Abstract

The matrix descriptions for three types of free-space regular optical interconnection networks, the perfect shuffle, the banyan, and the crossover, are given. Some new properties of the perfect shuffle are presented in matrix equations, and the equivalence of the three types of interconnection networks is simply shown.

© 1994 Optical Society of America

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References

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  1. A. Huang, “Parallel algorithms for optical digital computer,” in Proceedings of IEEE Tenth International Optical Computing Conference, IEEE catalog no. 83CH1880-4 (Institute of Electrical and Electronics Engineers, New York, 1983), p. 13.
  2. M. E. Prise, N. Streibl, M. M. Downs, “Optical considerations in the design of digital optical computers,” Opt. Quantum. Electron. 20, 49–77 (1988).
    [CrossRef]
  3. K.-H. Brenner, A. Huang, “Optical implementation of the perfect shuffle interconnection,” Appl. Opt. 27, 135–137 (1988).
    [CrossRef] [PubMed]
  4. A. W. Lohmann, W. Stork, G. Stuck, “Optical perfect shuffle,” Appl. Opt. 25, 1530–1531 (1986).
    [CrossRef] [PubMed]
  5. J. Jahns, “Optical implementation of the banyan networks,” Opt. Commun. 76, 321–324 (1990).
    [CrossRef]
  6. J. Jahns, M. J. Murdocca, “Crossover networks and their optical implementation,” Appl. Opt. 27, 3155–3160 (1988).
    [CrossRef] [PubMed]
  7. M. Cao, H. Li, “Optical hardware for perfect shuffle interconnection,” Opt. Computer Process. 1, 23–27 (1991).
  8. M. Cao, F. Luo, H. Li, “Optical perfect shuffle–exchange interconnection network using liquid crystal spatial light switch,” Appl. Opt. 31, 6817–6819 (1992).
    [CrossRef] [PubMed]
  9. M. Cao, Hongpu Li, F. Luo, “Optical implementation of perfect shuffle/exchange omega interconnection network,” Acta Opt. Sin. 12, 1130–1134 (1992).
  10. A. A. Sawchuk, I. Glaser, “Geometrics for optical implementations of the perfect shuffle,” in Optical Computing ’88, P. Chavel, J. W. Goodman, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 270–282 (1988).
  11. M. J. Murdocca, A. Huang, J. Jahns, N. Streibl, “Optical design of programmable logic arrays,” Appl. Opt. 27, 1651–1660 (1988).
    [CrossRef] [PubMed]
  12. G. Eichmann, Y. Li, “Compact optical generalized perfect shuffle,” Appl. Opt. 26, 1167–1168 (1987).
    [CrossRef]
  13. G. E. Lohman, A. W. Lohmann, “Shuffle communication component for optical parallel processing,” J. Opt. Soc. Am. A 4, 106–108(1987).
  14. J. W. Goodman, F. I. Leonberger, S.-Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–853 (1984).
    [CrossRef]
  15. C. L. Wu, T. Y Feng, “The universality of the shuffle–exchange network,” IEEE Trans. Comput. C-30, 324–332 (1981).

1992 (2)

M. Cao, F. Luo, H. Li, “Optical perfect shuffle–exchange interconnection network using liquid crystal spatial light switch,” Appl. Opt. 31, 6817–6819 (1992).
[CrossRef] [PubMed]

M. Cao, Hongpu Li, F. Luo, “Optical implementation of perfect shuffle/exchange omega interconnection network,” Acta Opt. Sin. 12, 1130–1134 (1992).

1991 (1)

M. Cao, H. Li, “Optical hardware for perfect shuffle interconnection,” Opt. Computer Process. 1, 23–27 (1991).

1990 (1)

J. Jahns, “Optical implementation of the banyan networks,” Opt. Commun. 76, 321–324 (1990).
[CrossRef]

1988 (4)

1987 (2)

G. Eichmann, Y. Li, “Compact optical generalized perfect shuffle,” Appl. Opt. 26, 1167–1168 (1987).
[CrossRef]

G. E. Lohman, A. W. Lohmann, “Shuffle communication component for optical parallel processing,” J. Opt. Soc. Am. A 4, 106–108(1987).

