsets in H , it is sufficient to show that there is one-to-one correspondence between paths (between pairs of target nodes) in G and induced paths in H . Consider a path T = ( t , , e , , v I 2 , e,, . . . ,e,, v, ,+ , , e ,+ ,, . . . e,,,t ,) between two target nodes t , & t , in G; then Pr{n}

= f i p , , . From the construction of H , the set of vertices

{tl. e , , e,, . . . , e , , , t ,} induces a path T‘ in H with Pr{n’} I = I

From the construction of H , no induced path between two target nodes contains any other target node. Consider an induced path

between two target nodes t , & t , in H . Therefore, the set of vertices { e , , 02 , . . . , e,,} corresponds to the set of edges { e , , e , , . . . , e,,} in G , such that e , and e , , , , 1 s i s n , are adjacent in G , and e , is incident with t , , and e,, is incident with t , . In addition, since n is an induced path, e, and e, are adjacent if and only if j = i 2 1. Therefore they form the edges of a path n’ in G and Pr{n’} = Pr{n}. 0

Corollary 3.2: There is one-to-one correspondence between cutsets in G and cutsets in H .

Proof: Suppose that C E is a cutset in G and that the set of vertices C is not a cutset in H . Therefore, E-C

is not a pathset in G and E-C is a pathset in H , which contradicts proposition 3.1. 0

3.2. Reducing the General Cases Model to an RBN Model

The model in the general case is an undirected graph G = (V, E), a set of target nodes K V , and associated with each edge e E E an operation probability pe, and with every nontarget node v(v E V-K) an operation probability p,,. In the general case pathsets and cutsets are defined in terms of both edges and vertices.

Proposition 3.3. For any model G in the general case, there exists an RBN model H , such that there is one-to-one correspondence between pathsets in G and pathsets in H , and therefore Rel,(G) = Rel,(H), and for equal operation probabilities, the two reliability polynomials are identical.

Proof: We construct the RBN model H by subdividing all edges in G and associating with every new vertex e the operation probabilityp,. Again, i t is easy to verify that there is one-to-one correspondence between pathsets (cutsets) in

0 For both WN and the general case, the reduction to the

RBN model can be done in O(IE() time (assuming a connected graph). To transform a WN problem to an RBN problem, one can apply either proposition 3.1 or 3.3. The graph produced by proposition 3.1 can be obtained from the graph of proposition 3.3 by contracting all nontarget nodes and adding an edge between every two vertices adjacent to the same target node.

G and pathsets (cutsets) in H .

4. 2-TERMINAL RELIABILITY FOR WN CASE

This section briefly describes some relevant bounding techniques for the 2-terminal reliability problem for WN. We describe the reliability polynomial that is used by Kruskal- Katona bounds. Then, we describe edge-packing bounds

Every pathset in WN corresponds to a unique operating state of the network where all links represented by the set of edges E‘ operate and all other links fail. The network states corresponding to the set of pathsets form disjoint events. Therefore, the 2-terminal reliability of the network is the sum of the occurrence probabilities of pathsets; a pathset occurs when all of its edges operate and all remaining edges fail. Next we use a well-known result (no proof is given here):

~ 3 1 .

Theorem 4.1 [ 13,23,31]: Computing Rel, is #P-complete. 0

In order to cope with this complexity, two approaches are taken.

1. Consider restricted classes of graphs where efficient algorithms can be developed by exploiting their special structures. For example, efficient algorithms have been developed for series-parallel graphs and k-trees [5,28,34].

542

2. Develop efficient algorithms for computing lower and upper bounds on the reliability measures.

The Kruskal-Katona bound is the best applicable bound known that is based on sub-graph counts. This bound is obtained by computing bounds on the coefficients of the reliability polynomial. The second bound uses a difference technique that is based on edge-packing.

4 .l. The Reliability Polynomial

Often we augment our basic assumptions (1)-(4):

Equal-Probability Assumption

All edges (for WN) or repeater nodes (for RBN) have the same operation probability p , and failure probability q = 1 - p .

IEEE TRANSACTIONS ON RELIABILITY, VOL. 38, NO. 5, 1989 DECEMBER

Notation

N , F ,

c c, 1

number of pathsets with i edges number of sets of i edges whose complement is a pathset E number of sets of i edges containing no cutset size of smallest cutset number of cutsets of size c size of smallest pathset; d m - 1

binf(*;p,n) binomial Cdf

Other, standard notation is given in “Information for Readers & Authors” at the rear of each issue.

Then under the Equal-Probability Assumption

Rel,(G,p) = pIE’lqIE-E’I;

The sum is taken over all pathsets E‘. The reliability has the following polynomial representation:

rn

Rel,(G,p) = E N , p‘qrn-’ ,=O

(4-1)

Polynomial (4-1) is the reliability polynomial of G.

pathsets: Another formulation is in terms of complements of

m

Rel,(G,p) = F, qlprn-‘ ,=O

F , has the following properties [32]:

(4-2)

F , = (7) - C,

F , = N , F , = 0 for i > d.

The cardinality of the minimum edge cutset (c) can be

computed efficiently using network flow techniques [21]. However, computing the number of minimum cardinality cutsets C , , and consequently F , , is #P-complete [23]. The I can be computed efficiently using breadth-first search; N , can be computed efficiently [8,11].

4.2. Kruskal-Katona Bounds

Kruskal [20] and Katona [ 191 developed a theorem that applied to any hereditary family of sets. A family of sets D E { D l , D,, . . . , D,} is hereditary if for every S E D and S‘ S , S ’ E D. This theorem was used by Frank & Van Slyke [32] to obtain bounds on the reliability polynomial. We first define certain combinatorial structures as in [91. For any non-negative integer m, the k-canonical representation of m is:

m = ( y ) + (mk+l) k - 1 + . . . +(yyi) + (:) where mk > mk-l > . . . 3 m, 2 1 3 1 . The m,’s are chosen successively so that:

m,=max{x:(i)Sm- , = ! + I 2 ( y ) ) , i = k - 1+1

mk = maxix: (;) Sm}.

The k-canonical representation is unique and always exists.

