Cybernetics and Control Theory

I. E. Maizlin

On One Method of Information Retrieval and Its Application in Implementing on a Computer an Algorithm for Finding the Critical Path

(Presented by Academician A. I. Berg on 27 III 1964)

1. Suppose there are \(n\) mutually distinct \(k\)-digit binary codes
\(X_0, X_1, \ldots, X_{n-1}\). To each code \(X_i\) \((i = 0, 1, \ldots, n - 1)\) we assign a “pseudonumber” — a number \(N_i^*\) satisfying the following conditions:

a) \(N_0^*, N_1^*, \ldots, N_{n-1}^*\) are independent random variables, each of which assumes with equal probability any integer value on the interval \([0, n - 1]\).

b) The value \(N_i^*\) is uniquely determined by the code \(X_i\) \((i = 0, 1, \ldots, n - 1)\).

The following process of obtaining \(N_i^*\), while satisfying condition b), ensures with sufficient accuracy the fulfillment of condition a). Let \(y_0, y_1, \ldots, y_{r-1}\) be a sample from a set of numbers distributed uniformly on the interval \([0, 1]\), \(r = 2^m\), \(m = k/q\), where \(q\) is an integer. We divide the \(k\)-digit code \(X_i\) \((i = 0, 1, \ldots, n - 1)\) into \(q\) \(m\)-digit codes \(x_i^1, x_i^2, \ldots, x_i^q\), and, considering \(x_i^j\) \((j = 1, \ldots, q)\) as an integer, set

\[ N_i^* = \left[ n \left\{ \sum_{j=1}^{q} y_{x_i^j} \right\} \right], \tag{1} \]

where \(\{ \}\) denotes the fractional part of a number, and \([ \ ]\) denotes the integer part.

Suppose that in the sequence \(N_0^*, N_1^*, \ldots, N_{n-1}^*\) there are \(\nu\) distinct numbers
\(N_{\alpha_1}^*, N_{\alpha_2}^*, \ldots, N_{\alpha_\nu}^*\), and that the pseudonumber \(N_{\alpha_j}^*\) \((j = 1, \ldots, \nu)\) is possessed by \(\beta_j\) codes of the sequence \(X_0, X_1, \ldots, X_{n-1}\). The quantity

\[ \beta = \frac{1}{n} \sum_{j=1}^{\nu} \beta_j^2 \]

may naturally be called the average number of codes that have received the same pseudonumber; \(\beta\) is a random variable.

We shall show that \(M\beta = 2 - 1/n\). Indeed:

\[ \beta = \frac{1}{n} \sum_{j=1}^{\nu} \beta_j^2 = \frac{1}{n} \sum_{l=0}^{n-1} \gamma_l^2, \]

where \(\gamma_l\) \((l = 0, 1, \ldots, n - 1)\) is the number of elements of the sequence
\(N_0^*, N_1^*, \ldots, N_{n-1}^*\) equal to \(l\); \(\gamma_l\) is a random variable with the binomial distribution \(\left(p = \frac{1}{n}\right.\) by virtue of condition a)\(\left.\right)\). Therefore

\[ M\beta = \frac{1}{n} \sum_{l=0}^{n-1} M\gamma_l^2 = \frac{1}{n} \sum_{l=0}^{n-1} \left[ D\gamma_l + (M\gamma_l)^2 \right] = \frac{1}{n} \sum_{l=0}^{n-1} \left[ n \frac{1}{n}\left(1 - \frac{1}{n}\right) + \left(n \frac{1}{n}\right)^2 \right] = 2 - \frac{1}{n}. \]

Let us formulate the information problem. There are \(n\) mutually distinct \(k\)-digit binary codes \(X_0, X_1, \ldots, X_{n-1}\), arranged consecutively in the memory of a computer, with \(n \leqslant 2^k\). It is required to find the numbers of those codes

sets \(\{X_i\}\) \((i=0,1,\ldots,n-1)\), which are equal to the given codes \(Y_1,\ldots,Y_\theta\). The proposed algorithm for solving this problem consists of two parts (let us call them the \(A\)-algorithm and the \(B\)-algorithm).

The \(A\)-algorithm processes information about the codes \(X_0, X_1,\ldots,X_{n-1}\). For its operation \(2n\) auxiliary quantities \(u_0,u_1,\ldots,u_{n-1}\) and \(p_0,p_1,\ldots,p_{n-1}\) are needed, each of which will take values from \(0\) to \(n-1\). Initially \(u_0=u_1=\cdots=u_{n-1}=0\), and \(p_i=N_i^*\) \((i=0,1,\ldots,n-1)\), where \(N_i^*\) is the pseudonumber of the code \(X_i\), obtained by formula (1). Next a recurrent process is carried out, at whose \((n-k)\)-th step \((n-k=1,2,\ldots,n)\) \(p_k\) is equated to \(u_{N_k^*}\), and the new value of \(u_{N_k^*}\) is set equal to \(k\). Here \(p_k\) is the number of the nearest code with pseudonumber equal to \(N_k^*\). The equality \(p_k=0\) means that \(N_j^*\ne N_k^*\) \((j>k)\). In implementing the \(A\)-algorithm on a computer, \(cn\) (\(c\) is a constant number) machine operations are required.

The \(B\)-algorithm solves the stated problem successively for each code \(Y_j\) \((j=1,\ldots,\theta)\). For the code \(Y_j\), by formula (1) we determine the pseudonumber \(N^*(Y_j)\). We compare the code \(X_{z_1}\), where \(z_1=u_{N^*(Y_j)}\), with the code \(Y_j\). If \(X_{z_1}=Y_j\), then the required number is \(z_1\). If \(X_{z_1}\ne Y_j\), we compare the code \(X_{z_2}\), where \(z_2=p_{z_1}\), with the code \(Y_j\), and so on until the equality \(X_{z_k}=Y_j\) \((z_k=p_{z_{k-1}}\), if \(k>1)\) is satisfied, from which we conclude that the required number is \(z_k\). The number of operations in implementing the \(B\)-algorithm on a computer turns out to be equal to \(c\theta\beta\), where \(c\) is a constant and \(\beta\) is the random variable defined above, i.e.
\[ M[c\theta\beta]=c\theta\left(2-\frac{1}{n}\right)<c_1\theta . \]

Let us note that the \(B\)-algorithm realizes Johnson’s proposed \((^3)\) method of indirect chained search for information in a card file.

1. The problem of finding the time characteristics of a plan in the planning of the development of large systems by the critical path method (pert-time) is solved. The statement of the problem, the reduction to logical networks, and some of the terms used in the note are described in detail in \((^1)\). In the algorithm described below, in contrast to \((^1)\), the initial information is information about activities, not about events. This eliminates the renumbering process, which requires either a large number of machine operations or substantial restrictions on the independence of information about different parts of the system.

We introduce two “fictitious” activities of zero duration \(d_{\alpha_0}\) and \(d_{\hat{\alpha}}\). Completion of activity \(d_{\alpha_0}\) corresponds to the event “beginning of activities,” and for the beginning of activity \(d_{\hat{\alpha}}\) the occurrence of the event “end of activities” is necessary.

In the information about activity \(d_\alpha\) its code \(X_\alpha\), duration \(l_\alpha\), and the number \(k_\alpha\) of activities whose completion is necessary for the beginning of activity \(d_\alpha\), as well as their codes \(X_{i_1},\ldots,X_{i_{k_\alpha}}\), are specified. The totality of the codes \(X_{i_1},\ldots,X_{i_{k_\alpha}}\) for all \(\alpha\) will be called the \(I\)-information.

The process of finding the critical activities is divided into a forward-code algorithm and a reverse-code algorithm. In the forward-code algorithm we associate with each activity \(d_\alpha\), in addition to its duration \(l_\alpha\), three more quantities \(\pi_\alpha\), \(q_\alpha\), and \(t_\alpha\), whose values will change in the course of the algorithm. Initially \(t_\alpha=0\), \(q_\alpha=k_\alpha\), and \(\pi_\alpha=0\) for all activities except \(d_{\alpha_0}\), for which \(\pi_{\alpha_0}=1\). After applying the \(A\)-algorithm to the \(I\)-information, we pass to the forward-code algorithm, one cycle of which is as follows. We take any activity \(d_\beta\) for which \(\pi_\beta=1\). We consider all activities \(d_{\beta_1},\ldots,d_{\beta_p}\) for whose start the completion of activity \(d_\beta\) is necessary. To find these activities, the \(B\)-algorithm is applied \(p\) times over the \(I\)-information. For all \(j=1,\ldots,p\), if \(t_{\beta_j}<t_\beta\), we set \(t_{\beta_j}=t_\beta\). Then we decrease \(k_{\beta_j}\) \((j=1,\ldots,p)\) by one and, if \(k_{\beta_j}\) has become zero, increase \(t_{\beta_j}\) by \(l_{\beta_j}\) and set \(\pi_{\beta_j}=1\). After this we set \(\pi_\beta=2\) and begin the next cycle by considering the next activity \(d_\gamma\), for whi-