1986 (1)

1984 (1)

J. W. Goodman, F. I. Leonberger, S.-Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–853 (1984).
[CrossRef]

1981 (1)

C. L. Wu, T. Y Feng, “The universality of the shuffle–exchange network,” IEEE Trans. Comput. C-30, 324–332 (1981).

Athale, R. A.

J. W. Goodman, F. I. Leonberger, S.-Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–853 (1984).
[CrossRef]

Brenner, K.-H.

Cao, M.

M. Cao, F. Luo, H. Li, “Optical perfect shuffle–exchange interconnection network using liquid crystal spatial light switch,” Appl. Opt. 31, 6817–6819 (1992).
[CrossRef] [PubMed]

M. Cao, Hongpu Li, F. Luo, “Optical implementation of perfect shuffle/exchange omega interconnection network,” Acta Opt. Sin. 12, 1130–1134 (1992).

M. Cao, H. Li, “Optical hardware for perfect shuffle interconnection,” Opt. Computer Process. 1, 23–27 (1991).

Downs, M. M.

M. E. Prise, N. Streibl, M. M. Downs, “Optical considerations in the design of digital optical computers,” Opt. Quantum. Electron. 20, 49–77 (1988).
[CrossRef]

Eichmann, G.

Feng, T. Y

C. L. Wu, T. Y Feng, “The universality of the shuffle–exchange network,” IEEE Trans. Comput. C-30, 324–332 (1981).

Glaser, I.

A. A. Sawchuk, I. Glaser, “Geometrics for optical implementations of the perfect shuffle,” in Optical Computing ’88, P. Chavel, J. W. Goodman, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 270–282 (1988).

Goodman, J. W.

J. W. Goodman, F. I. Leonberger, S.-Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–853 (1984).
[CrossRef]

Huang, A.

M. J. Murdocca, A. Huang, J. Jahns, N. Streibl, “Optical design of programmable logic arrays,” Appl. Opt. 27, 1651–1660 (1988).
[CrossRef] [PubMed]

K.-H. Brenner, A. Huang, “Optical implementation of the perfect shuffle interconnection,” Appl. Opt. 27, 135–137 (1988).
[CrossRef] [PubMed]

A. Huang, “Parallel algorithms for optical digital computer,” in Proceedings of IEEE Tenth International Optical Computing Conference, IEEE catalog no. 83CH1880-4 (Institute of Electrical and Electronics Engineers, New York, 1983), p. 13.

Jahns, J.

Kung, S.-Y.

J. W. Goodman, F. I. Leonberger, S.-Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–853 (1984).
[CrossRef]

Leonberger, F. I.

J. W. Goodman, F. I. Leonberger, S.-Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–853 (1984).
[CrossRef]

Li, H.

Li, Hongpu

M. Cao, Hongpu Li, F. Luo, “Optical implementation of perfect shuffle/exchange omega interconnection network,” Acta Opt. Sin. 12, 1130–1134 (1992).

Li, Y.

Lohman, G. E.

G. E. Lohman, A. W. Lohmann, “Shuffle communication component for optical parallel processing,” J. Opt. Soc. Am. A 4, 106–108(1987).

Lohmann, A. W.

G. E. Lohman, A. W. Lohmann, “Shuffle communication component for optical parallel processing,” J. Opt. Soc. Am. A 4, 106–108(1987).

A. W. Lohmann, W. Stork, G. Stuck, “Optical perfect shuffle,” Appl. Opt. 25, 1530–1531 (1986).
[CrossRef] [PubMed]

Luo, F.

M. Cao, Hongpu Li, F. Luo, “Optical implementation of perfect shuffle/exchange omega interconnection network,” Acta Opt. Sin. 12, 1130–1134 (1992).

M. Cao, F. Luo, H. Li, “Optical perfect shuffle–exchange interconnection network using liquid crystal spatial light switch,” Appl. Opt. 31, 6817–6819 (1992).
[CrossRef] [PubMed]

Murdocca, M. J.

Prise, M. E.

M. E. Prise, N. Streibl, M. M. Downs, “Optical considerations in the design of digital optical computers,” Opt. Quantum. Electron. 20, 49–77 (1988).
[CrossRef]

Sawchuk, A. A.

A. A. Sawchuk, I. Glaser, “Geometrics for optical implementations of the perfect shuffle,” in Optical Computing ’88, P. Chavel, J. W. Goodman, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 270–282 (1988).

Stork, W.