Example:

loo = (9 + (3 + (;) For any integer sequence (mk,. . . , m,), k 3 1 3 1 and any i 3 k, the (k, i)Ih lower pseudopower of (mk, . . . , m,) is:

= (m,) + p - 1 1 + . . . + ( i - k + l m‘ ) i - 1

Example:

(9,6,1)‘4’3’ = 146

The F,’s form an f-vector of a hereditary family of sets, the F-complex. To bound the polynomial Rel,(G,p) defined in (4-21we look - for vectors of coefficients - ( F o , F l , -- . . . ,&) and (C,FI, . . . ,Fe ) such that F , < F , S F,. Since for 0 < p < 1 , q, )”-‘ 2 0 such vectors lead directly to bounds on Rel,(G,p). U

ABOELFOTOH/COLBOURN: COMPUTING 2-TERMINAL RELIABILITY FOR RADIO-BROADCAST NETWORKS 543

The theorem developed by Kruskal [20] and Katona [I91 states that (no proof is given here):

Theorem 4.2 [13,19,20]. Let (F,,, F , , . . . , F d ) be the f-vector for a hereditary family D of sets. Then for each 0

0 < k < d , F k - , 2 F : k - ' / k ) . Equivalently,

Ft'k' > F,, when i 2 k Ft/k ' < F,, otherwise.

Thus the lower bound of Rel@) can be developed using the - F , vector as 5 = F:'"k' for some k 2 i in the range of i where the value of F , is unknown. Knowing F,, we can compute the following lower bound on Rel,(G,p):

d

Rel,(G,p) > binf(c - 1; q ,m) + F:/@ qbrn-' , = c

Similarly, the upper bound on Re1 can be developed using the E vector as = F:"k' for some k S i in the range of i where the value of F , is unknown. Knowing Fc- ,, we can compute the following upper bound on Rel,:

The k-canonical representation and the pseudopower can be computed efficiently using a simple recursive procedure.

4 . 3 . Edge-Disjoint Subgraph Bounds

These bounds use a different strategy that require edge- packings of graphs. An edge-packing of G is obtained by partitioning its edge set into disjoint subsets. A lower bound is obtained by an edge-packing by s,t-paths [ 11,131. Suppose that G has an edge-packing by U s,t-paths I , , . . . , 1,. Then-

The upper bound is obtained by edge-packing by s , t - cutsets.

Lemma 4.3 [13]: Let G = ( Y E ) be a graph and Re1 a reliability measure. Let C,, . . . , C, be an edge-packing of G by cutsets. Then-

0

The maximum number of s,t-cuts in an edge-packing equals the minimum length of an s,t-path [26]. Investiga- tions by Colbourn [ 13,141 suggest that edge-packing bounds are quite competitive with Kruskal-Katona bounds, and in many test examples lead to a practical improvement. Section 6 gives the counterpart of these bounds for RBN. But first,

section 5 shows that computing the RBN 2-terminal reliability exactly is hard, thus making the bounding methods important.

5. COMPLEXITY OF RBN 2-TERMINAL RELIABILITY

Reliability Polynomial

Notation

M Ni

1

nc

number of repeaters, n - 2 number of induced subgraphs of i repeaters that contain an s,t-path number of vertices in a minimum length s,t-path

number of s,t-cutsets of size c C cardinality of a minimum s,t-vertex-cutset

Rel, has the following polynomial representation:

M

Rel, = N , p ' f l - ' , , = I

N , = 0 for i < 1 .

(5- 1)

The coefficient N, is the number of shortest s,t-paths that can be computed efficiently for general graphs [8].

Since c repeaters must be removed to separate s from

t , N , = (7) for i > M - c. The coefficient NMPc =

We show that computing n, for arbitrary networks is #P-complete by a reduction from MINIMUM BIPARTITE VERTEX COVER; the latter is #P-complete [23].

Theorem 5.1. MINIMUM BIPARTITE VERTEX COV- ER is polynomial time reducible to counting the minimum cardinality s,t-vertex cutsets.

Proof Given an instance of MINIMUM BIPARTITE VERTEX COVER, G = (V = X U Y, E), where X and Y are the two partitions of V, we construct a new graph G' by adding two nodes s and t to V, and an edge from s to every node in X, and an edge from t to every node in Y . Now there is a one-to-one correspondence between a vertex cover in G and an s,t-cut in G' . A minimum cardinality vertex cover in G corresponds to a minimum cardinality s , t - cut in G' and vice versa. o

Corollary 5.2. For any random network with perfect edges and unreliable nodes, computing the 2-terminal reliability is #P-complete.

Proof: Suppose that we have an algorithm for computing Rel,. By substituting of p in (5-1) by m distinct values, (0 < p < l ) , we obtain a system of m linear equations. Since the coefficient matrix is nonsingular this system has


a unique solution. Therefore the unknowns {N , } can be found in polynomial time [31]. This contradicts theorem 5.1. 0

We show that the RBN 2-terminal reliability remains #P-complete when restricted to the grid graph model. The complexity of RBN Rel, follows immediately since grid graphs are a subset of UDG.

Theorem 5.3. Counting Hamiltonian cycles on grid graphs is aP computing the RBN 2-terminal reliability.

Proof: See Appendix. 0 This implies the #P-completeness of RBN 2-terminal

reliability on grid graphs.

6. UPPER & LOWER BOUNDS FOR RBN 2- TERMINAL RELIABILITY

6.1. Kruskal-Katona Bounds for the RBN Case

The RBN Rel, has the property of statistical coherence, since the failure of a node cannot transform a failed state to an operating one; ie, any superset of a cutset is a cutset. Similarly, the operation of a node cannot result in the de- struction of an s,t-path. Therefore, bounds similar to Krus- kal-Katona bounds can be developed under the Equal Prob- ability Assumption.

The reliability polynomial (5-1) can be rewritten in terms of the complements of the pathsets as follows:

M

Rel, = Fi q‘PM-‘; ,= 1

For RBN, F , the number of induced subgraphs of i repeaters not containing an s,t-vertex-cutset and C, = the number that contain an s,t-cutset. That is,

F , = (7) - C,.

The coefficients F,’s form an f-vector. Therefore theorem 4.2 applies and can be used to derive upper and lower bounds on the unknown coefficients as described in section 4.3.

One important coefficient is F,. Since counting minimum cardinality vertex s, t-cuts is #P-complete for arbitrary networks, as in theorem 5.1, computing F , for arbitrary networks is #P-complete. The complexity of computing F , for UDG is open. However, for s,t-planar graphs F , can be computed efficiently. A graph is s,t-planar if it has a planar embedding with s and t on the exterior face. Ball & Provan [8] have described a method for counting minimum-cardinality s, t-edge-cutsets in s,t-planar networks by counting shortest paths in the s,t-dual graph. To count minimum cardinality s, t-edge-cutsets they count shortest paths in the s,?- dual graph. We describe a method for counting s,t-vertex- cutsets in s,t-planar graphs.

Computing nL in s,t-planar graphs

Start with a planar embedding of G 3 (v E ) with s

F , and F2, using two lines at s and t (see figure 6-1 for example).

