Enumeration of pathsets of reliability graphs by repeated indexing

Microelectron. Reliab., Vol. 33, No. 4, pp. 481--487,1993. Printed in Great Britain.

0026-2714/9356.00+ .00 © 1993PergamonPress Ltd

E N U M E R A T I O N OF PATHSETS OF RELIABILITY G R A P H S BY REPEATED I N D E X I N G M. A. AzIz and M. A. SOBHAN Department of Applied Physics and Electronics, University of Rajshahi, Rajshahi, Bangladesh and M. A. SAMAD Department of Electrical and Electronic Engineering B.I.T., Khulna, Bangladesh

(Received for pubfication 14 November 1991) Al~traet--Two algorithms have been proposed for the enumeration of pathsets of complex reliability graphs. In algorithm- 1, the indices of the connection matrix and higher order row vectors are repeatedly made and subsequently used in each step of matrix operation for the enumeration of pathsets of reliability graphs. In algorithm-2, the indices of the connection matrices are made and used subsequently, each time after every node removal step. In both algorithms, multiplications that are 100% useless are eliminated, thereby saving a substantial amount of CPU time. Both the algorithms have been elucidated by suitable example illustrations. To justify the effectiveness of these algorithms, they have been implemented in computer programs which enumerate pathsets much faster in comparison to existing algorithms.

1. NOTATIONS AND ABBREVIATIONS C cardinality [C] connection matrix % an element of [C] i i=l(1)N

ith element / = 1 to N in steps of l

(j) jth node k average number of subentries per column in higher order row vector R(i) ith order row vector R(i,j) reliability of a network between nodes (i) and

(J)

R(i)j jth column of ith order row vector R(i)yk kth subentry of jth column of ith order row vector

r(i) index of R(i) r(i)j jth column of ith index RLD ta tc tm u

v(i) v(i)j w IX] xij y

z

reliability logic diagram computation time required for an addition total computation time computation time required for a multiplication average number of non-zero entries in a matrix; average number of non-zero entries per column/row in [C] ith column vector of the index of [C] jth entry of ith column vector average number of subentries per matrix incidence matrix an element of [X] average number of non-zero entries in last row/column of [C] number of zeros in R(1) 2. I N T R O D U C T I O N

Reliability evaluation of communication networks and other systems through the enumeration of pathsets and cutsets are the most widely used approaches. Considerable works have been reported [1-14] to date, providing numerous methods based MR 33/4---C

481

on these concepts. The reliability evaluation procedure, for these approaches, comprises mainly of two steps. In the first step, the minimal pathset/cutset of the reliability graph is enumerated by any suitable algorithm: in the second step, the system reliability is calculated by choosing a conventional method. Danielson [1] and Fratta and Montanari [2] have provided techniques for path and cut enumerations by using the method of symbolic solution for simultaneous equations. Krishnamurthy and Komissar [3] described a direct procedure for reducing the complexity of the network by decomposing it into smaller subnetworks which only have one terminal pair; Kim et al. [4] and Aggarwal et al. [5] used an operator which reduced the number of terms in the reliability polynomial. Aggarwal et al. [5] have presented a state removal algorithm to find all possible paths. The algorithm does not require matrix multiplication and the size of the matrix reduces in every step. A state removal algorithm is also used by Rai and Aggarwal [6] to determine all simple paths of a graph. The method does not require matrix multiplications, however it does require rearrangement of the paths in ascending order of cardinality. A three-step procedure is presented by Beigel [7] for determining the tiesets (paths) of a graph by using the connection matrix. Heidtmann [8] derived the paths of two state systems from their cuts and vice versa. References [3, 4, 9-11] utilized (N - 1) powers of the connection matrix for enumerating all simple paths of the reliability graph. The main disadvantage of this method lies in the difficulty of repeated whole matrix multiplication and of increased complexity with increasing size of the reliability graph.

482

M.A. AzIz et al.

