simple enumeration of minimal cutsets of acyclic directed graph

484 IEEE TRANSACTIONS ON RELIABILITY, VOL. 37, NO. 5,1988 DECEMBER

Simple Enumeration of Minimal Cutsets of Acyclic Directed Graph

S. Hasanuddin Ahmad King Abdul Aziz University, Jeddah

Key Word8 - Directed graph, cutset, algorithm

Reader Aid8 - Purpose: Widen state of art Special math needed for derivations: Elementary probability Special math needed to use results: Same Results useful to: Reliability theoreticians and analysts

Abstract - There are many methods to enumerate cutsets of acyclic directed graph. All of these involve advanced mathematics. This paper gives 2 methods which use combinations of nodes to enumerate all minimal cutsets. One simply has to enumerate all combinations of orders 1 to N - 3 of nodes from 2 to N - 1, where N is the total number of nodes. Collecting only those symbols of links of first row of adjacency matrix and in the rows given in a combination which are not in the columns of the combination a cutset of an acyclic directed graph, having all adjacent nodes, is obtained. To obtain the cutsets of a general acyclic directed graph, four rules are given for deletion of those combinations which yield redundant and nonminimal cutsets. These rules provide a reduced set of combinations which then gives rise to minimal cutsets of a general graph. Three examples illustrate the algorithms.

1. INTRODUCTION

There are many uses of enumerating cutsets of a graph. Though techniques are being designed which do not use tiesets or cutsets [ l , 91, the enumeration of all minimal cutsets separating a specified node pair is a fundamental step in many algorithms for evaluating the reliability of networks. Minimal cutsets are also used in calculating the maximal flow through a network.

Many methods have been proposed to generate cutsets of directed [2, 4, 7, 81, and undirected graphs [2, 3, 51. Biegel [4] has used transpose and sum of matrices to enumerate all minimal cutsets for directed acyclic graphs. Pearson [8] has developed branching procedures for enumerating all minimal cutsets for directed acyclic graphs. Locks [7], treating tiesets as Boolean polynomials, uses deMorgan's theorem to invert and minimalize, and thus to get cutsets. Arunkumar & Lee [2] have also dis- cussed a case where the arcs are directional.

This technique does not require any knowledge of mathematics except that one should be able to represent a graph in the form of a table (or a matrix as set up by Biegel [4]). Some notations and definitions are given in section 2. In section 3, an algorithm enumerates all minimal cutsets of an acyclic directed graph in which all nodes are directly

linked with each other (or in which all nodes are adjacent). The algorithm is illustrated using the network in figure 2. In section 4, an algorithm enumerates all minimal cutsets of an acyclic directed graph in which at least one pair of nodes is non-adjacent. Examples illustrate the algorithms.

2. NOTATION & DEFINITIONS 1, 2, 3, ..., N node numbers where 1 is the source node

and N is the terminal node; also row and columns numbers.

A, B, C, ... . links between two nodes (symbols of contributive or distributive links).

Poor (Dummy) node: A node into which only one link enters and only one link goes out of it (figure 1, node 2).

Fig. 1. Poor Node

Contributive (distributive) link: A link which enters into (or goes out of) a node.

Order of combination: The number of different columns in a combination. When only one column is considered, the order is 1. When a combination of two columns, say 3 & 4, is taken the order is 2; etc.

Cutset: Set of directed links of a graph which contains at least one link of every path from the source to the sink node.

Minimal Cutset: A cutset is minimal if it does not contain a subset which itself is a cutset of the graph.