We then construct a new graph H whose set of vertices consists of V - {s,r} plus two vertices s” and t”. The set of edges of H consists of an edge between every two vertices in V - {s,t} if they belong to the same face in G, and an edge between st’ and every vertex that belong to F , in G, and similarly, an edge between t“ and every vertex that belong to F z . Thus there is one-to-one correspondence between paths from s” to t” in H and s,t-vertex-cutsets in G. There- fore the number of the shortest (s”,t”)-paths in H is exactly the number of minimum cardinality s,t-vertex-cutsets in G , C,. The number of shortest paths can be computed efficiently for arbitrary graphs, for example using a modified version of breadth first search [8,11,22].

Fl

Fig. 6-1. An Example of a Graph G and the Corresponding and t on the exterior face. We cut the exterior face into two, Graph H.

ABOELFOTOH/COLBOURN: COMPUTING 2-TERMINAL RELIABILITY FOR RADIO-BROADCAST NETWORKS

~

545

6.2. Vertex-Packing Bounds

These bounds are the counterpart of the edge-packing bounds that are competitive for the edge case [ 141. A vertex- packing of a graph G = ( Y E ) is obtained by partitioning V into disjoint subsets V I , V,, . . . , V,.

6.2.1. Vertex-packing Lower Bounds

To obtain a lower bound we use a packing by s,t-paths. The s,t-paths in the RBN Rel, problem have a different inter- pretation from s,t-paths for WN. Any two operating nodes that are in the range of each other force any path going through them to go through the edge connecting them. Thus we are interested in induced s,t-paths:

RBN s,t-path: A sequence v, , v2, . . . , v,, v,,,, . . . , vk, such that v, = s, vk = t , (v,, v ,+ , ) E E , for all i , and (v,, v,) E , for all j f i + 1.

Suppose that G has a vertex-packing by m s, t-paths l , , . . . , 1,. Then-

m

Rel,(G) 2 1 - n(1 , = I - n p J . vel,

Generally, we would like to increase the number of paths and the operation probability of each path. However, the maximum number of vertex-disjoint s, t-paths equals the size of the minimum cardinality s, t-vertex-cutset. For equal operation probabilities the set of strongest paths is the set of shortest ones. For unequal operation probabilities, weighted shortest path algorithm or minimum cost network flow algorithms with node constraints can be used to select the set of most reliable paths, by assigning every vertex a weight equal to -lnp,.

6.2.2. Vertex-packing Upper Bounds

An upper bound is obtained by vertex-packing by s,t-

For C,, . . . , C, a vertex-packing of G by s,t-vertex- cutsets.

cutsets,

Since every cutset contains at least one vertex from every s,t-path the maximum number of s,t-vertex-cutsets in a vertex-packing is one less than the length 1 of the shortest s,t- path.

We can find a vertex-packing by 1 - 1 cutsets as follows. Starting at vertex s, label the vertices using breadth first search with their distance from s. Then a cutset C, (1 d i 6 1 - 1) consists of all vertices with label i. These cutsets are made minimal by removing all vertices that cannot belong to any RBN s,t-path.

The vertex-packing method for upper bounds can be viewed as the determination of a graph G' whose set of ver-

tices V' = fi C, U { s , t } , and whose set of edges E'

= {(u,v)Iu E C, and v E C,+,, 1 d i d k - I } U {(s,v)lv E C , } U {(v,t)lv E C,}. Therefore, the set of vertices in every cutset C , can be reduced to one vertex with op-

eration probabilityp = 1 - U (1 - p,,). Afterevery cutset

is reduced to one vertex G' becomes an s,t-path. Hence its reliability is obtained by reducing G ' to one vertex with operation probability equal to the product of operation probabilities of all the vertices on the path. Similarly, the lower bound method can be viewed as the determination of a graph that consists of a set of vertex-disjoint s,t-paths. Then every path is reduced to one vertex, and the set of resulting vertices is reduced to one vertex whose operation probability is the reliability of G' . This motivates a method for bounding via approximation of G by a graph reducible to one vertex by simple node reductions; this is the subject of the next section.

, = I

"€C,

7. SERIES-PARALLEL BOUNDS

We present a new method, producing bounds that are uniformly better than vertex-packing bounds. The basic strategy is to produce from G a new graph G' by a set of operations, such that G' can be reduced by simple node series and parallel reductions to a simple path (s,v,t). Thus the reliability of G' equals the operation probability of v. To obtain an upper bound, the set of operations used must not decrease the reliability of G, and for a lower bound the set of operations must not increase the reliability of G . Cau- tion: The terms series and parallel do not have their usual hardware meanings, but are specially defined logic terms in section 7.1.

71. Node Series and Parallel Reductions

Notation & Nomenclature:

Adj(x): Series:

set of nodes adjacent to x in G . Two nodes x and y are in series if Adj(x) = (y,w} and Adj(y) = {.,U} for some two distinct nodes U and w in G.

Parallel: Two nodes x and y are in parallel if Adj(x) -y = Adj(y) - x .

Series reduction: For any two non-target nodes x and y in series, replace them by a single node v with operation probability p,, = p r p r (see figure 7-1-a for example). The series reduction can also be applied to two adjacent nodes x and y with degree greater than two if the edges incident with x and y can be assigned directions such that all edges incident with x are going into x except for the edge ( x , y ) going outward, and vice versa for edges incident with y (see figure 7-1-b for example).

546

Fig. 7-1.

S 2 Y t 0 v 0

S 1) t c (7-1-a)

(7- 1-b)

Node Series Reduction.

a 2 h

W

a l h

W >( Fig. 7-2. Node Parallel Reduction.

Parallel reduction: for any two nodes x and y in parallel, replace them by a single node v with operation probability p y = p x + pr - pxpy (see figure 7-2 for example).

Example: Consider the RBN graph shown in figure 7- 3, where s and t are the source and destination nodes. We use reductions to obtain Rel,(G). Two parallel reductions,

IEEE TRANSACTIONS ON RELIABILITY, VOL. 38, NO. 5 , 1989 DECEMBER

2 U

t - c S W

Fig. 7-3 Parallel .?3hd Series Reductions Example.

= ‘ . U 3 t c _D-- -O

c c

0 2

Fig. 7-4. Forbidden Horneornorph of K, - e With Respect To s and t.

the first on { x , y} and the second on {U, v}, leave two nodes in series. Use a series reduction to obtain w. Thus,

Z2. Series-Parallel Graphs and Series-Parallel Reducible Graphs

Nomenclature

homeomorphic: G , is homeomorphic to G , if G, can be obtained from G , by a (possibly empty) sequence of contractions of edges incident with nodes of degree - 2.