Samad [12] proposed a different technique for enumeration of multiterminal paths and terminal paths in which the paths are enumerated with the help of a connection/adjacency matrix using a theoretic graph approach. Paths are generated in ascending order of cardinality. Both the suggested methods are applicable to oriented, non-oriented or mixed types of reliability graphs. Samad [13] also proposed an algorithm to enumerate simultaneously all the minimal paths between any single terminal pair, minimal paths for multiterminal pairs as well as cuts for the specified terminal pairs of any communication network. The proposed method is simple and computationally efficient for large networks when the reliability for many terminal pairs is to be evaluated critically for both success as well as failure frame of references. Aziz et al. [14], presented an algorithm which enumerates pathsets of reliability graphs using the method of indexing. In this algorithm, an index of the connection matrix [C] is prepared at the outset which gives the locations of all the non-zero entries of the connection matrix. During the process of enumeration of pathsets using this index, only the non-zero entries of [C] are picked up and useful multiplications are performed, thereby eliminating the useless multiplications which are present in [12, 13]. It is shown in [14] by a comparative study that the number of multiplications is reduced significantly. Due to the fact that the connection matrix of a digraph contains many more zero entries than nonzero entries, the higher order row vectors generated in [12-14] also contain a significant number of zeros. Thus as a fact, though a greater part of the multiplications by zeros are avoided by the indexing of [C] in [14], a few multiplications with zeros still remain due to the zero entries in the higher order row vectors. In this paper, two algorithms have been proposed for enumeration of pathsets of complex reliability graphs. Algorithm-1 is developed by modification of the algorithm presented in [14]. In this algorithm the multiplications with zeros are eliminated 100% by repeatedly making and using indices of the higher order row vectors. The method of repeated indexing is also used in the proposed algorithm-2. This algorithm is also a modified and extended form of an existing algorithm previously proposed [5]. The method makes use of a connection matrix and is based on the state removal algorithm. To use the method conveniently, the reliability graph is reduced in such a way that there only exists one element between a pair of nodes and the graph is free from self loops. The input node is labelled as (1), the output node is labelled as (2) and the other nodes are labelled arbitrarily. The main feature of this method is that it removes the last row and last column of [C] with new entries as c0(new) = c0(old) + cjk " Ckj, for i , j = 1,2 . . . . . (k - 1),

(1)

where the kth row and column are the last of the rows/columns in [C]. The formation of a new connection matrix involves the addition and multiplication of entries of the previous one and since the connection matrix contains a large number of zeros (especially for the digraph), so computational time and efforts become useless while multiplying with the zero entries. These useless multiplications and additions are avoided in the present algorithm by making an index of [C] each time, before formation of a new matrix. While forming the new c0, only the non-zero entries are picked up from the old [C] and thus, the multiplications with zeros are avoided. The operation is repeated (N - 2) times to obtain a matrix of 2 × 2 which has entries corresponding to input and output nodes only. The effectivenesses of the proposed algorithms are demonstrated by examples. Algorithm-1 is given below.

3. ALGORITHM-I Step 1. Reduce the R L D in such a manner that there exists only one element between a pair of nodes. S t e p 2. Label the elements and the nodes in an arbitrary manner. S t e p 3. (a) Read and store the entries cifi i = I(I)N, j = I(1)N. (b) Read and store source node (s). S t e p 4. For making the index of [C], construct new N column vectors v(k), (k = I(1)N), corresponding to each column of [C] by the following procedure: (a) scan the first column of connection matrix [C] from row 1 through N, identify row number where the first non-zero entry exists. If, for example, the first nonzero entry is found at ck~, put v(l)~ = k, if the second non-zero entry is found at Cpl, put v(l)2 = p , and so on until all the locations of all non-zero entries of column 1 are recorded in v(1). (b) Apply step 4(a) to columns 2 through N of [C] to put the locations of all non-zero entries in the respective column vectors. In this step the preparation of the index of [C] will be completed. (c) Locate the first order row vector R(1) from [C] corresponding to the source node (s). Scan R(1) from column 1 through N and make an index r ( l ) of R(1) by the location of the non-zero entries. If, for example, the first non-zero entry is found in Csl, put r(1)l = 1, if the second non-zero entry is found at cs4, put r(l) 2 = 4 and so on until all the locations of all non-zero entries of R(i) are recorded in r(i). S t e p 5. For calculating the second order row vector R(2)j, use the following procedure: (a) put the entry R(2) 1 = 0; (b) for the entry/entries R(2)2, read r(1)~, let it be equal to p. Scan v(2), let p be found at v(2)i. Then put R(2)21 = %2 x Csp. Next read r(I)2, let it be equal to q; scan v(2) and let q be found at v(2)j. Then put R(2)22 = Cq2 × Csq. Repeat the process until all the entries of r ( l ) are exhausted; (c) for the entries R(2)3, follow the same procedure as 5(b) but scan the

