Computational Aspects of Ordered Integer Partitions with Bounds

Abstract

This paper is dedicated to the counting problem of writing an integer number z as a sum of an ordered sequence of n integers from n given intervals, i.e., counting the number of configurations $(z_1,\ldots ,z_n)$ with $z = z_1 + \cdots + z_n$ for $z_i \in [x_i, y_i]$ with integers $x_i$ and $y_i$ and $1 \le i \le n$. We show an algorithm computing this number in $\mathop {}\mathopen {}{\mathcal {O}}\mathopen {}\left( n z^{\lg n}\right)$ average time, and a data structure computing this number in $\mathop {}\mathopen {}{\mathcal {O}}\mathopen {}\left( n\right)$ time, independently of z. The data structure is constructed in $\mathop {}\mathopen {}{\mathcal {O}}\mathopen {}\left( 2^n n^3\right)$ time. Its construction algorithm only depends on the intervals $[x_i,y_i]$ ($1 \le i \le n$). This construction algorithm can be parallelized with $\pi = \mathop {}\mathopen {}{\mathcal {O}}\mathopen {}\left( n^3\right)$ processors, yielding $\mathop {}\mathopen {}{\mathcal {O}}\mathopen {}\left( 2^n \frac{n^3}{\pi }\right)$ construction time with high probability.