A cut-based method for terminal-pair reliability

IEEE TRANSACTIONS ON RELIABILITY, VOL. 45, NO. 3, 1996 SEPTEMBER 413

A

Cut-Based Method for

Yu

G . Chen

Maria C.

Yuang, Member

IEEE

National Chiao Tung University, Hsinchu

National Chiao Tung University, Hsinchu

Key Words - Terminal-pair reliability, Factoring, Partition- ing, Network reduction

Summary & Conclusions - This paper assesses two categories of partition techniques for computing terminal-pair reliability (path- based and cut-based algorithms) by experimenting on published benchmarks; the criteria are the number of subproblems and the computation time. The cut-based algorithm is superior to the path- based algorithm with respect to the computation time for most benchmarks. A refinement of the cut-based algorithm (using net- work reduction) profoundly outperforms the path-based algorithm (with reduction) for all benchmarks.

1. INTRODUCTION Acronyms

D&G

CB, PB [cut, path] based

CBR, PBR [CB, PB] algorithm with reduction R&P

RKP R-CB

Dotson & Gobien algorithm [4]

Deo & Medidi algorithm [3]

Rai, Kumar, Prasad - CB partition algorithm [lo] refined cut-based algorithm (presented in this paper). The analysis of network reliability has been given con- siderable attention. In particular, terminal-pair reliability [ 1 -

6, 8 - 141 deals with the determination of the reliability between two nodes (source and sink) of a network, given failure prob- abilities of all links. Existing terminal-pair reliability algorithms, based on the partition technique, fall into two categories: PB or CB. In the PB technique, D&G, the most efficient PB algorithm [ 141

,

used a partition algorithm based on the shortest path from the source to the sink. R&P further combined D&G with network-reduction [8] and improved performance even more. On the other hand, RKP performed the partition by a source-cut

-

separating the source from the remaining network nodes.

This paper assesses the PB and CB partition algorithms in terms of the number of subproblems and computation time, by experimenting on published benchmarks. In both D&G and R&P, the numbers of subproblems generated by partitioning are locally minimized at the expense of executing the path- searching algorithm for finding the partition basis in each subproblem [3]. On the other hand, RKP makes no attempt to

'The singular & plural of an acronym are always spelled the same.

Terminal-Pair Reliability

minimize locally the number of subproblems, but greatly reduces the computation time for the partitioning of each subproblem. Our experimental results show the superiority of the CB algorithm over the PB algorithm with respect to the computa- tion time for most benchmarks. This paper also refines the CB algorithm using network reduction. R-CB profoundly outper- forms the PB algorithm (with reduction) for all benchmarks. Section 2 gives an overview of the D&G, R&P, and RKP, and presents R-CB. Section 3 compares the performance be- tween the PB and CB algorithms.

2. OVERVIEW OF PB & CB ALGORITHMS Four algorithms are summarized:

D&G - PB algorithm, R&P - PBR algorithm, RKP - CB algorithm, R-CB - CBR algorithm. Notation

Rel(G) terminal-pair reliability of network G s, t source, sink

ei set of links, i = l ,

...,

Z

p i , qi [success, failure] probability of ei; p i

+

qi

=

1

* - [contracting, deleting] operation of links.

Assumptions (for these algorithms)

1. The network is modeled as directed graph. 2a. The pi, qi, i = l,..

. ,I are known for the links.

2b. Nodes are perfect (do not fail).

3. All failure events are mutually statistically independent. 2.1 PB Algorithm - D&G

D&G computes Rel( G) from s to t in G by Boolean algebra. Given a set of links {el,ez, ... ,el> constituting an s-t

path (a path from s to t) , and based on the factoring theorem [8] : Rel(G) = ql.Rel(G-el)

+

p1.q2.Rel(G*el-e2)

+

...

+

[

@

p i ] ql. Re1 ( G*el *e2*.

. . *

el-l-el)

r 1 i

+

1

g p i ] .Rel(G*el*e2*...*el).

To minimize the number of subproblems generated by parti- tioning, the shortest s-t path is chosen as the basis for the parti- tion. The subproblems are recursively processed until the source & sink are contracted or disconnected.

414 IEEE TRANSACTIONS ON RELIABILITY, VOL. 45, NO. 3,1996 SEPTEMBER

2.2 PBR Algorithm - R&P

To gain better performance, R&P combined D&G with network reduction. The network is simplified by using network reduction. R&P used 6 reduction rules [SI, including remov- ing valueless links (such as entering the source) and series- parallel link reduction. The PB partition is in turn performed based on a shortest s-t path. Each generated subproblem is recur- sively processed by reduction & partition until the source & sink are contracted or disconnected.

2.3 CB Algorithm - RKP

Instead of partitioning based on the shortest s-t path, the CB partition uses the source-cut consisting of all links emanating from the source. Given source-cut (el, e2, . .

.

, el>, then Rai

et al, recursively factor Rel(G) analogous to D&G until the source & sink are contracted or disconnected.

2.4 CBR Algorithm - R-CB

A CB algorithm can incur more generated subproblems owing to the consideration of valueless links during the parti- tion. This difficulty can be eliminated by using network reduc- tion. In CBR, network reduction is always performed prior to the partition of each subproblem.

adjacency matrix representing the connections in an input net- work with n nodes, the worst-case computation time to find a shortest s-t path by a breadth-first search [7] is O(n2). By con- trast, the computation time to determine the source-cut is only O ( n ) . To justify this, we implemented the 4 algorithms (see section 2 ) in the C language and executed them in IBM RISC System/6000 using a collection of input-network benchmarks [3, 4, 8, 11 - 141, as shown in figure 1. Each input network with n nodes is represented by an n x n adjacency matrix denoting the connections in the network.