Enumeration of pathsets index v(3) instead of v(2). Repeat the procedure until all the entries up to R(2)N are found. Step 6. F o r calculating the entries o f R(3), the following procedure is followed. (a) Following the same procedure of step 4(c), m a k e an index r(2) o f R(2) which will record the location of all non-zero entries of R(2). (b) Read r(2)~, let it be p a n d scan v(1), if p is found in v(1) then put R(3)H = Cpl x R(2)p I. If R(2)p contains other subentries, p u t R(3)12 = Cp~ × R(2)p 2, a n d so on. (c) Read R(2)2 to all the non-zero entries and follow the same procedure as in 6(b). W h e n all the non-zero entries are exhausted, the m a k i n g of the third order row vector R(3) will be complete. Step 7. The m a k i n g of other higher order row vectors up to R ( N -- 1) is similar to those described in steps 6(a-c). In every step, a n index r(i) is to be m a d e from R(i) to locate the non-zero entries o f R(i), so that only multiplications with non-zero entries can be performed. Step 8. After the construction of all the higher order row vectors, the p a t h s p(s, i), i = I(I)N, and i ~ s o f cardinality 1 to ( N - 1 ) will be found by scanning c o l u m n i of row vectors o f order 1 t h r o u g h (N -- 1). Step 9. Stop.

483

Since the source node is (I), the first order row vector is given by R ( I ) = [ 0 1 2 0 0 0]. Since R(I)2 a n d R(I)3 are the only non-zero entries, then the index r ( l ) is given by r(1) = [2 3]. The second order row vector R(2) is calculated as follows: since v ( l ) is empty, put R ( 2 ) I = 0 . As r(I)l = 2 a n d there is no entry equal to 2 in v(I), it is therefore skipped. Next r(l)2 = 3, so R(2)2 = cl3 x c32 = 23. Similarly,

R(2)3 = cl2 x c23 = 13 R(2)4 = cl2 x c24 = 14 R(2)5 = q3 x c35 = 25 R(2) 6 = 0.

Hence the second order row vector R(2)=[0 23 13 14 25 0 ] . N o w , dex r(2) of R(2) will be r ( 2 ) = [ 2 3 v(1) is empty, again R(3)l = 0. Similarly the o t h e r entries of R(3)

is given by again the in4 5]. Since are given by

R ( 3 ) 2 = R(2) x c32= 23 x 3 = 0 R ( 3 ) 3 = R(2) x c23= 23 x 3 = 0 R(3)41 = R(2)2 x c24 = 23 × 4 = 234 R(3)42 = R(2)5 x c54 = 25 × 6 = 256

3.1. Example 1

R(3)51 = R(2)3 x c35 = 13 × 5 = 135

E n u m e r a t e all the minimal pathsets for the multiterminal pairs from node (1) to node (6) o f the A R P A c o m m u n i c a t i o n network s h o w n in Fig. 1. Solution. The connection matrix o f the digraph is s h o w n below.

R(3)s: = R(2)4 x c45 = 14 x 6 = 146

0 0 3 4 0 0 0 3 0 0 5 0 0 0 0 0 6 7 0 0 0 6 0 8

R(4) I = 0

0 0 0 0 0 0

R(4)2 = 0

The six c o l u m n vectors of [C] are shown below. Since the first column o f [C] contains no non-zero entry, v(1) is void. v(l)

R(3)62 = R(2)5 x c56 = 25 x 8 = 258. Hence, the third order row vector R(3) is given by R(3)=[0 0 0 234, 256 135, 146 147, 258]. Again the index r(3) of R(3) is given by r(3) = [4 5 6]. The entries of the fourth order row vector R(4) are calculated as

- 0 1 2 0 0 0 -

[C] =

R(3)61 = R(2)4 x c46 = 14 x 7 = 147

v(2)

v(3)

v(4)

v(5)

v(6)

1

1

2

3

4

3

2

5

4

5

R (4)3 = 0 R(4)41 = R(3)51 x c~ = 135 x 6 = 1356 R(4)42 = R(3)s2 x c54 = 146 x 6 = 0 R(4)sl = R(3)41 x c45 = 234 x 6 = 2346 R(4)52 = R(3)a 2 x c45 = 256 x 6 = 0

®

,

R(4)61 = R(3)41 x c46 = 234 x 7 = 2347

®

R(4)62 = R(3)42 x c46 = 256 x 7 = 2567 1

(~)

7

3

6

R(4)63 = R(3)s I x c56 = 135 x 8 = 1358 R(4)64 = R(3)52 x c56 = 146 x 8 = 1468.