( ) columns (or nodes or rows) combination { } set of column combinations

3. ALGORITHM SHAl For a GRAPH WITH ADJACENT NODES

Consider the following graph

c: .- 1 '. +---

Fig. 2. A Graph with Adjacent Nodes

0018-9529/88/1200-0484$01 .000 1988 IEEE

AHMAD: SIMPLE ENUMERATION OF MINIMAL CUTSET OF ACYCLIC DIRECTED GRAPH 485

Steps

1. Set up an adjacency matrix of figure 2 with symbols of links instead of 1

TABLE 1 Adjacency Matrix of Figure 2

1 2 3 4 5 6

1 A B C D E 2 F G H I 3 J K L 4 M P 5 - Q 6 -

2. Collect all the symbols in row 1. Here we have ABCDE. It is a cutset because it isolates the source node.

3. Collect all the symbols in the last column N. It is a cutset as it isolates the sink node. Here it is EILPQ.

4. Form a set of all column combinations of order 1 to N - 3 only; using only column numbers 2 to N - 1 , as a set of all column combinations of order N - 2 yields the same cutset as derived in step 3. Here orders of combinations are 1, 2, 3, because N = 6. These combinations are now formed using 2, 3, 4, 5 . The set is {(2),(3),(4),(5), (2 , 3) , (2,4) 9 (2 , 5) , (3,4) 9 (3 9 5) , (4 9 5) ,(2,3 $4) 9 (2,3,5),(2,4,5) , (3,431. This set is formed to generate all possible cuts alternate to the cutsets derived in steps 2 and 3.

5. Pick any one combination from the set formed in step 4. Now consider, along with row 1, all the rows of the numbers given in the combination. Collect all the symbols in these rows, including the first one, except the ones which are in the columns given in the combination. This collection of symbols forms a cutset. Exclusion of symbols in the columns of the combination is to reduce the cutset to a minimal one. Step 5 is illustrated by picking a combination of each order from the set in step 4.

Combination of Order 1:

Pick (2). Consider now the first row and the second row. Collect all the symbols in the first and the second row except the one in the second column to get BCDEFGHI. It is a cutset.

Combination of order 2:

Pick (33). Now consider rows 1, 3, 5 . Collect all the symbols in these rows except the ones in columns 33, to get ACEJLQ. It is a cutset.

Combination of order 3:

Pick (2,4,5). Here rows 1,2,4, 5 are considered. The symbols in these rows, excluding the ones in columns 2, 4, 5 form a cutset. BEFIPQ is thus a cutset.

Repeat step 5 till all the combinations in the set of step 4 have been exhausted. The cutsets thus formed in steps 2,3, 5 are all the possible minimal cutsets of a graph in which all nodes are adjacent. For figure 2, the 16 cutsets are:

ABCDE ABCEQ ACDEJKL ACEJLQ ADEKLMP AELPQ BCDEFGHI BCEFGIQ BEFIPQ BDEFHIMP CDEGHIJKL CEGIJLQ DEHIKLMP EILPQ

ABEPQ ABDEMP

For a general acyclic directed graph which contains one or more non-adjacent nodes -

a. One can remove the symbols of non-existing links from the cutsets obtained by SHAl algorithm;

b. Compare the new cutsets obtained in #a with each other;

c. Delete those cutsets which contain all the symbols of some other cutset.

The remaining cutsets are then the minimal cutsets of the general graph. For example, suppose node 1 is not directly connected to nodes 4 & 5. This means links C & D do not ex- ist. Thus remove C & D from all the cutsets obtained above. Compare all new cutsets. All the symbols of cutset ABE are in ABEQ, ABEPQ, ABEMP, which are then deleted.

However, algorithm SHAl is indirect, time consum- ing, and inefficient. The next section presents a direct algorithm for a general acyclic directed graph.

4. GENERAL GRAPH ALGORITHM, GG

Consider the graph in figure 3.

Fig. 3. General graph

Steps

1. Set up a table representing the network in figure 3, in the same way as in step 1 of SHA1, except that the symbols of distributive links of poor nodes are replaced by those of contributive links. For figure 3, A&R are inserted in place of G&W in table 2. We thus have:

TABLE 2 General Graph Adjacency Matrix

1 2 3 4 5 6 1

I O A B O O E O 2 0 0 A 0 0 0 0 3 0 0 0 L O P Q 4 0 0 0 0 R T O 5 0 0 0 0 0 0 R 6 0 0 0 0 0 0 X 7 0 0 0 0 0 0 0

486 IEEE TRANSACTIONS ON RELIABILITY, VOL. 37, NO. 5,1988 DECEMBER

2. (Same as step 2 of SHA1). Thus ABE is a cutset. 3. (Same as step 3 of SHA1). Thus QRX is a cutset.