A graph is series-parallel with re- Series-parallel graph:

Chord:

spect to s and t (s,t-series-parallel) if it can be reduced to an edge ( s , t ) by applying edge series and parallel reductions [ 151. The necessary and sufficient condition for a graph to be s, t-series-parallel is that it contains no subgraph homeomorphic to K4-e (see figure 7-4) with respect to s and t (that is, if we add an edge ( s , t ) we form a homeomorph to K 4 , the complete graph of four vertices

The cross path of a homeomorph to K4-e (for example, the path from v , to v, in figure 7-4). By K4-e we mean a subgraph homeomorphic to K4- e with respect to s and t .

[151).

ABOELFOTOH/COLBOURN: COMPUTING 2-TERMINAL RELIABILITY FOR RADIO-BROADCAST NETWORKS

~

547

Series-Parallel Reducible Graphs: an undirected graph G , with two distinguished nodes s and t , is series- parallel reducible if it can be reduced to a path (s,v, t) using node series and parallel reductions.

A subgraph S is irrelevant if none of its vertices (and edges) can belong to any induced (RBN) s,t-path.

Irrelevant Subgraphs:

Fact 7.1. Any s,t-series-parallel graph G = (VT E ) with no irrelevant vertices is reducible by node series and parallel reductions (series-parallel reducible).

Proof: The proof is by induction on number of vertices (see [ I ] Section 5.4.2). 0

7.3. Upper Bounds via Approximation by Series-Parallel Reducible Graphs

The objective is to find a graph G’ such that Rel,(G’) 2 ReI,(G). We impose on G‘ the condition that every cutset in G used in the vertex-packing method (section 6.2.2) is a cutset in G’ . Hence, the reliability of G’ is at most the reliability of the graph obtained from the vertex-packing method, and the new bounds are uniformly no worse than the vertex-packing bounds.

7.3 .l. Definitions, Notation, and Preliminaries

Main s,t-paths:

p ,

1 Free vertex:

Free edge: Free path: Isolating irrelevant subgraphs:

A set of internally vertex-disjoint s, t-paths. Z number of main paths

main path, chosen so that P, is no longer than P,+ I

(if it exists). length of shortest main path P I .

path. Any vertex that does not belong to any main

An edge that does not belong to any main path. A path that contains only free edges.

An irrelevant subgraph S, whose vertices are connected to both s and t through only one vertex v is an irrelevant subgraph attached to v . An irrelevant subgraph S, whose vertices are connected to both s and t through two adjacent vertices v and w is an irrelevant subgraph attached to the edge (v,w). Since no edge and no vertex in an irrelevant subgraph belongs to any RBN s, t-path, the deletion of irrelevant subgraphs does not affect the reliability of the graph.

Identibing a vertex w with another vertex v means- a . Edges incident to w are made incident to v. b . loops and irrelevant subgraphs attached to v are

deleted. c . any multiple edge created is replaced by a sin-

gle edge. d. If w is a free vertex or v and w belong to the

same main path P,, w is deleted from G. Oth- erwise, if v E P, and w E P,, j # i , the new vertex is taken to be both v on P, and w on P,.

A set of vertices that are identified

Identifying vertices:

Identified in parallel:

into one vertex with operation probability equivalent to a parallel reduction. Identifying a set of nodes in parallel is equivalent to adding edges such that all the nodes have the same adjacency and then applying node parallel reductions to the nodes.

The node resulting from identifying two nodes is assigned an operation probability equivalent to a series reduction.

The I - 1 minimal cutsets obtained by the vertex-packing method from section 6.

vertices in the same component as s when cutset C, is removed.

Identijied in series:

Main vertex cutsets:

c, main vertex cutset i . x, B,,+I x,+,\x,; B,-l,l EE V\(X,-, U it)>. Vertices Between: X , + contains vertices between C, and

c,+ 1. Cut sequence: Let v be a vertex in C, such that v is not

adjacent to any vertex in C,+ , (or t for i = 1 - 1 ) . We define a cut sequence between v and C,, , as follows. Assume that all vertices C, - v are tem- porarily deleted and that all vertices of C, + are identified in one vertex (say e ,+ , , with = t for i = 1 - 1). Hence, v is connected to e,,, through only vertices from Bt , , + , . Let the distance from v to c , + ~ be d. The cut sequence between v and C,,, contains d - 1 cutsets and is obtained by taking all vertices (in B,.,+,) adjacent to v as the first cutset (csI), then taking all vertices adjacent to cs, as cs,+ ,, for 1 < i < d - 1 . In obtaining the cut sequence (as in all algorithm steps) we consider only relevant vertices. We start with a graph that has no irrelevant subgraphs and we always delete irrelevant subgraphs as they are created.

If a graph contains a cutvertex, its 2-terminal reliability is easily computed from 2-terminal reliabilities of the bi- connected components; hence we always treat 2-connected graphs.

7.3.2. An Overview of the Method

Starting with a graph G we obtain a series-parallel graph G’ by vertex identification and deletion of irrelevant subgraphs. Thus, node reductions can be used to reduce G‘ to one vertex. The following steps outline the construction of G’ from G .

0. If G is series-parallel graph delete all irrelevant

1. Determine main s,t-paths and free vertices. 2 . Determine main vertex cutsets c , , 1 s i s 1 - 1. 3. For every main vertex cutset C , , 1 s i =S 1 - 1,

identify all vertices that belong to the same main s, t-path. Now every cutset contains exactly one vertex from every main path and possibly one or more free vertices. Let C,, = the vertex from P, in C,. Since identifying vertices from the same main cutset C, = ( X , , x , ) does not add any edge

subgraphs and reduce using node reductions.


between a vertex in X , and x,, C, remains a cutset after step 3.

In step 4 we process vertices between two main cutsets (B, , + ,) to determine the cut sequences between vertices of C, and C,, I . The vertices of every cutset in a cut sequence are identified in parallel. Therefore after step 4 every vertex in C, is either adjacent to at least one vertex in C,,, or adjacent to exactly one vertex in B, ,+ I .

In processing a cutset C, , we first process C , ) ; then we process free vertices in C,.

4 . For all vertex cutsets C, i = 1 to i = 1 - 1. a. Copy G to H . b. Identify all vertices from C,+ , into one vertex c, + I

in H . ( c ,+ , = t for i = 1 - 1). c. For every vertex v in C, that is not adjacent to c,, I

in H determine B , , + I .

i . Delete all vertices C,- v. ii. Find the cut sequence that separates v from

c, , ,. Identify the vertices of each cutset in H , and reduce the vertices of each cutset A in G to one vertex with op-

eration probability 1 - rI( 1 - p, , ) (parallel identification). U 6 4

b. set p , = p , p u . c. Repeat at step 5a for all the vertices in the cut

For example, consider the subgraph of C' shown in figure 7-5 after performing step 4. In step 5 when we process vertex b, e is identified in series with b, then c is identified in parallel with 6 , then f is identified in series with b. At processing the vertex d, j is identified with d in series then k is identified with d in series.