®

;

®

Fig. 1. ARPA communication network of Example 1.

Thus the entries o f the fourth order row vector R(4) are given by R(4)=[0

0 0

1356 2346 2347,2567, 1358, 1468].

M . A . AzIz et al.

484 Again the index r(4) of R(4) is given by r(4)=[4

5

Saving in computation time = 86.6%.

6].

The entries of the fifth order row vector are calculated as

Thus, it is seen that more than 85% of the computation time is saved by using the proposed method.

R(5), = 0

R(5)2 =

4. ALGORITHM-2

0

R(5) 3 = 0 R(5)4 = R(4)5 x e54 = 2346 x 6 = 0 R(5)5 = R(4)4 x c45 = 1356 x 6 = 0 R(5)61 = R(4)4 × c46 = 1356 x 7 = 13567 R(5)62 = R(4)5 x c56 = 2346 x 8 = 23468. Hence, the fifth order row vector is given by R(5)=[0

(6)

0

0

0

0

13567,23468].

The multiterminal pathset of the digraph, from node (1) to node (j), j = 2 to 6, for cardinality 1 to 5 is found by scanning the columns of the above five row vectors. e(1, 2) = 1, 23 P(1, 3) = 2, 13 P(1, 4) = 14, 234, 256, 1356 e(1, 5) = 25, 135, 146, 2346 P ( I , 6 ) = 147,258, 2347, 2567, 1358, 1468, 13567, 23468. 3.2. Comparative study with existing method Let u, z, k, N, to, tm and ta (t~ = 10 tQ) have the same meanings as in Section 1. By the existing method, the total number of multiplications= N2(k2 + i).

(2) Total number of additions for existing method

=N(N- 1)[k(N-2)].

(3)

Step 1. Reduce the graph in such a manner that it does not contain any self loop or any parallel elements. Step 2. Label the source node as (!) and the sink node as in (2). Label all elements and other nodes arbitrarily. Step 3. Read and store the entries c~/= I(I)N, j = I(1)N. Step 4. For making the index of [C], construct new N column vectors v(k), (k = I(I)N), corresponding to each column of [C] by the following procedure: (a) scan the first column of connection matrix [C] from row 1 through N, identify row number where the first non-zero entry exists. If, for example, the first nonzero entry is found at c~1, put v(1)~ = k, if the second non-zero entry is found at Cp], put v(I): = p , and so on until all the locations of all non-zero entries of column 1 are recorded in v(1). (b) Apply step 4(a) to columns 2 through N of [C] to put the locations of all non-zero entries in the respective column vectors. In this step the preparation of the index of [C] will be completed. Step 5. Initialize the new matrix with zero entries. Put the non-zero entries in the new matrix following the procedure below: (i) set i = l ; j = 1; (ii) read ith entry o f j t h column vector. Let the ith entry = x, if x = 0 or k go to (iv), otherwise, put c~ ( n e w ) = c~, (old); (iii) set i = i + 1, go to (ii); (iv) set j = j + 1, i = 1, i f j = k go to step 6, otherwise go back to (ii). Step 6. (i) Let the ith e n t r y = x , scan all the ( k - 1)th column vectors, if k is found in the nth column vector, then cx, = existing entry + C~k" ck, ; (ii) set i = i + 1, if all the entries of the kth column are exhausted go to step 7, otherwise go back to (i). Step 7. Stop.

In Example 1, for N = 6, k = 3: 4.1. Example 2

total number of multiplications

The reliability graph of the A R P A communication network shown in Fig. 1 is redrawn in Fig. 2 with a little modification in the node numbering as required for the algorithm. The pathsets are enumerated by

=N2(k + 1) = 36 x 4 = 144 total number of additions = 6 x 3 x 3 x 4 = 360. By the proposed method: total number of multiplications = 24

®

4

®

total number of additions = 0 total savings in multiplications = 1 4 4 - 24 = 120

®

total computation time in existing method =t,. = 144 t,, + 360 ta = 1800 t~

(4) ®

total computation time in the proposed method

=t,. = 24 t,, + 0 = 240 ta.