The performance of PB & CB are compared in terms of the number of subproblems and computation time. Figure 2 shows that the number of subproblems generated by PB is un- surprisingly less than that generated by CB for most of the benchmarks. As for the computation time, however, CB outper- forms PB in most of the benchmarks, as shown in Figure 3. Figures 2 & 3 also compare performance of PBR & CBR for published benchmarks. Figure 2 shows that the numbers of subproblems generated by CBR have been greatly reduced, and become comparable to those of PBR. As for the computation time shown in Figure 3 , CBR outperforms PBR for all benchmarks.

3. PERFORMANCE COMPARISOlNS

The CB algorithm results in lower complexity of deter- mining the partition basis than the PB algorithm. Given an n x n

- -I - (24) NSFNET A complete network with 10 nodes Figure 1. Benchmarks

CHENIYUANG: A CUT-BASED METHOD FOR TERMINAL-PAIR RELIABILITY 41 5

Figure 2. Number of Subproblems for the Benchmarks

416 IEEE TRANSACTIONS ON RELIABILITY, VOL. 45, NO. 3, 1996 SEPTEMBER

REFERENCES

[l] J.A. Abraham, “An improved algorithm for network reliability”, IEEE Trans. Reliability, vol R-28, 1979 Apr, pp 58-61.

[2] M.O. Ball, “Computational complexity of network reliability analysis: an overview”, IEEE Trans. Reliability, vol R-35, 1986 Aug, pp 230-239. [3] N. Deo, M. Medidi, “Parallel algorithm for terminal-pair reliability”,

IEEE Trans. Reliability, vol 41, 1992 Jun, pp 201-209.

[4] W.P. Dotson, J.O. Gobien, “A new analysis technique for probabilistic graphs”, IEEE Trans. Circuits & Systems, vol GAS-26, 1979 Oct, pp

S. Hariri, C.S. Raghavendra, “SYREL: A symbolic reliability algorithm based on path and outset methods”, IEEE Trans. Computers, vol C-36, M. Macgregor, W.D. Grover, U.M. Maydell, “Connectability: A per- formance metric for reconfigurable transport networks”, IEEE J. Selected Areas in Communications, vol 11, 1993 Dec, pp 1461-1469. [7] U. Manber, Introduction To Algorithms, 1989; Addison-Wesley. [8] L.B. Page, J.E. Perry, “Reliability of directed networks using the fac-

toring theorem”, IEEE Trans. Reliability, vol38, 1989 Dec, pp 556-562. [9] S. Rai, A. Kumar, “Recursive technique for computing system reliabili-

ty”, IEEE Trans. Reliability, vol R-36, 1987 Apr, pp 38-44. [lo] S. Rai, A. Kumar, E.V. Prasad, “Computer terminal reliability of com-

puter network”, Reliability Engineering, vol 16, 1986, pp 109-119. [ll] S. Soh, S. Rai, “CAREL: Computer aided reliability evaluator for

distributed computing networks”, IEEE Trans. Parallel & Distributed Systems, vol 2, 1991 Apr, pp 199-213.

[12] D. Torrieri, “An efficient algorithm for the calculation of node-pair reliability”, Proc. IEEE MILCOM ’91, 1991 Nov, pp 0187-0192. [13] D. Torrieri, “Calculation of node-pair reliability in large networks with

unreliable nodes”, IEEE Trans. Reliabilio, vol43, 1994 Sep, pp 375-377.

[14] Y.B. Yoo, N. Deo, “A comparison of algorithms for terminal-pair reliability”, IEEE Trans. Reliability, vol 37, 1988 Jun, pp 210-215. 855-865.

[5]

1987 Oct, pp 1224-1232. [6]

AUTHORS

Yu G. Chen; Dept. Computer Science and Information Eng’g; National Chiao Tung Univ; 1001 Ta Hsueh Rd; 30050 Hsinchu, TAIWAN - R.O.C. Internet (e-mail): [email protected]

Yu G. Chen (born in Taiwan, 1970) received the BS (1992) in Com-

puter Science and the MS (1994) in Information Engineering from the Na- tional Chiao Tung University, Hsinchu. He is a PhD candidate in the same department. His current research interests include network reliability analysis, ATM network management, high speed networking, and multimedia communications.

Dr. Maria C. Yuang; Dept. Computer Science and Information Eng’g; Na- tional Chiao Tung Univ; 1001 Ta Hsueh Rd; 30050 Hsinchu, TAIWAN Internet (e-mail): [email protected],edu.tw

Maria C. Yuang received the BS (1978) in Applied Mathematics from the National Chiao Tung University, Hsinchu; the MS (1981) in Com- puter Science from the University of Maryland, College Park, and the PhD (1989) in Electrical Engineering and Computer Science from the Polytechnic University, Brooklyn. From 1981 to 1990, she was with AT&T Bell Laboratories and Bell Communications Research (Bellcore), where she was a member of technical staff working on high-speed networking and protocol engineering. She has been an Associate Professor in Computer Science and Information Engineering at the National Chiao Tung University since 1990. Her current research interests include high speed networking, ATM network management, performance modeling & analysis, multimedia communica- tions, and protocol engineering.

- R.O.C.

Manuscript received 1996 June 25

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