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ASYMPTOTIC OF CYCLY EXISTENCE IN ORIENTED GRAPH WITH LOW RELIABLE EDGES
G. Tsitsiashvili
Russia, Vladivostok, IAM FEB RAS, FEFU
e-mail: guram@iam. dvo. ru
ABSTRACT
In this paper a power asymptotic of a probability that there is a cycle in a random oriented graph with n nodes and low reliable edges is constructed. An accelerated algorithm for a calculation of asymptotic coefficients with 0(s(n)ln n) products, where s(n) is an amount of products in a multiplication of two matrixes with a size nxn, is constructed.
Keywords: a cycle, an oriented graph, an edge, a probability.
1. INTRODUCTION
We consider a problem of power asymptotic construction for a probability of a cycle existence in a random graph with low reliable edges. A presence of cycles in a deterministic oriented graph allows to factorize it by a relation of a cycle equivalence [1], [2]. A calculation of an amount of cycles with minimal length may be applied in an investigation of free scale networks which receive large spread last years. [3, Theorems 10 - 12]. An algorithm of a calculation of power asymptotic coefficients with products amount 0(s(n)ln n), where s(n) is an amount of products for a multiplication of two matrixes with a size nxn, is constructed.
2. ASYMPTOTIC OF CYCLE EXISTENCE
Consider an oriented graph G with nodes 1, ...,n, without loops and fold edges. Denote A = aij II- •_ its adjacency matrix, D - minimal cycle length, C - an amount of cycles with minimal
l,J 1
length in the graph G. Construct a model of an oriented random graph with nodes 1, ...,n in which only edges of the graph G may enter. The edge (i,j) enters with the probability ptj = h, h ^ 0 (it is low reliable). Random events that different edges enter the graph are independent. Denote S the event that there is a cycle in the graph and put P(S) its probability.
Theorem 1. The limit relation P(S)~ChD, h ^ 0 is true.
Proof. As = Ui<k<n^k , where Sk is the event that there is simple (without repetitions of nodes) cycle with the length k in the graph so P(S) satisfies the relation
P(S) = P(U1<k<nSk) = P(UD<k<nSk)~P(SD)~C(hD), h^0.
Theorem 1 is proved.
Define ck = trAk and calculate asymptotic constants D, C.
Theorem 2. If min (k: ck > 0) < n then D = min (k: ck >0), C =
Proof. It is well known that the element a^ of the matrix Ak equals the amount of ways (i = h, ., ifc-1, ifc, 0 with the length k in the oriented graph G. If k = D then all cycles with the length k
contain k different nodes. Indeed if not there is a cycle with the length k passes through some node more than one time. So this cycle has length smaller than k.
Consequently the equality D = min (k: ck > 0) is true and all cycles with the length D are simple. So the cycle (i = i1,..., ik-t, ik, i) adds units in k diagonal elements of the matrix Ak and
the equality C = ^ takes place. Theorem 2 is proved.
Assume that the constant D is known and = min (k: 2k > n). Represent the constant D in the binary-number system and write it in the form
D = 2l* + 2l? + — + 2lr, 0 < < l2 < — < lr < kt.
Calculate now the matrixes A2± =AxA, A2" = A2± x A2±A2*1 = A2*1 1 x A2*1 1, using k1s(n) = 0(s(n)ln n) products. Then the constant C may be calculated by the formula
C = —-(1)
D v '
using 0(s(n)ln n) products. The constant D = min (k: trAk > 0) may be found by a sequential calculation of the matrixes Ak, 1 < k < n, using 0(s(n)n) products. So there is a problem to accelerate an algorithm of the constant D calculation.
3. ACCELERATED ALGORITHM OF CONSTANT D CALCULATION
Put B = A + I where I is the unit matrix and denote dk = trBk — n. Theorem 3. If Theorem 2 condition is true then
D = min (k: bk >0), 0 = b1 < b2 < — < bn . (2)
Proof. The relation (2) is a corollary of the equality
bk = tr(A + I)k — n = If=1 C}k trAi , where C^ is a number of combination from k by j.
Using Theorem 3 and an idea of a dichotomy dividing for a search of a root of monotonically increasing and continuous function construct the following algorithm of the constant D definition. Using the formulas B2+1 = B2 • B2 , t > 0, calculate by s(n) products. If d2u1 = 0 then we stop calculation and put formally D = o>, C = 0. If not define q1 = min (k: d2k > 0), q1 < Wog2n] + 1, where [a] is an integer part of a real number a.
Denote P = 2qi, Q = and construct the following recurrent procedure: if dQ+2qi-2 > 0 then P -=Q + 2qi~2, else Q ■= Q + 2^~2, if dQ+2qi-3 > 0 then P ■= Q + 2^"3, else Q ■= Q + and so on. This procedure continues q^ — l steps till we obtain the equality P — Q = 1. Then the relation D = P is true. To fulfill this recurrent procedure it is necessary to make 0(s(n)lnn) products. Theorem 3 is proved.
Consequently the asymptotic constants D, C may be calculated by 0(s(n) ln n) products.
4. CONCLUSION REMARKS
For the standard algorithm of the multiplication of two matrixes with the size nxn s(n) = 0(n3), for F. Strassen algorithm s(n) = 0(n2S1), for D. Coppersmith and Sh. Winograd algorithm s(n) = 0(n2 3755) and for V. Williams algorithm s(n) = 0(n2 3727) [4]. But main part of calculators consider that the F. Strassen algorithm is the most applicable among algorithms accelerated in a comparison with the standard one.
II wn
Assume that elements of the matrix V = |pi_/||. ._ , vtj > 0, characterize weights of the graph G
1,1 1
edges and in the model of the random graph G„ the probability pi7~fi7h, h ^ 0. Then it is not complicated to obtain that the probability of the cycle existence in the graph G„ satisfies the relation
P(S)~L—,. h^0. And the
matrix VD is calculated similar to the matrix AD by 0(s(n)lnn)
products. REFERENCES
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