(5)

5

®

Fig. 2. ARPA communication network of Example 2.

Enumeration of pathsets proposed algorithm-2. The connection matrix of the reliability graph is given below:

Cl2(new) = ct2(old) + c~5" cs2 = 0 + 258

0

0 0 0 0

cu(new ) = Cl3(Old) +cl5"c53 = 1 + 2 3

o

0 4 3 0

0

7

0 0 0 6

0

0

3 0 0 5

c32(new) = c32(old) + c35' c52 = 0 + 358

0

8

0 6 0 0

c33(new) = c33(old) + c35" c53 = 0 + 3.3 = 0

0

0 0

cl4 (new) = ct4 (old) + cls" c~ = 0 + 256

The six column vectors are shown below. Since the first column of [C] contains no non-zero entries, v(1) is void. v(l)

The following products are added to the existing entries:

1020-

-0 [Cl=

485

v(2)

v(3)

v(4)

v(5)

v(6)

4

1

3

1

4

6

5

6

3

5.

Initialize the new 5 x 5 matrix with zero entries. Then put the non-zero entries from the index as c42(new) = c42(old) = 7;

cu(new) = cl3(old) = 1

c53(new) = c53(old) = 3;

c34(new) = c34(old ) = 4

c15(new) = cls(old) = 2;

c35(new) = c35(old) = 3.

c34(new) = c34(old) + c35" c~ = 4 + 356. The new 4 x 4 matrix is given below:

[C] =

358

0

7468

0

4 4356

The new index of [C] is given by v(l)

v(2)

v(3)

v(4)

1

1

1

3 The following products are added to the existing entries:

The non-zero entries of 3 x 3 matrix are as follows (calculated as before): cl2 = 258; c32 = 358; Cl3 = 1 + 23.

c~ (new) = c44(old) + C 4 6 " C ~ = 0 + 6 . 6 = 0

The following products are added:

c52(new) = Csz(old) + C56"C62= 0 + 58

C12 = C12 4 ¢14" C42 =

[Ol0l

c54(new) = c~ (old) + C56' C64= 0 + 56. Hence the new matrix is given by

0

0

0

0

0

4

7+68

0

0

58

3

56

.

258 + 2567

c32 = c32 + c~" c42 = 358 + 47 + 468 + 3567. The new 3 x 3 matrix is therefore

[Cl =

0

258 + 2567

0

0

0

4 7 + 358 + 4 6 8 + 3567

v(I)

v(3) 1.

v(3)

v(4)

v(5)

1

4

1

3

1

3

5

5

5

3.

By the same procedure the non-zero entries of new 4 x 4 matrix are calculated as follows: c42(new) = c42(old) = 7 + 68

The non-zero entry of the 2 × 2 matrix is c~2 = 258 + 2567. The following products will be added ct2 = cl2 + cl3 • c3z = 258 + 2567 + 147

c13(new) = cl3(old) = 1

rc,_i

.

v(2)

v(2)

c34(new) = c34(01d) = 4.

1 !23 ]

The new index of the connection matrix is

The new index of [C] is given below: v(l)

3.

4

c42(new) = c42(old) 4 C46"C62~- 7 + 68

[C] =

"

+ 1358 + 1468 + 13567 + 2347 + 23468. The new 2 x 2 martix is given by

258+,47+2567+ 1358+1468+13567+2347+234681 0

486

M . A . AzIz et al.

Hence, the paths from node (1) to node (2) are given by P ( I , 2) = 147, 258, 1358, 2567, 1468, 2347, 13567, 23468. 4.2. C o m p a r a t i v e s t u d y with existing m e t h o d Since in the proposed method, the matrix reduces in size in every operation, the following study has been done on average assumption. Let N, tc, t,,, t, (tm = 10 ta) have the same meaning as mentioned in Section 1, and w = average number of subentries per matrix, u = average number of non-zero entries per matrix, y = average number of non-zero entries in last row/column of each matrix. In the existing method, total multiplications - 1)2 + (N - 2) 2 + - . . + (3) 2}

=w{(N N

l

=w ~ x ~

(7)

~= 3 N-[

total additions = w ~ x 2.

(8)

x=3

In the proposed method, total multiplications = y w ( N - 2)

(9)

total additions = u ( N - 2).

