Научная статья на тему 'Asymptotic of cycly existence in oriented graph with low reliable edges'

Asymptotic of cycly existence in oriented graph with low reliable edges Текст научной статьи по специальности «Математика»

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a cycle / an oriented graph / an edge / a probability

Аннотация научной статьи по математике, автор научной работы — G. Tsitsiashvili

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 n × n, is constructed.

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Текст научной работы на тему «Asymptotic of cycly existence in oriented graph with low reliable edges»

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

1. Mirzayanov M.R. 2006 . On minimal strongly connected congruencies of a directed path. Izvestiiia Saratovskogo Universiteta. Mathematika. Mekhanika. Informatika. Vol. 6 (1/2) P. 9195 (In Russian).

2. Karmanova E.O. 2012. Congruences of Paths: Some combinatory properties. Prikladnaya diskretnaya mathematika. Vol. 2 (16). P. 86-89. (In Russian).

3. Raigorodskiy A.M. 2010. Models of random graphs and their application. Trudi MFTI. Vol. 2:4. P. 130-140. (In Russian).

4. Cohn H., Kleinberg R., Szegedy B. and Umans Ch. Group-theoretic algorithms for matrix multiplication. Arhiv: math. Gr/0511460. Proceedings of the 46th Annual symposium on foundations of Computer Science, 23-25 October 2005, Pittsburgh, PA, IEEE Computer Society. P 379-388.

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