Related papers: Partition and sum is fast

The joint cumulative distribution function for order statistics arising from several different populations is given in terms of the distribution function of the populations. The computational cost of the formula in the case of two…

As two fundamental problems, graph cuts and graph matching have been investigated over decades, resulting in vast literature in these two topics respectively. However the way of jointly applying and solving graph cuts and matching receives…

We consider the positive divisors of a natural number that do not exceed its square root, to which we refer as the {\it small divisors\/} of the natural number. We determine the asymptotic behavior of the arithmetic function that adds the…

The unit cost model is both convenient and largely realistic for describing integer decision algorithms over (+,*). Additional operations like division with remainder or bitwise conjunction, although equally supported by computing hardware,…

Our main result shows that for every odd natural number n > 1 there is a partition of the rational numbers, x/2n in lowest terms, in the unit interval, the product of whose Gamma values takes the form of powers of 2 and pi.

This article concerns the resolution of impartial combinatorial games, and in particular games that can be split in sums of independent positions. We prove that in order to compute the outcome of a sum of independent positions, it is always…

In this article, we show how the finding the number of partitions of same size of a positive integer show up in caching networks. We present a stochastic model for caching where user requests (represented with positive integers) are a…

The partition functions $P(n,m,p)$, the number of integer partitions of $n$ into exactly $m$ parts with each part at most $p$, and $Q(n,m,p)$, the number of integer partitons of $n$ into exactly $m$ distinct parts with each part at most…

We consider the problem of sequencing a set of positive numbers. We try to find the optimal sequence to maximize the variance of its partial sums. The optimal sequence is shown to have a beautiful structure. It is interesting to note that…

We propose a new method, probabilistic divide-and-conquer, for improving the success probability in rejection sampling. For the example of integer partitions, there is an ideal recursive scheme which improves the rejection cost from…

The integer division of a numerator n by a divisor d gives a quotient q and a remainder r. Optimizing compilers accelerate software by replacing the division of n by d with the division of c * n (or c * n + c) by m for convenient integers c…

Approximating Subset Sum is a classic and fundamental problem in computer science and mathematical optimization. The state-of-the-art approximation scheme for Subset Sum computes a $(1-\varepsilon)$-approximation in time…

In the context of big data analysis, the divide-and-conquer methodology refers to a multiple-step process: first splitting a data set into several smaller ones; then analyzing each set separately; finally combining results from each…

Let A be a nonempty finite set of relatively prime positive integers, and let p_A(n) denote the number of partitions of n with parts in A. An elementary arithmetic argument is used to obtain an asymptotic formula for p_A(n).

Let b $\ge$ 2 be an integer and let s b (n) denote the sum of the digits of the representation of an integer n in base b. For sufficiently large N , one has Card{n $\le$ N : |s 3 (n) -- s 2 (n)| $\le$ 0.1457205 log n} \textgreater{} N…

In this paper, we study the "sum composition problem" between two lists $A$ and $B$ of positive integers. We start by saying that $B$ is "sum composition" of $A$ when there exists an ordered $m$-partition $[A_1,\ldots,A_m]$ of $A$ where $m$…

In the study of random access machines (RAMs) it has been shown that the availability of an extra input integer, having no special properties other than being sufficiently large, is enough to reduce the computational complexity of some…

We propose two algorithms to factor numbers using Gauss sums and entanglement: (i) in a Shor-like algorithm we encode the standard Gauss sum in one of two entangled states and (ii) in an interference algorithm we create a superposition of…

We derived the sum identities for generalized harmonic and corresponding oscillatory numbers for which a sieve procedure can be applied. The obtained results enable us to understand better the properties of these numbers and their…

We consider space-saving versions of several important operations on univariate polynomials, namely power series inversion and division, division with remainder, multi-point evaluation, and interpolation. Now-classical results show that…