Two asymptotic formulae on the k + 1-power free numbers

The main purpose of this paper is to study the distributive properties of k + 1-power free numbers, and give two interesting asymptotic formulae

A Faster Algorithm to Build New Users Similarity List in Neighbourhood-based Collaborative Filtering

Neighbourhood-based Collaborative Filtering (CF) has been applied in the industry for several decades, because of the easy implementation and high recommendation accuracy. As the core of neighbourhood-based CF, the task of dynamically maintaining users' similarity list is challenged by cold-start problem and scalability problem. Recently, several methods are presented on solving the two problems. However, these methods applied an $O(n^2)$ algorithm to compute the similarity list in a special case, where the new users, with enough recommendation data, have the same rating list. To address the problem of large computational cost caused by the special case, we design a faster ($O(\frac{1}{125}n^2)$) algorithm, TwinSearch Algorithm, to avoid computing and sorting the similarity list for the new users repeatedly to save the computational resources. Both theoretical and experimental results show that the TwinSearch Algorithm achieves better running time than the traditional method

An Accuracy-Assured Privacy-Preserving Recommender System for Internet Commerce

Recommender systems, tool for predicting users' potential preferences by computing history data and users' interests, show an increasing importance in various Internet applications such as online shopping. As a well-known recommendation method, neighbourhood-based collaborative filtering has attracted considerable attention recently. The risk of revealing users' private information during the process of filtering has attracted noticeable research interests. Among the current solutions, the probabilistic techniques have shown a powerful privacy preserving effect. When facing $k$ Nearest Neighbour attack, all the existing methods provide no data utility guarantee, for the introduction of global randomness. In this paper, to overcome the problem of recommendation accuracy loss, we propose a novel approach, Partitioned Probabilistic Neighbour Selection, to ensure a required prediction accuracy while maintaining high security against $k$NN attack. We define the sum of $k$ neighbours' similarity as the accuracy metric alpha, the number of user partitions, across which we select the $k$ neighbours, as the security metric beta. We generalise the $k$ Nearest Neighbour attack to beta k Nearest Neighbours attack. Differing from the existing approach that selects neighbours across the entire candidate list randomly, our method selects neighbours from each exclusive partition of size $k$ with a decreasing probability. Theoretical and experimental analysis show that to provide an accuracy-assured recommendation, our Partitioned Probabilistic Neighbour Selection method yields a better trade-off between the recommendation accuracy and system security.Comment: replacement for the previous versio