Step 6 transforms G into a series-parallel graph. We use parallel identification to eliminate any existing chord. In step 6-a every free vertex in a main cutset is identified with a vertex on the main paths. In step 6-b-i we eliminate any free path between two vertices that belong to two different main paths. Thus steps 6-a and 6-b-i eliminate all chords formed by free paths. Step 6-b-ii eliminates all chords formed by edges of the main paths (see figures 7-6

sequence between v and C,, I .

& 7-7).

6. Transform G' into a series-parallel graph. a. For every C , 1 S i 4 1 - 1-

Reduce in parallel every free vertex v in C, with C, , where P, is the highest order path to which v is connected.

iii. Any main vertex that does not belong to a cut sequence is marked as a free vertex and all incident edges are marked as free edges.

b. For every C,- i . If there exists a free path from C,,,, j = 1 to p - 1, to a vertex U on path P , , k > j , identify all vertices from C,,, to Cl, , inclusive in parallel.

ii. Identify C J , k with C,,) ,k > j , if any vertex on P, has been identified with a vertex on p,, k > x > j (figure 7-6-a). Identify C,,.r with C,,,,x > j , if C,,, has been identified with C,,,, k > x > j and there exists a vertex on P, that is identified with P., and not identified with P, (figure 7-6-b).

7. At this stage C' is a series-parallel graph. Using node series and parallel reductions we can reduce G' to one vertex between s and t that has probability equal to the reliability

Figure 7-5 shows an example of step 4. In step 5 , node series and parallel identifications re-

move vertices between main cutsets {B,,,+ I } that belong to a cut sequence. Vertices in B,, , , that do not belong to a cut sequence become irrelevant after step 6 and are deleted. We show in section 7.3.3 that series identifications in step 5 cannot decrease the reliability when combined with the parallel identifications in step 6.

5 . For all vertex cutsets C,, i = 1 to i = 1 - 1- For every vertex v in C, that is not adjacent to a vertex in

a. If the next vertex ( U ) in the cut sequence between v and C , , , is not adjacent to any vertex in C, - v identify U with v and set p , = pvpu (series identification). Otherwise, make U adjacent to the subset A of C, - v. Identify the

c,+ I--

vertices of A with v and set p,, = 1 - (1 - p v ) n(1 - p a ) . Then identify U with v and us4

I I

a\-: I

I I

Ci+l Ci

Fig. 7-5. Example of Step 4.

of G ' .

7.3.3. Correctness

We must establish:

1. Rel, (G') 2 Rel,(G) 2 . Rel,(G') S Rel,(G,,), where G, is the graph ob-

3. After step 6 G' is reducible by node series and par-

According to fact 7.1 it is sufficient to show that G' is a series-parallel graph.

We construct G' from G by identifying vertices in parallel, identifying vertices in series, and deleting irrelevant edges and vertices. Identifying two vertices in parallel is identical to adding edges such that both vertices have the same adjacency, then reducing them using a simple node parallel reduction. This cannot decrease the reliability. The deletion of irrelevant vertices and edges also does not affect the reliability of G ' . Therefore, it remains to show that identifying vertices in series in step 5 does not decrease the reliability of G' .

tained by the vertex-packing method

allel reductions.


Ci,k ............... (7-6-a)

Fig. 7-6. E1imir:ating Chords of K, - e in Srep 6-b-ii

.............

h pk ............... : a c h o r d

Fig. 7-7. The First Case of a Chord of K4 - e.

Lemma 7.2: The series identification of vertices in step 5 does not decrease the reliability of G’ when combined with the parallel identification of vertices in step 6.

Proof Details appear in [ 11; we outline the proof here. Consider two vertices v and U to be identified in series in step 5 . v E C , and v is not adjacent to any vertex in C, + I

( v is not adjacent to c , , ~ in H ) . U is the vertex adjacent to v in the cut sequence between v and C,, I . If both v and U

have degree two, the series identification of U with v is equivalent to simple series reduction of v and U . Suppose that v and U have degree three or more. We show that some edges incident with U and v are either eliminated in a parallel identification or become irrelevant after adding the edges implied by the parallel identification in step 6. Whether we add the edges implied by the parallel identification before doing the series reduction, or after, does not change the resulting graph. 0

Lemma 7.3.

Rel,(G’) d RelJG,,) = fi , = I (1 - 9 (1 - p, ) ).

Proof: Since the parallel identifications and reductions do not imply adding edges between vertices in two different partitions with respect to any cutset C,, every cutset C, corresponds to a cutset C, in G’. Identifying a set of vertices in series results in a vertex with at most the same probability of operation as the vertex with the least probability. There- fore, every vertex v’ in C’, (before the parallel identification) has a probability of failure (1 - p’ , ) 2 (1 - p , ) where v is the corresponding vertex in C,. Hence, for every failed

: a chord (7-6-b)

state of Cop, there exists a failed state of G’ with at least U

Lemma 7.4. After step 6, G’ is reducible by node series and parallel reductions to a path (s,v, t) .

Proof: It is sufficient to show that G’ (after step 6) is a series-parallel graph. We show that after step 6 , G’ contains no subgraph homeomorphic to K, - e with respect to s and r . After step 6 , G’ contains only main paths with all vertices in the main cutsets. Suppose that G’ has a subgraph homeomorphic to K , - e. The two ends of the chord (a,b) either belong to the same main path i (figure 7-7) or belong to two different main paths j and x , x < j (figure 7-6). The first case contradicts step 6-b-i since if b is identified with a vertex from Pk and k > j then b should be identified with a vertex from j ( e ) . The second case contradicts step 6-b- ii. U

the same probability of failure.

7.3.4. Complexity

Finding the shortest paths and labeling the vertices with their distance from s requires O ( m ) time using breadth first search. Therefore steps 1 & 2 require time O(m) . We can identify vertices on the same path (step 3) as soon as we add them to the main vertex cutset. This way we need not sort the vertices within every cutset. Since we add and delete O(m) edges, step 3 requires O ( m ) time. In step 4 we process O(n) cuts in the cut sequences, Processing each cutset involves two main operations: labelling using breadth first search and identification of vertices. Both operations require O(m) time. Therefore, the total time required by step 4 is O(nm). Both steps 5 & 6 identify O(n) vertices, requiring total time O(mn). A simple method to calculate the reliability of G’ is to iterate between doing all possible parallel reductions at every cutset and all series reductions for every path. Since we perform O(n) iterations, and each iteration takes O ( m ) time, the total time for step 7 is O(mn). There- fore, the complexity of the algorithm is O(nm) .

