DiffNodesets: An Efficient Structure for Fast Mining Frequent Itemsets

Mining frequent itemsets is an essential problem in data mining and plays an important role in many data mining applications. In recent years, some itemset representations based on node sets have been proposed, which have shown to be very efficient for mining frequent itemsets. In this paper, we propose DiffNodeset, a novel and more efficient itemset representation, for mining frequent itemsets. Based on the DiffNodeset structure, we present an efficient algorithm, named dFIN, to mining frequent itemsets. To achieve high efficiency, dFIN finds frequent itemsets using a set-enumeration tree with a hybrid search strategy and directly enumerates frequent itemsets without candidate generation under some case. For evaluating the performance of dFIN, we have conduct extensive experiments to compare it against with existing leading algorithms on a variety of real and synthetic datasets. The experimental results show that dFIN is significantly faster than these leading algorithms.Comment: 22 pages, 13 figure

Congruences concerning Legendre polynomials

Let $p$ be an odd prime. In the paper, by using the properties of Legendre polynomials we prove some congruences for $\sum_{k=0}^{\frac{p-1}2}\binom{2k}k^2m^{-k}\mod {p^2}$. In particular, we confirm several conjectures of Z.W. Sun. We also pose 13 conjectures on supercongruences.Comment: 16 page

Generalized Legendre polynomials and related congruences modulo p2p^2

For any positive integer $n$ and variables $a$ and $x$ we define the generalized Legendre polynomial P_n(a,x)=\sum_{k=0}^n\b ak\b{-1-a}k(\frac{1-x}2)^k. Let $p$ be an odd prime. In the paper we prove many congruences modulo $p^2$ related to $P_{p-1}(a,x)$. For example, we show that P_{p-1}(a,x)\e (-1)^{_p}P_{p-1}(a,-x)\mod {p^2}, where $_p$ is the least nonnegative residue of $a$ modulo $p$. We also generalize some congruences of Zhi-Wei Sun, and determine $\sum_{k=0}^{p-1}\binom{2k}k\binom{3k}k{54^{-k}}$ and $\sum_{k=0}^{p-1}\binom ak\binom{b-a}k\mod {p^2}$, where $[x]$ is the greatest integer function. Finally we pose some supercongruences modulo $p^2$ concerning binary quadratic forms.Comment: 37 page