4.1 Enumeration: (Same as step 4 of SHAl)

The orders of combination are 1, 2, 3, 4 because N = 7. The set S is:

4.2 Deletion

Delete those combinations of set S, which give rise to redundant or non-minimal cutsets according to following rules.

a. If the combination consists of those columns only which have zeros in the first row; because if not deleted this will generate a cutset which contains the cutset generated in step 2; here 4,5, (43) are deleted.

b. In collecting symbols to form a cutset from a combination one excludes the symbols in columns of the combination. However if a combination contains columns having any one symbol of contributive link of a poor node only once then the collection of all other symbols from the rows of combinations will include, from some other column, the symbol of the contributive link of the poor node which should have been excluded. This would generate a non-minimal or redundant cutset. Thus such combinations should be deleted from S. (2),(3),(5),(2,4),(2,5),(2,6),(3,4), (3,5),(3,6),(4,5),(5,6),(2,3,5),(2,4,5),(2,4,6),(2,5,6),(3,4,5),

are deleted. c. Now consider a combination of order higher than

1, and having the last column with non-zero symbol in the first row only, viz, (N - 1). This, in step 5, generates a cutset different from the one generated in step 1. Here the non-zero symbol in the first row in column N - 1 is replaced by the non-zero symbol in row (N - 1) in the last column. However, if a combination which contains column N - 1 alongwith those columns which have zeros in the first row is considered one does not get the same cutset as generated by (N - 1) alone because symbols from the rows of other columns of the combination are also added. One thus gets a cutset which is not minimal. Thus combinations having only columns which have zeros in the first row alongwith the last column, N - 1, should be deleted from S, here (4,6),(5,6),(4,5,6) are deleted;

d. If the combination consists of all those rows which have non-zero symbols in the last column because this generates a cutset which contains the cutset already generated in step 3; here (3,5,6),(2,3,5,6),(3,4,5,6) are deleted.

Apply all four rules (#a-#d) on the set S obtained in step 4.1; a reduced set S is obtained:

(3 94, 6) (3 5,6), (4, 5 $6) (293 94, 5) (2,495 6)s (2 3 5,6) (3 9 4 9 5 36)

5. Step 5 of SHAl applies to the reduced set S obtained in step 4.1. Repeat this step till all the combinations are taken care of.

6. Find new cutsets by replacing symbols of contributive links of poor nodes by the symbols of distributive ones, if any, in the cutsets obtained in steps 2, 3, 5. End

5. EXAMPLE 1

Example 1 is continued from the previous sections. All references are to algorithm GG.

From the reduced set obtained in step 4.1, consider the combination (6). Applying step 5 gives ABX. Similarly, the remaining combinations give ELPQ, EPQRT, LQX.

By step 6, replace A and R by G and W respectively in the cutsets obtained in steps 2, 3, 5 to get new cutsets.

In all there are 10 cutsets:

ABE, QRX, ABX, ELPQ, EPQRT, LQX, GBE, QWX, GBX, EPQWT.

6. EXAMPLE2

Consider the following adjacency matrix of an acyclic directed network, where F is symbol of contributive link of a poor node 3, and J is symbol of its distributive link. The steps refer to algorithm GG

1 2 3 4 5 6

1 0 A 0 0 D E 0 2 0 0 F

3 0 0 0 F 0 0 4 0 0 0 O M P 5 0 0 0 0 O Q 6 0 0 0 0 0 0

G H

By-step 2, ADE is a cutset. Also by step 3, EPQ is a cutset. In step 4.1 the set of combinations of columns is already obtained. Apply the rules in step 4.2 to obtain a reduced set.

By rule #a: Combinations (3), (4), (3,4) are deleted.

By rule #b: Combinations (3), (4), (2,3), (2,4), ( 3 3 , ( 4 3 , (2,3,5), (2,4,5) are deleted.

By rule #c: Combination (3,4,5) is deleted. By rule #d: Combinations ( 4 3 , (2,4,5), (3,4,5) are deleted.

The reduced set is then {(2),(5),(2,5),(2,3,4)}. Apply step 5 to each of these combinations in the reduced set; the result is: DEFGH, AEQ, EFGQ, DEHMP as cutsets.

By step 6, new cutsets are DEJGH, EJGQ. The total minimal cutsets are 8. One can easily check it by removing the symbols B, C, I, K, L of non-existing links from the cutsets obtained in section 3, and then by deleting those cutsets which contain other cutsets (ie, by deleting non- minimal cutsets).

AHMAD: SIMPLE ENUMERATION OF MINIMAL CUTSET OF ACYCLIC DIRECTED GRAPH 487

7. CONCLUSIONS [31

Many algorithms have been designed to enumerate cutsets of network having a source and a terminal node. These algorithms generally involve mathematics. For example, one has to know matrix theory to understand Biegel’s [4] work; and to understand and use Lock’s [7] method one should know advanced mathematics. To understand and use algorithm GG one has simply to enumerate all combinations and then delete unwanted combinations using certain rules. In fact for a network having a few nodes one can only enumerate those combinations of nodes which can not be deleted. These algorithms are simple to use in class room lectures for il- lustration purposes as well.

