BDD与FDD中‘path’的含义及重要意义是什么？

Hey there, let’s break down what "path" means in Binary Decision Diagrams (BDD) and Functional Decision Diagrams (FDD), and why it matters—especially tying back to the paper you referenced.

一、What is a "path" in BDD/FDD? #

Both BDD and FDD are graphical tools for representing boolean functions. A path here is a sequence of nodes starting at the root node and ending at one of the terminal nodes (typically labeled 0 or 1). Each step along the path corresponds to assigning a binary value (0 or 1) to a boolean variable as you move from a parent node to its child.

For a quick example: If you have a BDD for the function f(x,y) = x ