A Smart Backtracking Algorithm for Computing Set Partitions with Parts of Certain Sizes

TL;DR

This paper introduces an efficient backtracking algorithm for generating all set partitions with parts of certain sizes, significantly reducing the computational complexity compared to naive methods.

Contribution

The paper presents a novel backtracking algorithm that computes all -partitions of a set in linear time, improving over the exponential naive approach.

Findings

Algorithm computes all -partitions in (n) time

Naive methods require (nB_n) time, where B_n is the Bell number

Efficient enumeration of set partitions with size constraints

Abstract

Let $\alpha=\{a_1,a_2,a_3,...,a_n\}$ be a set of elements, $\delta < n$ be a non-negative integer, and $\Gamma: \alpha \to \{0, 1, 2, ..., n\}$ be a total mapping. Then, we call $\Gamma$ a \emph{partition} of $\alpha$ if and only if for all $x \in \alpha$, $\Gamma(x) \neq 0$. Further, we call $\Gamma$ a $\delta$-\emph{partition} of $\alpha$ if and only if $\Gamma$ is a partition of $\alpha$ and for all $i \in \{1, 2, 3, ..., n\}$, $|\{x: \Gamma(x)=i\}| > \delta$. We give a non-trivial algorithm that computes all $\delta$-partitions of $\alpha$ in $\Omega(n)$ time. On the opposite, a naive generate-and-test algorithm would compute all $\delta$-partitions of $\alpha$ in $\Omega(nB_n)$ time where $B_n$ is the Bell number.