\(\pi_\gamma = 1\). Under the assumption that the initial logical network is a graph without circuits, with a single initial and a single terminal vertex\({}^{2}\), it is not difficult to show that, if \(\pi_{\hat{\alpha}} \ne 1\), there will be found an operation \(d_\gamma\) with \(\pi_\gamma = 1\). If \(\pi_{\hat{\alpha}} = 1\), then for all operations \(d_\gamma\) \((\gamma \ne \hat{\alpha})\) we have \(\pi_\gamma = 2\), and \(t_\gamma\) is equal to the earliest completion time of operation \(d_\gamma\). In this case \(t_{\hat{\alpha}}\) is equal to \(L\), the length of the critical path being sought.

Next the backward-pass algorithm is carried out. To each operation \(d_\alpha\) there correspond, besides its duration \(l_\alpha\), three numbers \(P_\alpha\), \(Q_\alpha\), and \(T_\alpha\). First, the initial information about the operations is processed using the \(A\)- and \(B\)-algorithms, as a result of which the information about operation \(d_\alpha\) is supplied with the numbers of the computer memory cells in which information is stored about the operations \(d_{i_1}, \ldots, d_{i_{K_\alpha}}\) needed for the execution of operation \(d_\alpha\), and the number \(K_\alpha\) is determined—the number of operations for whose start the execution of operation \(d_\alpha\) is necessary. Before the algorithm starts, \(T_\alpha = 0\), \(Q_\alpha = K_\alpha\), and \(P_\gamma = 0\) for all operations except \(d_{\hat{\alpha}}\), for which \(P_{\hat{\alpha}} = 1\). One cycle of the backward-pass algorithm consists in the following. We take an operation \(d_\beta\) for which \(P_\beta = 1\). Consider the operations \(d_{\beta_1}, \ldots, d_{\beta_r}\) needed for the start of operation \(d_\beta\). If \(T_{\beta_j} < T_\beta + l_\beta\) \((j = 1, \ldots, r)\), set \(T_{\beta_j} = T_\beta + l_\beta\). Next decrease the value \(Q_{\beta_j}\) \((j = 1, \ldots, r)\) by one and, if \(Q_{\beta_j}\) has become equal to zero, set \(P_{\beta_j} = 1\). Then set \(P_\beta = 2\) and begin a new cycle by considering the next operation \(d_\gamma\) with \(P_\gamma = 1\). Under assumptions analogous to those indicated above, it can be proved that, if \(P_{\alpha_0} \ne 1\), there will be found an operation \(d_\gamma\) with \(P_\gamma = 1\). If \(P_{\alpha_0} = 1\), then for all operations \(d_\alpha\) \((\gamma \ne \alpha_0)\) \(P_\gamma = 2\), and \(L - T_\gamma\) is equal to the latest execution time of operation \(d_\gamma\), provided that the entire complex of operations is executed in time \(L\).

The operations \(d_{i_1}, \ldots, d_{i_g}\), for which \(t_{i_k} = L - T_{i_k}\) \((k = 1, \ldots, g)\), are critical. Any critical path consists of critical operations.

If the size of the computer’s main memory is sufficient to store all the information about the operations, then the total number of operations in the implementation of the described algorithm is a random variable with mathematical expectation less than \(CMn\), where \(C\) is a constant depending on the type of machine, \(n\) is the number of operations in the system, and \(M\) is the greatest degree of a vertex\({}^{2}\) of the corresponding logical network.

A program has been written that implements the described algorithm on a computer. The author expresses deep gratitude to Corresponding Member of the Academy of Sciences of the USSR L. A. Lyusternik for his guidance in carrying out this work.

Received
24 III 1964

CITED LITERATURE

\({}^{1}\) G. S. Pospelov, A. I. Teiman, Izv. AN SSSR, Technical Cybernetics, No. 4, 60 (1963). \({}^{2}\) C. Berge, Theory of Graphs and Its Applications, Moscow, 1962. \({}^{3}\) L. R. Johnson, Commun. Assoc. Computing Machinery, 4, No. 5, 218 (1961).