Streibl, N.

M. E. Prise, N. Streibl, M. M. Downs, “Optical considerations in the design of digital optical computers,” Opt. Quantum. Electron. 20, 49–77 (1988).
[CrossRef]

M. J. Murdocca, A. Huang, J. Jahns, N. Streibl, “Optical design of programmable logic arrays,” Appl. Opt. 27, 1651–1660 (1988).
[CrossRef] [PubMed]

Stuck, G.

Wu, C. L.

C. L. Wu, T. Y Feng, “The universality of the shuffle–exchange network,” IEEE Trans. Comput. C-30, 324–332 (1981).

Acta Opt. Sin. (1)

M. Cao, Hongpu Li, F. Luo, “Optical implementation of perfect shuffle/exchange omega interconnection network,” Acta Opt. Sin. 12, 1130–1134 (1992).

Appl. Opt. (6)

IEEE Trans. Comput. (1)

C. L. Wu, T. Y Feng, “The universality of the shuffle–exchange network,” IEEE Trans. Comput. C-30, 324–332 (1981).

J. Opt. Soc. Am. A (1)

G. E. Lohman, A. W. Lohmann, “Shuffle communication component for optical parallel processing,” J. Opt. Soc. Am. A 4, 106–108(1987).

Opt. Commun. (1)

J. Jahns, “Optical implementation of the banyan networks,” Opt. Commun. 76, 321–324 (1990).
[CrossRef]

Opt. Computer Process. (1)

M. Cao, H. Li, “Optical hardware for perfect shuffle interconnection,” Opt. Computer Process. 1, 23–27 (1991).

Opt. Quantum. Electron. (1)

M. E. Prise, N. Streibl, M. M. Downs, “Optical considerations in the design of digital optical computers,” Opt. Quantum. Electron. 20, 49–77 (1988).
[CrossRef]

Proc. IEEE (1)

J. W. Goodman, F. I. Leonberger, S.-Y. Kung, R. A. Athale, “Optical interconnections for VLSI systems,” Proc. IEEE 72, 850–853 (1984).
[CrossRef]

Other (2)

A. Huang, “Parallel algorithms for optical digital computer,” in Proceedings of IEEE Tenth International Optical Computing Conference, IEEE catalog no. 83CH1880-4 (Institute of Electrical and Electronics Engineers, New York, 1983), p. 13.

A. A. Sawchuk, I. Glaser, “Geometrics for optical implementations of the perfect shuffle,” in Optical Computing ’88, P. Chavel, J. W. Goodman, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 270–282 (1988).

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Figures (11)

Fig. 1
Fig. 1

(a) Left and (b) right perfect-shuffle permutation of eight elements.

Fig. 2
Fig. 2

Complex polarized prism for implementation of (a) a left and (b) a right PS permutation of eight elements. Part I is a cubical polarized prism, part II is a triangular prism, and part III is the optical compensator plate used to keep the path lengths of all optical channels the same.

Fig. 3
Fig. 3

Input arrays labeled A, B, C, D, E, F, G, and H.

Fig. 4
Fig. 4

Output permutation of 8 × 2: (a) output of the left PS permutation, (b) output of the right PS permutation.

Fig. 5
Fig. 5

Optical setup for implementation of an inverted left PS of eight elements. Part I is a cubical polarized prism, part II is a triangular prism and part III is the optical compensator plate.

Fig. 6
Fig. 6

Left PS of Fig. 2(a) (i.e., the two consecutive left PS’s for vector (0, 1, 2, 3, 4, 5, 6, 7). Part I is a cubical polarized prism, part II is a triangular prism, and part III is the optical compensator plate.

Fig. 7
Fig. 7

Crossover connection for N = 8; the index i denotes the number of a particular stage.

Fig. 8
Fig. 8

Banyan connection for N = 8; the index i denotes the number of a particular stage.

Fig. 9
Fig. 9

PS interconnection networks (in which the right PS and the left PS are not considered separately) for N = 8; the input ports are labeled A, B, C, D, E, F, G, and H.

Fig. 10
Fig. 10

Crossover interconnection networks for N = 8; the input ports are labeled A, B, C, D, E, F, G, and H.

Fig. 11
Fig. 11

Banyan interconnection networks for N = 8; the intput ports are labeled A, B, C, D, E, F, G, and H.

