# Binary Decision Diagrams: from Tree Compaction to Sampling

## TL;DR

This paper introduces a novel enumeration and sampling method for reduced ordered binary decision diagrams (ROBDDs), enabling efficient generation and analysis of these compact Boolean function representations.

## Contribution

It presents a new approach to enumerate ROBDDs without full tree expansion and introduces an unranking and uniform sampling procedure for ROBDDs of fixed size.

## Key findings

- Developed a method for enumerating ROBDDs by size
- Created an unranking procedure for ROBDDs
- Provided a uniform sampler for ROBDDs of given variables and size

## Abstract

Any Boolean function corresponds with a complete full binary decision tree. This tree can in turn be represented in a maximally compact form as a direct acyclic graph where common subtrees are factored and shared, keeping only one copy of each unique subtree. This yields the celebrated and widely used structure called reduced ordered binary decision diagram (ROBDD). We propose to revisit the classical compaction process to give a new way of enumerating ROBDDs of a given size without considering fully expanded trees and the compaction step. Our method also provides an unranking procedure for the set of ROBDDs. As a by-product we get a random uniform and exhaustive sampler for ROBDDs for a given number of variables and size.