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AST ALGORITHMS OF ASYMPTOTIC ANALYSIS OF NETWORKS WITH UNRELIABE EDGES
G.Sh. Tsitsiashvili
A.S. Losev
690041, Vladivostok, Radio St. 7, Institute for Applied Mathematics Far Eastern Branch of RAS
A problem of a reliability in networks with unreliable elements naturally origin in technical applications [1]. But a direct calculation of the reliability demands a number of operations which increases geometrically dependently on a number of edges. So it is necessary to use approximate methods and particularly asymptotic one. In [2] a reliability asymptotic is calculated in analogous asymptotic suggestions on the network edges. Main parameters in these asymptotic are a shortest way length and a maximal flow in a network. In this paper different partial classes of networks are considered and effective algorithms of their parameters calculations are suggested. These networks are networks originated by dynamic systems, networks with integer-valued lengths of edges, superposition of networks and bridge schemes.
1. Preliminaries
Define the graph r with the finite nodes set U and the set W of edges w = (u, v). The graph r may contain cycles or not, its edges may be oriented or not. Denote by R (u) the set of all ways R of the graph r, which connect the nodes u0, u, and assume that R(u) ^ 0, u e U . Suppose that r(u) is the sub-graph of the graph r, which consists of the ways R e R(u) . Consider the sets
A (u) = {A e U: u0 e A, u g A}, L = L (A) = {(u, u'): u e A, u' g A}
And the set L(u) = {L(A), A e A(u)} ofall sections of the sub-graph r(u).
Characterize each edge w e W of the graph r by the logic number a(w) = I (the edge w works), where I(B) is the indicator function of the event B . Denote
P(u )= V A a( w)
ReR(u) weR
the characteristic of the nodes u0, u connectivity in the graph r. Suppose that a(w), w e W, are independent
random variables, P (a(w) = 1) = pw (h), qw (h) = 1 - pw (h),
where h is small parameter: h ^ 0. In [2] the following statements are proved.
Theorem 1. Suppose that pw (h) ~ exp(-h"c(w)), h ^ 0, where c(w) > 0, w e W. Then
-lnP((u) = l)~ h~D(u), D(u)= min max c(w) . (1)
V V 7 7 V ' ReR(u) weR V '
Theorem 2. Suppose that qw (h) ~exp(-h_ci(w)), h ^ 0, where c1 (w)> 0, w e W. Then
- ln P(p(u) = 0h~A(u), D1 (u)= max min c1 (w). (2)
ReR(u) weR
Theorem 3. Suppose that pw (h) ~ hg(w), h ^ 0, where g(w) > 0, w e W. Then
- ln P (p(u) = 1)~ T (u) ln h, T (u)= min £ g (w). (3)
ReR (u)weR
Theorem 4. Suppose that qw (h) ~ hg(w), h ^ 0, where g (w) > 0, w e W. Then
- ln P (P(u) = 0)~ T (u) ln h, Tx (u)= m i(n E g (w) . (4)
LeL(u ) weL
Statement 1. Suppose that all c (w) (all c1 (w)), w e W, are different. Then there is the single edge w (u) (there is the single edge w1 (u)), so that c (w (u)) = D (u) (c1 (w1 (u)) = D1 (u)). It is called the critical edge.
2. Graphs generated by dynamic systems
Suppose that the set U consists of non-intersected subsets U0, U1,...,Um, and the set U0 contains the single vertexu0, which is called initial. All edges of the oriented graph r are represented as (,uj), 1 < i < j < m, ut eUi, uj eUj, and each vertexis accessible from the initial vertexu0. Described graphs are generated by dynamic systems with a delay. In this section we calculate D(u),D1 (u),T(u) and find critical edges w (u), w1 (u) for a fixed u0 .
A main idea of this section is an application of the Floyd algorithm [3], when a solution is calculated for all u eU . To construct fast algorithms it is natural to constrict a class of considered graphs. An idea of such a constriction is illustrated in [4] but for a fixed u .
Suppose that D (u0 ) = D1 (u0 ) = T (u0 ) = 0, for all u e U1 put
D(u) = D1 (u) = T(u) = c(u), w(u) = w 1 (u) = (u0,u) .
