BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI - Universitatea ...

then, under the principle of optimality:
(5) V i ( V j + mij
)
=
j≠i
Bul. Inst. Polit. Iaşi, t. LVI (LX), f. 2, 2010 207
min , ( i = 0, n −1
; j = 0,
n ) and V = 0 .
To solve system (5) shall be iterative, noting Vi the value of Vi
obtained at
iteration k, namely:
(6) V i = min
Calculate:
0
(7) V i ( V j + mij
)
and then:
0
1
= min , ( = 0, n −1
j≠
i
k
k −1
(8) V i = ( V j + mij
)
j≠
i
( i = 0, n −1
); V 0 .
k
0
n =
i ; j = 0,
n ) and V 0 .
n
1
n =
min , ( i = 0, n −1
; j = 0,
n ) and V = 0 .
Ordinal iteration of k expressed by relations (8) gives values only for the
finite length of roads at most k + 1 arriving at xn
, choosing between them is the
minimum.
From iteration to the next:
(9)
k
−1
≤ k k
i Vi
V , ∀ j .
Numbers Vi
( i ≠ n ; k = 0,
1,...
) monotone decreasing pattern formed that
reach to the minimum necessary, after a finite number of iterations which not
exceeding n −1.
So algorithm stops when it reaches an iteration k, such that
+ 1
= , (
k k
V V i = 0,
n ), and the minimum between the peaks road and x is
i i
k k + 1
0 = V0
V
. To identify which roads have minimum values founded, are
derived from (8) that have them at the last iteration we have:
k
n
x0 n