(10)

For Example 2, in the existing method, w=1.18,

N=6

The proposed algorithm-1 is a modified form of the algorithm proposed in [14]. In [14], though more than 95% of the useless multiplications have been avoided, a very small number of them still remain because of the presence of zero entries in higher order row vectors. These useless multiplications will be a little more numerous in digraphs, because the number of non-zero entries in a digraph is more than that of a non-directed graph. In the proposed algorithm-l, useless multiplications have been eliminated 100% by making the indices of the higher order row vectors in every step of operation. This is also applicable to the enumeration of cutsets and simultaneous enumeration of pathsets and cutsets of reliability graphs. This algorithm has also been implemented in a computer program to enumerate pathsets quite effectively. The proposed algorithm-2 is a modification of the method proposed by Aggarwal et al. [5]. This algorithm also used a connection matrix and enumerates pathsets from the source node to sink node by a node removal process. The proposed algorithm eliminates the unnecessary multiplications 100% by making the indices of matrices after each step of matrix operation. It is seen in the comparative study of Section 4.2 that the computation time is 80% less in the proposed method than that in the existing method. This algorithm has also been tested for a number of digraphs for small as well as large sizes and found quite superior to the existing one. This algorithm is also applicable to the enumeration of cutsets of reliability graphs.

total multiplications = 59 REFERENCES

total additions = 50. In the proposed method, y=!.75,

u=6.25,

w=1.18,

N=6

total multiplications = 9 total additions = 25. Thus, the total computation time in the existing method is given by t, = 59 t,, + 50 t~ = 640 t~,

(11)

whereas the total computation time in the proposed method is t,=9tm+25t,=

115 t,,.

(12)

Thus, the computation time in the proposed method is more than 80% less in comparison to the existing method. 5. RESULTS AND DISCUSSION Two algorithms have been proposed for the enumeration of pathsets of reliability graphs.

1. G. H. Danielson, On finding the simple paths and circuits in a graph, IEEE Trans. Circuit Theor. CT-I$, 294-295 (1968). 2. L. Fratta and U. Montanari, All simple paths in a graph by solving a system of linear equations, Nota Interna No. B 71-11 of Consigloi Nazionale Dells Ricerche, lnstitut di Elaborazions della Intermazione, Pisa, Oct. 1971. 3. E. V. Krishnamurthy and G. Komissar, Computeraided reliability analysis of complicated networks, IEEE Trans. Reliab. R-21, 86-89 (1972). 4. Y. H. Kim, K. E. Case and P. M. Ghare, A method for computing complex system reliability, IEEE Trans'. Reliab. R-21, 215 219 (1972). 5. K. K. Aggarwal, J. S. Gupta and K. B. Misra, A new method for system reliability evaluation, Microelectron. Reliab. 12(5), 435-440 (1973). 6. S. Rai and K. K. Aggarwal, An efficient method for reliability evaluation of a general network, IEEE Trans. Reliab. R-27, 206-211 (1978). 7. J. E. Beigel, Determination of tie-sets and cut-sets for a system without feedback, IEEE Trans. Reliab. R-26, 39-42 (1977). 8. K. D. Heidtmann, Inverting paths and cuts of 2-state systems, IEEE Trans. Reliab. R-32, 469-471 (1983). 9. K. B. Misra, An algorithm for the reliability evaluation of redundant networks, IEEE Trans. Reliab. R-19, 146-151 (1970).

Enumeration of pathsets 10. R. B. Misra, Symbolic reliability evaluation of reducible networks, Microelectron. Reliab. 19, 253-257 (1979). 11. R. B. Misra and K. B. Misra, An algorithm for reliability evaluation of complex systems, J, Inst. Engrs (India) 60, pt. IDGE, 77-80 (1980). 12. M. A. Samad, An efficient method for terminal and multiterminal pathset enumeration, Microelectron. Reliab. 27(3), 443~,46 (1987).

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13. M. A. Samad, An efficient algorithm for simultaneously deducing minimal paths as well as cuts of a communication network, Microelectron. Reliab. 27(3), 437-441 (1987). 14. M. A. Aziz, M. A. Sobhan and M. A. Samad, Reduction of computations in enumeration of terminal and multiterminal pathset by the method of indexing, Microelectron. Reliab. 32, 1067-1072 (1992).