7.4. Lower Bounds via approximation by Series-Parallel Reducible Graphs

We present a method for obtaining lower bounds that are uniformly better than vertex-packing lower bounds. The

550

vertex-packing method can be viewed as the determination of a simpler graph G' with Rel,(G') S Rel,(G) by deleting edges and vertices until G' consists of a vertex-disjoint set of s,t-paths. Deleting edges and vertices cannot increase the reliability of G. We also delete some of the edges and vertices of the graph. However, we try to retain as much as possible of the graph structure while making it reducible by node series-parallel reductions.

Z4.1. More Nomenclature & Notation:

P r i m a r y p a t h s : A se t of ver tex-dis joint s , t -pa ths P I , . . . , P z ; z = the number of primary paths. These paths are ordered by non-increasing relia-

bility ( g p . ) . If all edge probabilities are equal,

paths are ordered by non-decreasing length. If v E P , , the vertex order of v with respect

to P , order (v , i ) , is the position of v on P , (order ( s , i ) = 0, for all i ) .

A vertex v is shared if it is in common between two or more s,t-paths, or if there exists an edge-disjoint segment between v and another vertex on the same s,t-path.

Two paths P, & P, are incompatible if they share at least two vertices U and v, and order( v, i ) < order( U , i ) and order( v,j) < order( u, j ) (see figure 7-8 for example). Vertex v is a shared vertex on PI with a reversed order with respect to P , . For example, in figure 7-8-a v and w are shared vertices with reversed orders. Two paths P, and P, are compatible if they share at most one vertex or, if they share more than one vertex and the set of shared vertices have the same relative ordering.

Two paths P , & P, are crossing at the two vertices v and w if each of them shares a vertex with some path P k , v and w respectively, v # w, v E P, and w P , , and there exists no vertex U

such that order(v,k) < order(u,k) < order(w,k) and U E P , U P, U P , (see figure 7-9-a). Any two

Vertex order:

Shared vertices:

Incompatible paths:

Crossing paths:

IEEE TRANSACTIONS ON RELIABILITY, VOL. 38, NO. 5, 1989 DECEMBER

(7-8-a) Two Incompatible Paths

(7-8-b) Two Compatible Paths.

Fig. 7-8. Illustration of Incompatible Paths

compatible paths and any set of paths that are pairwise noncrossing can be reduced using node series and parallel reductions.

For P an s,t-path containing U , v and w with order(u) < order(v) < order(w), and II = {u ,x , . . . ,y, w } a vertex-disjoint u,w-path, we shift II left to v by replacing the edge (y ,w) by the edge ( y , v ) . We shift I1 right to v by replacing the edge (x ,u) by the edge ( x , v ) (see figure 7-10).

For x and y two vertices on an s,t-path, a segment between x and y exclusive is written ( x - y ) . Consider the subgraph in figure 7-10. We find conditions to ensure that the path shifting of TI to v does not increase the reliability of G .

Derivation of the condition for the shift-left operation. The condition for the shift-right case can be derived

similarly. Let H be the subgraph before shifting and H' be the subgraph after shifting.

Let S,, 1 s i s 7, be the segments ( s -U) , (u-v) , II excluding U and w, ( w - t ) , ( v - w ) , ( s - v ) , ( v - t ) , respectively. The only states that are operating in H' while failed in H occur when segments SI, S3, S, operate, u and v operate, and segments S,, S6 fail, and either w fails or both S, & S, fail. Let Pr, be the probability that segment S,operates. The probability of this set of states is:

Path shifting operation:

Prob, = pup,Pr,Pr,Pr,(l - Pr,) . (1 - h6) . ( 1 - p w ) + p,(l -PrJ ( 1 - Pr,)). (1)

The states that are operating in H while failed in H' are defined by: the segments S, and S, operate, U and w operate, and, either:

1) S, is operating and v is not, or 2 ) SI and v are operating, and S, and S, are not, or 3) S,, s6 and v are operating and S, and S, are not.

The total probability of these states is: Prob, =

Compare ( 1 ) and (2); if P , 3 P , and q y p w P , 2 qw p y P , U then Prob, S Prob, and Rel,(H') s Rel,(H).

Z 4 . 2 . An Overview of the method

Obtain primary paths {Pi } from a vertex-packing method. Initially, G' contains only primary paths. G' is aug- mented in two phases:

1. Add any remaining s,t-paths in G , in which case the paths have some vertices shared with the initial set of paths. We must check for incompatible and crossing paths. When we add a path to G', and it shares some vertices with a primary path P , it is a secondary path associated with P .

11. Check the unused segments in G , and try to add them to G ' .

ABOELFOTOH/COLBOURN: COMPUTING 2-TERMINAL RELIABILITY FOR RADIO-BROADCAST NETWORKS 55 1

(7-9-a) Pi a n d Pj a r e crossing at v , 20, 5 and y.

t S

(7-9-b) Pi, P, a n d Pk are pairwise non-crossing.

Fig. 7-9. Illustration of Crossing Paths

At the end of phase I, G' has a set of primary paths, and associated with each primary path is a set of secondary paths. Figure 7-11 shows an example of a graph G and the graph G' resulting from phase I. The repeaters in the initial set of paths are as follows. P , = {a ,b ,c} , P, = {h,f,i ,g,j}.

e

s6 s 7

Fig. 7-10. Example f the Path Shifting (left) Operation

Phase I

In copying an s,t-path P to G' , we check paths in G' , P,, 1 d j < i , in order, searching for the first primary path

(7-11-a) G.

e 0

that shares vertices with P,; one of the following cases s a b C t occurs.

I . P shares one or more vertices with a primary path P, and P, shares no vertices with any other path in G ' , ie has no secondaries. If P is compatible with P, we add it to G ' and mark P as a secondary path associated with P,.

2 . P shares one or more vertices with a primary path in G ' , P,, and P, has one or more secondary paths associated with it. If P is noncrossing with all secondary paths associated with P, we add it to G' and mark P as a secondary path associated with P,.

(7-11-LJ) C: afler Phase 1.

Fig. 7-11. Example of Phase I


G’ has two primary paths, associated with each of them is one secondary path. 0

Phase I I

After copying every path from G to G’ , we mark (delete) its edges in G. In phase I1 we try to add to G’ any remaining paths between pairs of repeaters on the primary paths, if they are vertex-disjoint from all other paths in G’. The method can be extended to pairs of vertices on the newly added segments to obtain a better approximation.