Jumah [6] has programmed algorithm SHAl on a computer.

In a separate paper algorithms have been shown to work on undirected acyclic graphs as well.

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P I

M. Belmore, P. A. Jensen, “An implicit enumeration scheme for proper cut generation”, Technornetrics, vol 12, 1970 Nov, pp

J. E. Biegel, “Determination of tie sets and cutsets for a system without feedback”, ZEEE Trans. Reliability, vol R-26, 1971 Apr, pp 3942. G. B. Jasmon, 0. S. Kai, “A new technique in minimal path and cutset evaluation”, ZEEE Trans. Reliability, vol R-34, 1985 Jan, pp

I. M. Jumah, “Computer programs for determination of tie sets and cutsets of a system”, BSc Project, Dept. of Industrial Engg., King Abdul Aziz University, Jeddah. 1986 Jun. M. 0. Locks, “Inverting and Minimalising Path sets and Cutsets”, ZEEE Trans. Reliability, vol R-21, 1978 Jun, pp 107-109. G. D. M. Pearson, “Computer program for approximating the reliability characteristics of acyclic directed graphs”, ZEEE Trans. Reliability, vol R-26, 1977 Apr, pp 32-38. A. Satyanarayana, A. Prabhkar, “New topological formula and rapid algorithm for reliability analysis of complex networks”, ZEEE Trans. Reliability, vol R-21, 1978 Jun, pp 82-100.

175-188.

136-141.

AUTHOR S. Hasanuddin Ahmad; Department of Industrial Engg, King Abdul Aziz University; Jeddah 21413 SAUDI ARABIA.

S . Hasanuddin Ahmad was born in Allahabad; he got his BSc and MSc in Math from Karachi, Pakistan; he got MA, and PhD in Industrial Engineering from Arizona State University. He has taught in several universities in the USA, Pakistan, and Nigeria; he is an Associate Pro-

ACKNOWLEDGMENT

1 am pleased to acknowledge the Comments and sug- gestions of referees and the Editor which have helped to improve the paper.

fessor in King Abdul Aziz University. He has several publications in the field of reliability, operations research, and statistics in IEEE Trans. Reliability, Microelectronics & Reliability, Int. J. Systems Science, Riazi, etc.

REFERENCES

S. Hasanuddin Ahmad, “A simple technique for computing network reliability”, ZEEE Trans. Reliability, vol R-31, 1982 Apr, pp 4145. S. Arun Kumar, S. H. Lee, “Enumeration of all minimal Cutsets for a node pair in a graph”, ZEEE Trans. Reliability, vol R-28, 1979 Apr, pp 51-55. IEEE Log Number 21896 4 TR b

Manuscript TR87-040 received 1987 March 26; revised 1988 April 4.

Hints and Kinks

(continued from page 483)

Until recently, the model second in popularity was the normal (Gaussian) distribution. This distribution and its characteristics are well-known, and it has been a reasonable adopt more flexible models. The Weibull distribution in particular has been embraced by observers of time- dependent failure rates in semiconductor devices and mechanical assemblies. The appeal of the Weibull and other multi-parameter distributions lies in their ability to fit a wide variety of data patterns convincingly. Unfor- tunately, this ability not only provides models of high descriptive capacity; it also tends to mislead the analyst, who may forget to look for the necessary physical relation- ships between the mathematical model and the phenomena it is purported to represent. The unpleasant tendency of

such models to suffer unpredictable changes in all parameters when stresses, materials, or processes differ from the original sample makes them very dangerous for extrapolation of any kind.

If the reader’s confidence in models has been shaken, a major purpose of this column has been achieved. The point, however, is not that models are useless; they are essential. The point is that the choice of model requires careful thought, and that the more sophisticated the model the greater the need to examine the data and implications. The natural tendency in any engineering field is toward cookbook solutions; that is precisely what we cannot afford to use.

[This “Hints and Kinks” is reprinted from the 1964 October issue of the IEEE Reliability Group Newsletter. Paul was at that time a Principal Scientist with Booz-Allen Applied Research Inc. He is now a consultant .] 4 TR b

simple enumeration of minimal cutsets of acyclic directed graph

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