Computational Aspects of Ordered Integer Partitions with Bounds


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.

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    In particular, we allow negative integer values.


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This work is partly funded by the JSPS KAKENHI Grant Number JP18F18120.

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Correspondence to Dominik Köppl.

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Parts of this work have already been presented at the 12th International Symposium on Experimental Algorithms [10].

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Glück, R., Köppl, D. Computational Aspects of Ordered Integer Partitions with Bounds. Algorithmica 82, 2955–2984 (2020).

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