Equations (39)

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k = F ( k ) , ( k = 0 , 1 , , N - 1 ) , ( k = 0 , 1 , , N - 1 ) .
K = ( k 1 , k 2 , , k N - 1 ) T , K = ( k 0 , k 1 , k 2 , , k N - 1 ) T .
X ( i , j ) = { 1 if i = F - 1 ( j ) ( i = 0 , 1 , 2 , , N - 1 ) 0 otherwise ( j = 0 , 1 , 2 , , N - 1 ) .
K = X K
X T X = I ,
k = ( 2 k + [ 2 k / N ] ) mod N ( k = 0 , 1 , 2 , , N - 1 ) , k = ( 2 k + [ 2 k / N ] ) Mod N ( K = 0 , 1 , 2 , , N - 1 ) .
P L ( i , j ) = { 1 if j = ( 2 i + [ 2 i / N ] ) mod N ( i = 0 , 1 , 2 , , N - 1 ) 0 otherwise ( j = 0 , 1 , 1 , , N - 1 ) ,
P R ( i , j ) = { 1 if j = ( 2 i + 1 - [ 2 i / N ] ) mod N ( i = 0 , 1 , 2 , , N - 1 ) 0 otherwise ( j = 0 , 1 , 2 , , N - 1 ) .
P L = [ 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 ] , P R = [ 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 ] .
P L log 2 N = I
P L ( log 2 N ) - 1 = P L - 1 .
P R log 2 N = I ,
P L P L T I ,             P R P R T I .
P L = P R P L - 1 P R ,             P R = P L P R - 1 P L .
C i ( 1 ) = [ I i 0 0 I i ] 2 i ,             I i = [ 0 1 1 0 ] 2 n - i ,
C i ( 2 ) = I .
C ( 1 ) ( n ) = C n - 1 ( 1 ) C n - 2 ( 1 ) C 1 ( 1 ) C 0 ( 1 ) .
C 0 ( 1 ) = [ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ] , C 1 ( 1 ) = [ 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 ] , C 2 ( 1 ) = [ 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 ] ,
C ( 1 ) ( 3 ) = C 2 ( 1 ) C 1 ( 1 ) C 0 ( 1 ) = [ 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 ] .
K = C ( 1 ) ( 3 ) A = ( 5 , 4 , 7 , 6 , 1 , 0 , 3 , 2 ) T .
B i ( 1 ) = [ b i 0 0 b i ] 2 i ,             b i = [ 0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0 2 n - i - 1 2 n - i - 1 ] 2 n - i ,
B i ( 2 ) = I ,
B ( 1 ) ( n ) = B n - 1 ( 1 ) B n - 2 ( 1 ) B 1 ( 1 ) B 0 ( 1 ) .
B 0 ( 1 ) = [ 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 ] , B 1 ( 1 ) = [ 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 ] , B 2 ( 1 ) = [ 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 ] ,
B ( 1 ) ( 3 ) = B 2 ( 1 ) B 1 ( 1 ) B 0 ( 1 ) = [ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 ] .
K = B ( 1 ) ( 3 ) K = ( 7 , 6 , 5 , 4 , 3 , 2 , 1 , 0 ) .