For u eU define S(u) = {v:(v,u)e W}, |S(u)| a number of elements in the finite set S(u) . Assume that for all u eU1,...,Uk the meanings D(u), D1 (u), T(u), w(u), w1 (u) are defined. Take u eUk+1 and in an accordance with the formulas (1), (2) put
D (u)= min max (c (v, u), D (v)), D1 (u)= max min (c (v, u), D1 (v)) , (5)
veS(u) veS(u)
T (u)= min (c (v, u) + T (v)), k > 1. (6)
veS(u)
To calculate each element from the set D(u), D1 (u), T(u), u e U it is necessary 2 |S(u)|-1 arithmetic operations and this number can not be decreased. So the algorithm (5), (6) is optimal. And if for fixed u eU D (u ), D (u ), T (u ) are calculated by the algorithm (5), then we find D (v), D1 (v), T(v) for all nodes v from which the vertexu is accessible.
To define critical edges it is necessary to complement the formulas (5) by
w (u ) = 1 (u )=
w(u) = w1 (u) = (u0,u), if u0 e S(u), w(v), if D(u) = max(D(v), c(v,u))> c(v,u), (v,u), if D(u) = max(d(v), c(v,u))> D(v), w1 (v), if D ( u) = max (D (v), c ( v, u )) < c ( v, u ), (v, u), if D (u) = max (( (v), c (v, u))< D (v).
(7)
(8)
3. Graphs with integer-valued lengths of edges
In this section we consider a calculation of T(u) for all u e U in graphs with integer-valued lengths of edges. Suppose that g (w), w eW, are natural numbers, g (w) < g and define
Gp = Z g(w). (9)
weW
Divide each edge of the graph r into edges with unit lengths by an introduction of intermediary nodes. As a
1 11 1 / 1 \ result obtain the graph r with the nodes set U , U e U and with the edges set W . Denote NI u ) the minimal
number of the graph edges in ways, which connect the nodes u0, u1. It is easy to obtain that
N(u) = G(u), u eU . (10)
Consider now an algorithm of N(u1), u1 e U1 calculation.
Suppose that all nodes of the graph. r1 are not marked. Mark the vertex u0, and put U1 ={u0}. Then
construct a recurrent procedure of non-intersected sets Ulk, k > 0, definition. Suppose that the sets Ulk,
Vk = U U1 are known and all nodes of the set Vk are marked and all other nodes are not marked. Define the set. 0<i< k
U1+1 as a set of all unmarked nodes from U1, which are connected directly with some vertexfrom the set Ul. By a definition the set U](+1 satisfies the formula
U1+1 ={ul: N (u1 ) = k +1} .
Mark all nodes of the set Ulk+1 and define the set = Vk U Ulk+1.
Estimate a number of operations which are necessary to calculate U]i+1 if each vertexof the graph r is connected directly with no more l nodes. Then a number of operations to define U]i+1 does not exceed l Ulk . Define M by the formula
V1 c V11 c... c VM = VM+1 =...,
then to construct the sequence U^,...,UM it is necessary no more lGr operations where lGr <l2g|U|. Compare these results with the results of Deikstra [4], in a case when c(w) is not integer-valued. To calculate D(u), u e U
in a general case it is necessary no more K1 |U|2 operations and for a dendriform graph - no more K2 |U| ln |U| operations, where K1, K2 .
4. Superposition of graphs
Fix in the graph r some vertexv0. Assume that r' is non-oriented graph with the nodes set U' = {1',...,m'}, UIU' = 0 and with the edges set W' (i', /),(i',iW'. Distinguish in the graph r' initial and final nodes
u0 ', v0 ' and in the set U - two nodes u, v so that w = (u, v)e W . Denote by R' the set of all ways R' of the graph r ' from u0' to v0'.
— w ,--v
Define the superposition T = ' of the graphs r, r ' with a replacement of the edge (u, v) from the
graph r by the graph r' and with an aliasing of the nodes u with u0' and of the nodes v with v0' correspondingly. Denote by U the nodes set, by W - the edges set and by R - the set of ways from the vertex u0 to the vertex v0 in the graph r. Put R the set of ways from u0 to v0 in the graph r, R' - the set of ways from u0' to v0' in the graph r '. Analogously define L, L, L ' the sets of sections in the graphs r, r, r ' with pairs of initial and final nodes (,vo),(uo,vo), (u^,v0) correspondingly. Define
P= v a a(w), P = _v_ w)
ReR weR ' ReR weR
characteristics of a connectivity between the nodes u0, v0 in the graphs r, r correspondingly. Then from the theorems 1-4 it is possible to obtain asymptotic formulas for the superposition r.
