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Effective Algorithm for Optimal K-Terminal Reliability of Distributed Systemw ( ε ) = 1 − ( q × p )i,jWherey i,j ≤ n − 2.yi,jΠ , (3)z = 1i, kz kz , jAssume not only that we have a selected set G k of nodes with reliability R ( G k),but also that the nodes in G k are all directly connected. If another set G k′ofnodes exists in which just one node is different from G k and G k′has one nodewhich is not directly connected with other nodes in G k ′ , then we saythat R ( G k) ≥ R ( Gk ′ ) for R( G k′ ) the reliability of set G k ′ .Fig. 1. (A) A selected set G k ={ v 1 , v 2 , v 3 } under a DS with six nodes and sevenlinks; (B) A selected set G k ′ ={ v 1 , v 2 , v 4 } under a DS with six nodes and sevenlinks.:denotes unselected node :denotes selected nodev 1 v 5 v 1v 5v 3v 4v 3 v 4v 2(A)v 6v 2v 6(B)By assuming that the 2-terminal reliability between v 1 and v 2 is R 1 , this relationcan be represented as R({v 1 , v 2 }) = R 1 , and R({v 1 , v 3 }) = R 2 , R({v 2 , v 3 }) = R 3 ,R({v 3 , v 4 }) = R 4 . In Fig. 1(A), we select nodes v 1 , v 2 and v 3 . Therefore, R({v 1 ,v 2 }) = R 1 , R({v 1 , v 3 }) = R 2 , R({v 2 , v 3 }) = R 3 . According to Fig. 1(B), we selectnodes v 1 , v 2 and v 4 , Therefore, R({v 1 , v 2 }) = R 1 , R({v 1 , v 4 }) = R 2 × R 4 andR({v 2 , v 4 }) = R 3 × R 4 . Because R 2 ≤ 1, R 3 ≤ 1 and R 4 ≤ 1, R 2 × R 4 ≤ R 2and R 3 × R 4 ≤ R 3 , the reliability of node v 1 , v 2 , v 3 ≥ the reliability of node v 1 ,v 2 , v 4 . Restated, R({v 1 , v 2 , v 3 }) ≥ R({v 1 , v 2 , v 4 }). However, this assumption isnot always true if (a) a path exists between v 4 and v 1 or between v 4 and v 2 , and(b) the reliability of the path is larger than the reliability between v 3 and v 1 andbetween v 3 and v 2 . For this reason, in some cases, the maximum reliabilitycannot be achieved using the proposed method.107