For any two vertices U and w on an s,t-path, if there is no shared vertex between them, any unused segments from U to w can be added as is to G’. However, if there is a shared vertex between U and w we use the path shifting operation to avoid creating a homeomorph to K4 - e. To ensure that the conditions for the path shifting operations are not violated while we are adding new segments, we first try to add new segments to secondary paths.

Figure 7-12 shows the graph G’ of figure 7-11 after applying phase 11. 0

7.5. Correctness

We need to establish the following: G’ is reducible by node series-parallel reductions, and Rel,(G’) s Rel,(G).

Any series-parallel graph with no irrelevant vertices or edges in the RBN sense is reducible by node series and parallel reductions (fact 7.1). Therefore we show that G’ is node reducible by showing that it is a series-parallel graph. The primary paths are vertex-disjoint and there are no edges in G’ between any two vertices that do not belong to the same s, t - path. Therefore, at the end of phase I G’ is a series- parallel graph, by the two conditions on associating secondary paths with primary ones, namely, every primary path is compatible with every associated secondary path, and every primary path and any two associated secondaries are pairwise non-crossing.

In phase I1 we add only segments between pairs of vertices on the same path with no shared vertices between them, and these segments are vertex-disjoint with all other paths in G’. This is exactly adding segments in parallel.

Lemma 7.5. Every two compatible paths can be reduced using series-parallel reductions.

e

Fig. 7-12. Example of Phase II

Proof: Let P, & P, be two compatible paths, and let v be the lowest order shared vertex on P,. There must be a segment from s to v on P, that shares no vertex with P,, and the two segments from P, and P, can be reduced using a parallel reduction. Otherwise, P, shares a vertex w with P, and order(w, i) > order(v, i) and order(w,j) < order(v,j), which is a contradiction. The same argument can be applied

0 Lemma 7.6. Every set of non-crossing secondary paths

associated with the same primary path can be reduced using series-parallel reductions.

Proof Let P, & P, be two non-crossing paths associated with the primary path P,, and let v be the lowest order shared vertex on P,. There must be a segment from s to v on P , that shares no vertex with P,, and the two segments from P, and P, can be reduced using a parallel reduction. Otherwise, P, shares a vertex w, # v, with Pp, and P, and P, are crossing. The same argument can be applied induc-

0 After completing phase 11, all unused edges in G are

deleted. Therefore, the method involves two operations: 1) deleting vertices and edges, and 2) path shifting. The first operation does not increase the reliability of G. The order in which we add the new segments ensures that after testing the conditions of path shifting when adding segments to primary paths, no more segments are added to secondary paths and if the condition was satisfied it remains so until the end of the algorithm.

7 6. Complexity

The method for series-parallel lower bounds requires obtaining primary paths. Here we discuss only the time corn- plexity of phases I & 11. For every vertex we keep track of the paths to which it belongs, and its order on each path. Therefore checking for compatible and crossing paths can be done in time proportional to the length of the path. In the worst case the new path (to be added to G‘) has to be checked against all paths in G’. Say at some stage G’ has i - 1 paths. Then the time required to add the ith path is O(il,), where I, is the length of P,. Therefore the total time

for phase I is 0 s 0 m c1,

inductively from v to t .

tively from v to t .

s O(m2). i,:, ) i ,:, ) Finding and adding the new segments in phase I1 can

be done in O(m) time; this is repeated for O(n) vertices for total time of O(mn) for phase 11.

ACKNOWLEDGMENT

We thank Mike Ball, Anna Lubiw, and an anonymous referee for helpful comments. Research of the second author is supported by NSERC Canada under grant A0579.

APPENDIX (Proof of Theorem 5.3)

Before the proof we restate a formula due to Satyana- rayana & Prabhakar [27], that applies to the general case


where both edges and nodes can fail. They showed that 2- terminal reliability in directed graphs (s, t-connectedness) can be written as:

where {G,} is the set of acyclic subgraphs of G with b edges and n vertices, in which every edge is in at least one simple path from s to t .

For the undirected case the formula can still be applied; every undirected edge is replaced by a pair of arcs. Provan [24] observed that the G,'s in the undirected case correspond to acyclic orientations of subgraphs of G for which every edge is in at least one simple s,t-path. Hence the G,'s are referred to as s,t-subgraphs.

For the purpose of this theorem, (A-1) is equivalent to:

R(G,s,t; p ) = 2 ( - l ) b - n + l p" b . n H&bn

{Gb,,} is the set of s,t-subgraphs with b edges and n vertices.

Dejnitions

In a grid graph G, a vertex cutset containing only degree-2 vertices is an orthogonal cutset if it can be partitioned into two or more subsets such that: 1 ) each subset is a cutset; 2) all vertices in each subset are incident with one type of edges, either horizontal (vertical cutset) or vertical (horizontal cutset); and 3) every vertical cutset consists of all vertices in G with x-coordinate i , for some i , and every horizontal cutset consists of all vertices in G with y-coordinate j , for some j . An example is shown in figure A-1.

For C an orthogonal cutset,

R" (G, S, t , C ; p ) = 2 ( - l ) b - n + l p"; b.n H c C g

Cc is the set of s,t-subgraphs with b edges, m vertices from C , and n vertices from V - C.

We use m p for polynomial time reducibility. Lemma A.1. Rm m p R , restricted to grid graphs. Proof Given a grid graph G, s, t , C , m and p , where

the cardinality of C is c, partition C into horizontal and vertical cutsets. Let C, be the set of vertical cutsets and C, the set of horizontal cutsets.

Construct a graph G, by replacing every vertex in C, by r vertices and r - 1 horizontal edges, and every vertex in C, by r vertices and r - 1 vertical edges. To show that G, is a grid graph for any positive integer value of r , it is sufficient to show that the graph obtained from replacing only the vertices of one of the subsets is a grid graph.

Consider a vertical cutset C, = {c,,c,, . . . ,cs}. Let X & Y be the two partitions of V after removing the vertices of C,. Since C,, contains only degree-2 vertices every vertex in C,, is adjacent to exactly one vertex in X and one vertex in Y . Let X, = {x,,xz,. . . , x,} and Y , = ( y , , y 2 , . . . ,y5} be the subsets of X & Y respectively that are adjacent to vertices

horizon tsl

horizontal

I vertical cutset

A

cutset

cutset

l

o : Orthogonal rutset vertices

Fig. A l . An Example of an Orthogonal Cutset

of C,. Assuming that the sets C,, X , , Y , are ordered according to one of the grid coordinates, therefore, every three nodes x,,c,,y, for i = 1 to s induce a path of two horizontal edges in the grid. Consider a mapping of G onto the grid. To construct G, every path (x,,c,,y,) is replaced by a path of r + 1 horizontal edges. And, every vertex in Y - Y , with grid coordinates (ij) is mapped into coordinates ( i + r - 1 , j ) . Similar steps are applied for other vertical and horizontal cutsets.

