Spiral hashing is a well known dynamic hashing algorithm. Traditional analysis of this search algorithm has been proposed under the assumption that all keys are uniformly accessed. In this paper, we present a discrete analysis of the average search cost in consideration of the frequency of access on each key for this spiral hashing algorithm. In the proposed discrete analysis, the number of probes itself is regarded as a random variable and its probability distribution is derived concretely. The evaluate formulae derived from the proposed analysis can exactly evaluate the average and variance of the search cost in conformity with any probability distribution of the frequency of access.

Mathematical analysis of the average behavior of the AVL balanced tree insertion algorithm has not ever been done completely. As the first step toward this analysis, we have proposed an approximate analysis based on the assumption that all AVL balanced trees with a given number of nodes and height are constructed with equal probability. In this paper, we propose a new analysis of the AVL balanced tree insertion algorithm in conformity with that all n! permutations of n keys occur with equal probability. First we derive the formulae to enumerate the number of permutations constructing the AVL balanced trees with a given number of nodes and height. Then, we propose the formulae to evaluate the average behavior of the AVL balanced tree insertion algorithm and show some results from the proposed formulae.

The linear hashing is a well-known dynamic hashing algorithm designed for internal main memory as well as external secondary memory. Traditional analysis of this search algorithm has been proposed under the assumption that all keys are uniformly accessed. In this paper, we present a discrete analysis of the average search cost of the linear dynamic hashing algorithm for internal main memory in consideration of the frequency of access on each key. In the proposed discrete analysis, the number of probes itself is regarded as a random variable and its probability distribution is derived concretely. Furthermore, the evaluate formula derived from the proposed analysis can exactly evaluate the average search cost in conformity with any probability distribution of the frequency of access. The proposed analysis is compared to the traditional one provided that the frequency of access on each key is uniform, and the differences are discussed.

In this paper we discuss a neural network model that can recognize patterns rotated at various angles. The model employs copy learning, a learning method entirely different from those used in conventional models. Copy-Learning is an effective learning method to attain the desired objective in a short period of time by making a copy of the result of basic learning through the application of certain rules. Our model using this method is capable of recognizing patterns rotated at various angles without requiring mathematical preprocessing. It involves two processes: first, it learns only the standard patterns by using part of the network. Then, it copies the result of the learning to the unused part of the network and thereby recognizes unknown input patterns by using all parts of the network. The model has merits over the conventional models in that it substantially reduces the time required for learning and recognition and can also recognize the rotation angle of the input pattern.