P L = P R B 0 ( 1 ) ,             P R = P L B 0 ( 1 ) ,
B ( 1 ) ( n ) = I = P R log 2 N .
S i = [ s i ( 0 ) 0 s i ( 1 ) s i ( N - 2 ) 0 s i ( N - 1 ) ] .
X i = X i ( 1 ) - X i ( 2 ) .
K i + 1 = S i X i K i = S i ( X i ( 1 ) - X i ( 2 ) ) K i .
K 0 = ( A , B , C , D , E , F , G , H ) T .
K 1 = S 0 X 0 K 0 = S 0 ( P L - P R ) K 0 = [ s 0 ( 0 ) ( A - E ) s 0 ( 1 ) ( A - E ) s 0 ( 2 ) ( B - F ) s 0 ( 3 ) ( B - F ) s 0 ( 4 ) ( C - G ) s 0 ( 5 ) ( C - G ) s 0 ( 6 ) ( D - H ) s 0 ( 7 ) ( D - H ) ] .
s 0 ( 0 ) ( A - E ) = { A if s 0 ( 0 ) = 0     ( corresponding to upper output ports ) E if s 0 ( 0 ) = 1     ( corresponding to lower output ports ) .
K 2 = [ s 1 ( 0 ) [ s 0 ( 0 ) ( A - E ) - s 0 ( 4 ) ( A - E ) ] s 1 ( 1 ) [ s 0 ( 0 ) ( A - E ) - s 0 ( 4 ) ( A - E ) ] s 1 ( 2 ) [ s 0 ( 1 ) ( A - E ) - s 0 ( 5 ) ( A - E ) ] s 1 ( 3 ) [ s 0 ( 1 ) ( A - E ) - s 0 ( 5 ) ( A - E ) ] s 1 ( 4 ) [ s 0 ( 2 ) ( A - E ) - s 0 ( 6 ) ( A - E ) ] s 1 ( 5 ) [ s 0 ( 2 ) ( A - E ) - s 0 ( 6 ) ( A - E ) ] s 1 ( 6 ) [ s 0 ( 3 ) ( A - E ) - s 0 ( 7 ) ( A - E ) ] s 1 ( 7 ) [ s 0 ( 3 ) ( A - E ) - s 0 ( 7 ) ( A - E ) ] ] , K 3 = [ s 2 ( 0 ) { s 1 ( 0 ) [ s 0 ( 0 ) ( A - E ) - s 0 ( 4 ) ( C - G ) ] - s 1 ( 4 ) [ s 0 ( 2 ) ( B - F ) - s 0 ( 6 ) ( D - H ) ] } s 2 ( 1 ) { s 1 ( 0 ) [ s 0 ( 0 ) ( A - E ) - s 0 ( 4 ) ( C - G ) ] - s 1 ( 4 ) [ s 0 ( 2 ) ( B - F ) - s 0 ( 6 ) ( D - H ) ] } s 2 ( 2 ) { s 1 ( 1 ) [ s 0 ( 0 ) ( A - E ) - s 0 ( 4 ) ( C - G ) ] - s 1 ( 5 ) [ s 0 ( 2 ) ( B - F ) - s 0 ( 6 ) ( D - H ) ] } s 2 ( 3 ) { s 1 ( 1 ) [ s 0 ( 0 ) ( A - E ) - s 0 ( 4 ) ( C - G ) ] - s 1 ( 5 ) [ s 0 ( 2 ) ( B - F ) - s 0 ( 6 ) ( D - H ) ] } s 2 ( 4 ) { s 1 ( 2 ) [ s 0 ( 1 ) ( A - E ) - s 0 ( 5 ) ( C - G ) ] - s 1 ( 6 ) [ s 0 ( 3 ) ( B - F ) - s 0 ( 7 ) ( D - H ) ] } s 2 ( 5 ) { s 1 ( 2 ) [ s 0 ( 1 ) ( A - E ) - s 0 ( 5 ) ( C - G ) ] - s 1 ( 6 ) [ s 0 ( 3 ) ( B - F ) - s 0 ( 7 ) ( D - H ) ] } s 2 ( 6 ) { s 1 ( 3 ) [ s 0 ( 1 ) ( A - E ) - s 0 ( 5 ) ( C - G ) ] - s 1 ( 7 ) [ s 0 ( 3 ) ( B - F ) - s 0 ( 7 ) ( D - H ) ] } s 2 ( 7 ) { s 1 ( 3 ) [ s 0 ( 1 ) ( A - E ) - s 0 ( 5 ) ( C - G ) ] - s 1 ( 7 ) [ s 0 ( 3 ) ( B - F ) - s 0 ( 7 ) ( D - H ) ] } ] .