Applying (A-1) to G, -

since every s,t-subgraph in G, corresponds to exactly one s,t-subgraph in G. Hence-

WG,, s, t ; p ) = C (-p')"R"(G, s, t , c; p ) . m = O

Given a polynomial time algorithm for evaluating R , we can substitute for r by values 1 to c + 1 to obtain a system of linear equations in R", form = 0 to c. This system of equa- tion has a unique solution and can be solved in polynomial

0 time even with rational coefficients [ 3 1 ] . For the next step, set -

R""(G, S, t , C) = ( - , ) ' + I b HcGG

GTn = set of s,t-subgraphs with b edges, m vertices of C , and n vertices of V - C.

554 IEEE TRANSACTIONS ON RELIABILITY, VOL. 38, NO. 5 , 1989 DECEMBER

Lemma A.2. R““‘(G, s, t , C) xcp R, restricted to grid

Proof: We first show that R”“ mp R’“, and then the lemma graphs.

follows from lemma A.1. R”’ can be rewritten as:

IV-CI

R y G , s, t, c; p ) = 2 2 ( - l)b-n+lp’i n = O b H G r n

I V - C I

= ’ 2’ Rmrr(G, s, t, C) ( -PI“ n = O

Substituting for p by (V - CI + 1 distinct values (0 < p < 1) we have a system of linear equations in R“”’s. 0

We now prove theorem 5.3. Theorem 5.3. Counting Hamiltonian cycles on grid

graphs is mp computing the 2-terminal reliability with perfect edges and imperfect nodes (#HC m p R(G,s,t; p ) ) .

Proof: Given a grid graph G = ( Y E ) with m vertices, construct the graph H by splitting every edge into two edges with the same orientation as the split edge with respect to the grid. That is every edge (u ,w) is replaced by a path (u,v,w). The graph H is still a grid graph. Let C be the cutset that consists of all vertices in H resulting from subdividing edges in G({v}). C is an orthogonal cutset since it can be partitioned into the following subsets:

e Vertices corresponding to vertical edges in G connecting vertices at grid coordinates (i,j) and (i,j + I ) , i,,, =s i S i,,,,

0 Vertices corresponding to horizontal edges in G that connect vertices at grid coordinates (ij) and ( i + I j ) , j,,, S j S j,,, (imln,imax,jm,n,jmax are the minimum and maximum x and y coordinates of the grid representation of G respectively).

Consider the function:

( H , s, t , C ) = ( - , ) ’ + I Rmm-1

b sCH;“‘- ’

{HY;’} = set of s,t-subgraphs in H with b edges, m vertices from V, and m - 1 vertices of C.

According to lemma A.2, R“-’(H,s,t,C) m p R(H,s,t). Take any vertex in G as s (without loss of generality we assume that the minimum vertex degree is 2), and take any of its neighbour vertices as t. The set of s,t-subgraphs in H with m vertices of V and m - 1 vertices of C is exactly the set of Hamiltonian paths in G from s to t. Therefore, the number of edges is exactly 2m - 2 and-

#HP,,I = ( - 1)+’

#HP,,, = the number of Hamiltonian paths from s to

Apply the same function to all neighbours of s in G;

t.

we have-

Thus, #HC mxD R”‘.’”-‘. 0

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ABOELFOTOH/COLBOURN: COMPU’MNG 2-TERMINAL RELIABILITY FOR RADIO-BROADCAST NETWORKS 555

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AUTHORS Dr. H.M.F. AboEIFotoh; Dept. of Mathematics; Kuwait University; PO Box 5969; 13060 KUWAIT.

Hosam M. F. AboEtFotoh (S ’82, M ’88) received the BSc (with honors) in Electrical Engineering (Computer Section) from Ain Shams University in Cairo, Egypt in 1977, and MMath and PhD degrees in Computer Science from the University of Waterloo in 1984 and 1989,

respectively. In 1989 September, he joined the Department of vathe- matics at the University of Kuwait as an Assistant Professor in the Com- puter Science group. His research interests include combinatorial algorithms, computer networks, and parallel and distributed computing and databases.

,

Dr. Charles J . Colbourn; Dept. of Combinatorics and Optimization; Uni- versity of Waterloo; Waterloo, Ontario, N2L 3G1; CANADA

Charles Colbourn (A ’89) earned his BSc (University of Toronto, 1976), MMath (University of Waterloo, 1978) and PhD (University of Toronto, 1980) degrees in computer science. From 1980 until 1983, he was a faculty member in Computational Science at the University of Saskatchewan; since 1984, he has been with the Departments of Com- binatorics and Optimization, and Computer Science, at the University of Waterloo, where he currently holds the rank of Professor. Dr. Colbourn is the author of The Combinatorics of Network Re1 Algorithms in Combinatorial Design Theory and C Theory. He is the author of over 100 refereed publi networks, graph algorithms, and combinatorial de search interests focus on the interplay between algorithms.

Manuscript TR88-156 received 1988 August 24; revised 1989 June 10.

IEEE Log Number 30684 4TRb

MANUSCRIPT RECEIVED MANUSCRIPTS RECEIVED MANUSCRIPTS RECEIVED MANUSCRIPTS RECEIVED

“A simple algorithm for computing network reliability”, Brijendra Singh 0 Dept. of Electronics & Communication 0 University of Allahabad Allahabad (U.P.)-211 002 0 INDIA. (TR89-213)

“Nonparametric confidence bounds, using censored data, on the mean residual life”, Dr. Frank Mitchell Guess 0 Dept. of Statistics 0 338 Stoke- ly Management Center 0 University of Tennessee Knoxville, Tennessee 37996-0532 USA. (TR89-214)

“Beta-expectation tolerance-intervals and sample-size determination for the Rayleigh distribution”, Dr. Mostafa S. Aminzadeh 0 Dept. of Mathematics 0 Towson State University 0 Towson, Maryland 21204 0 USA. (TR89-215)

“Factoring & reductions for networks wlth imperfect vertices”, Olympia R. Theologou 0 URA CNRS HEUDIASYC 0 Universite do TechnQlQgie de Compiegnc 0 BP 649 60206 Compiegne 0 FRANCE. (TR89-

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Computing 2-terminal reliability for radio-broadcast networks

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