Theorem 5. Suppose that pw (h) ~ exp(-h~c(w)), h ^ 0, where c (w) > 0, w e W. Then
- ln P ((= 1)~ h~D, D = minmax c (w),
V ' ReR weR
c (w) = c (w), w ^ w, c |w|= minmax c (w).
V ' V ' V / r'eR' weR' V '
Theorem 6. Suppose that qw (h) ~exp(-h_c1(w)), h ^ 0, where c1 (w)> 0, w eW. Then
-lnP( = o)~ h~Dl, Di = minmaxc, (w),
v / LeL weR 1 v '
c1 (w) = c, (w), w * w, c1 ( w) = minmax c, (w ).
V ' 1V ' V / L'eL' weL 1 V '
Theorem 7. Suppose that pw (h ) ~ hg(w), h ^ 0, where g ( w) > 0, w eW. Then
ln P (( = 1)~ T ln h, T = min 2 g(w),
V ' ReR weR
g(w) = g(w),w*w, g(w)= min 2 g(w).
V ' V ' V ' R eR weR'
Theorem 8. Suppose that qw (h) ~ hg(w), h ^ 0, where g (w) > 0, w e W. Then
ln P(( = 0)~ T1 ln h, T1 = min 2 g (w),
V ' LeL WeL 1
g1 (w) = g1 (w),w*w, g1 (w) = min 2 g1 (w).
LeL weL'
It is obvious that the formulas from these theorems allow calculating asymptotic of a reliability for superposition of networks with unreliable elements rationally. These formulas may be used to calculate a reliability in recursively defined networks which are widely used in the solid state physics and in the nanotechnology.
5. Asymptotic analysis of bridge scheme
The simplest superposition of graphs is parallel-sequential graphs. But there are graphs widely used in the reliability theory, which are not parallel - sequential. One of them is a bridge scheme.
Consider the non-oriented graph r with the nodes set U = {ui, i = 0,..., 3} and with the edges set
W = {wj, j = 1,..,5},where
"1 =(0, u ), w2 =(0, u2), w3 =(ui, u3), w4 =(u2, u3 ), w5 =(ui, u2).
w
The vertexu0 is initial and the vertexu3 is final. The edge w5 is a bridge element in the graph r. The graph r is called the bridge scheme in the reliability theory. Define the r1 by a deleting of the edge w5 from the graph r. Introduce the graph r2 by an aliasing of the nodes u1, u2 in the graph r1.
w1 w3
S>
w2 w4
Fig. 3. Graph r2 .
Suppose that the edges w1,..., w5 work independently and define positive numbers c(wi) = ci,1 < i < 5,
C1 = min(max(c1,c3),max(c2,c4)), C2 = max(min(c1,c2),min(c3,c4)), C2 < C1.
If random logical variables P, P1,P2 characterize the nodes u0,u3 connectivity in the graphs r,r1,r2, correspondingly, then from the complete probability formula we have:
P(P = 1)= Pw5 (h)P(P2 = 1) + (1 - Pw5 (h))P(P1 = 1), P(P1 = 1) < P(P2 = 1). (11)
From the theorem 1 and the equalities (11) obtain ) the statement which characterizes a role of the bridge element.
Theorem 9. If pw (h) ~ exp(-h_c(w)), h ^ 0, where c (w) > 0, w e W, then
-ln P(P = 1)~ h~D , D = min(C1, max(C2, c5)) . (12)
References
1. Riabinin I.A. . Reliability and safety of structural complicated systems. St. Petersburg: Edition of the Petersburg University. 2007. 276 p. (In Russian).
2. Tsitsiashvili G.Sh. Asymptotic Analysis of Logical Systems with Unreliable Elements// Reliability: Theory and Applications. 2007, Vol. 2, № 1. Pp. 34-37.
3. Floyd R.W, Steinberg L. An adaptive algorithm for spatial grayscale// SID 75 Digest. 1975. Pp. 36-37.
4. Kormen T., Leizerson Ch., Rivest R. Algorithms: construction and analysis. Moscow: Laboratory of basic knowledge. 2004. 955 p. (In Russian).