[ A if F = 000 B if F = 001 C if F = 010 D if F = 011 E if F = 100 F if F = 101 G if F = 110 H if F = 111 .
K 0 = ( A , B , C , D , E , F , G , H ) T .
K 3 = [ s 2 ( 0 ) { s 1 ( 0 ) [ s 0 ( 0 ) ( A - H ) - s 0 ( 3 ) ( D - E ) ] - s 1 ( 1 ) [ s 0 ( 1 ) ( B - G ) - s 0 ( 2 ) ( C - F ) ] } s 2 ( 1 ) { s 1 ( 0 ) [ s 0 ( 0 ) ( A - H ) - s 0 ( 3 ) ( D - E ) ] - s 1 ( 1 ) [ s 0 ( 1 ) ( B - G ) - s 0 ( 2 ) ( C - F ) ] } s 2 ( 2 ) { s 1 ( 2 ) [ s 0 ( 0 ) ( A - H ) - s 0 ( 3 ) ( D - E ) ] - s 1 ( 3 ) [ s 0 ( 1 ) ( B - G ) - s 0 ( 2 ) ( C - F ) ] } s 2 ( 3 ) { s 1 ( 2 ) [ s 0 ( 0 ) ( A - H ) - s 0 ( 3 ) ( D - E ) ] - s 1 ( 3 ) [ s 0 ( 1 ) ( B - G ) - s 0 ( 2 ) ( C - F ) ] } s 2 ( 4 ) { s 1 ( 4 ) [ s 0 ( 7 ) ( A - H ) - s 0 ( 4 ) ( D - E ) ] - s 1 ( 5 ) [ s 0 ( 6 ) ( B - G ) - s 0 ( 5 ) ( C - F ) ] } s 2 ( 5 ) { s 1 ( 4 ) [ s 0 ( 7 ) ( A - H ) - s 0 ( 4 ) ( D - E ) ] - s 1 ( 5 ) [ s 0 ( 6 ) ( B - G ) - s 0 ( 5 ) ( C - F ) ] } s 2 ( 6 ) { s 1 ( 6 ) [ s 0 ( 7 ) ( A - H ) - s 0 ( 4 ) ( D - E ) ] - s 1 ( 7 ) [ s 0 ( 6 ) ( B - G ) - s 0 ( 5 ) ( C - F ) ] } s 2 ( 7 ) { s 1 ( 6 ) [ s 0 ( 7 ) ( A - H ) - s 0 ( 4 ) ( D - E ) ] - s 1 ( 7 ) [ s 0 ( 6 ) ( B - G ) - s 0 ( 5 ) ( C - F ) ] } ] .
K 3 = [ s 2 ( 0 ) { s 1 ( 0 ) [ s 0 ( 0 ) ( A - E ) - s 0 ( 2 ) ( C - G ) ] - s 1 ( 1 ) [ s 0 ( 1 ) ( B - F ) - s 0 ( 3 ) ( D - H ) ] } s 2 ( 1 ) { s 1 ( 0 ) [ s 0 ( 0 ) ( A - E ) - s 0 ( 2 ) ( C - G ) ] - s 1 ( 1 ) [ s 0 ( 1 ) ( B - F ) - s 0 ( 3 ) ( D - H ) ] } s 2 ( 2 ) { s 1 ( 2 ) [ s 0 ( 0 ) ( A - E ) - s 0 ( 2 ) ( C - G ) ] - s 1 ( 3 ) [ s 0 ( 1 ) ( B - F ) - s 0 ( 3 ) ( D - H ) ] } s 2 ( 3 ) { s 1 ( 2 ) [ s 0 ( 0 ) ( A - E ) - s 0 ( 2 ) ( C - G ) ] - s 1 ( 3 ) [ s 0 ( 1 ) ( B - F ) - s 0 ( 3 ) ( D - H ) ] } s 2 ( 4 ) { s 1 ( 4 ) [ s 0 ( 4 ) ( A - E ) - s 0 ( 6 ) ( C - G ) ] - s 1 ( 5 ) [ s 0 ( 5 ) ( B - F ) - s 0 ( 7 ) ( D - H ) ] } s 2 ( 5 ) { s 1 ( 4 ) [ s 0 ( 4 ) ( A - E ) - s 0 ( 6 ) ( C - G ) ] - s 1 ( 5 ) [ s 0 ( 5 ) ( B - F ) - s 0 ( 7 ) ( D - H ) ] } s 2 ( 6 ) { s 1 ( 6 ) [ s 0 ( 4 ) ( A - E ) - s 0 ( 6 ) ( C - G ) ] - s 1 ( 7 ) [ s 0 ( 5 ) ( B - F ) - s 0 ( 7 ) ( D - H ) ] } s 2 ( 7 ) { s 1 ( 6 ) [ s 0 ( 4 ) ( A - E ) - s 0 ( 6 ) ( C - G ) ] - s 1 ( 7 ) [ s 0 ( 5 ) ( B - F ) - s 0 ( 7 ) ( D - H ) ] } ] .

Metrics