SCIENTIFIC ABSTRACT GRISHUKHIN, V.P. - GRODSKIY, V.A.

• Document Type: CREST
• Collection: Scientific Abstracts
• Document Number (FOIA) /ESDN (CREST): CIA-RDP86-00513R002201020013-0
• Release Decision: RIF
• Original Classification: S
• Document Page Count: 100
• Document Creation Date: November 2, 2016
• Document Release Date: September 1, 2001
• Sequence Number: 13
• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

USSR
GRISHUKHIN, V. P., 14at. metody resheniya ekon. zadach, No 3,
3-972, pp 93-105
C
n4l
Zh. Ansel' (Kibern. sbornik, vY--n. 5, Moscow, "Mir", 1968) showed that
X 1~tjI
,,;, . This result is compared with the number of checks for admis-
sability of the plan in the Balash algorithm.
It/it
- 73 -

USSR
QRISHUKHIN. V. P.
"The Mean Number of Iterations of a Balash Algorithm"
0
Issled. po diskretnoy mat. [Studies in Discrete Mathematics -- Collection
of Works], Moscow, Nauka Press, 1973, pp S8-68 (Translated from Refera-
tivn)7 Zhurnal - Kibernetika, No 8, 1973, Abstract No 8 VS12 1)), Yu. Finkel'-
shteyn)
Translation: An intqyer pi,olli-amming, problem is studied (problem p)
cixi -~ m in,
n
V Apq B. (Ai, BC-RI").
X1 - 0 ww 1. n.
1/7
- 66

USSR
GRISHUKHIN, V. P., Issled. po diskretnoy mat., Moscow, Nauka Press, 1973,
pp 58-68
Here Aj, B are m-dimensional vectors with components aij' bi I number cI and
vectors A.,B corresponj to a certain point in space ,n RnillRIII=R(n+l)(M-1 -1
i lijn+n+m
and vice versa: any vector zER corresponds to the coefficients of
a certain problem D, if we write z in the form of z=(C, A, B), where C=
(cil..., cn) R", A=(alp ... I An) R"', B( R'.
Thus, between problems p and points Rn+mn+m there is a mutually unambiguous
correspondence. Further It n+nm+m is -studied with ordinary Euclidean metrics
and the set of problems dp is assigned the measure
p (dp) V dcidizip1bl,
jC-1V ic-31
2/7

USSR
GRISHUKHIN, V. P., Issled, po diskretno), mat., Moscow, Nauka Press, 1973,
pp 58-68
where N= 1,2, ..., n , NI= 1,2, ..., m . Further, certain areas in space
Rn+Tim+m are studied, the measure of which is their volume. 'rhe set of
problems corresponding to a certain area G is called by the author problem.
class PG. For a certain function f(p) in problems from class pG' the me" n
for the class is defined
iGI (P) IL
where IGI is the volume of area G. Further, only one function ( 1) ~ i s-
studied -- the number of steps in the Balash algorithm. This Function
was introduced in a worY by the author (RZINat, 1973, SV652); this same
work studied area G, fixed by the inequalities
C/ > Cj' for a I I j~N,
P-j 0
A,
6

USSR
GRISHUMIN, V. P., Issled. po diskretnoy In at., Moscow, Nauka Press, 19731
pp 58-68
is shown that 45(p) inthe class of problems corresponding to this area (and
called PC (n)) is independent of vector C=(cl, ... ) c1) ) and satisfies certain
equations. Further, the mean ~; G(A,c) is studied, where C(A,c) is the area
r
of G limited by the conditions A4A, An',10 O'-B"IA- Here c is a number,
while A=(al, ... a ) is a vector. We note th
while A=(all ... aIn ) is a vector. We note that G(a,c)-(;(A) G(c), and
since (p) is independent of C, 1, G (A, c) =~ G (A) -It is shown that IG(A) is
- -n
write q =ri ather cumbersome
independent of A, so that we can G(A) In After r,
calculations, it is shown that
-. 31n
Irl ~ 2---+' (1 +
M -n
4/7

USSR
GRISHUMIN, V. P., Issled. po diskretnoy mat., Moscow, Nauka Press, 1973,
pp 58-68
This formula gives us the behavior of -.-il as n-~~with fixed m. It is further
shown that
(p (n) = I im (p, n + 1.
M
M -M
in (RZINkfat, 19"13, SV6S2) it was found thit
n n
3
-(1 -1 0('
I-C
q, (p) - C
Thus, the mean numbei- of iterations of the 11alash algorithm in arca P C (11)
(like the maximnm) increases exponentially with increasing Ptimber of variables
/-7

USSR
GRISHUMIN, V. P. , Issled. po diskretnoy mat., Moscow, Nauka Press, 197135,
pp SS-68
n (with fixed m). If n is fixed, as m increases this HICZATI eXpOnCiltial1V
approaches the limit (m+1). The author shows that the "exponential depen-
dence of the mean number of iterations of the Balash alporithni on n in area
P (n) results completely from the fact that it does not. consider the (:)al
C 9
function." In order to prove this Statement, lie analVl7es algorithm B from
(RZHNiat, 1973, SV652), which has a very simple rule for selection of namely
i,=min (and the goal function is not considered). Then, calculating the
jE NS-1
k
.n
mean1p, (B in the area
G- 10,-Ai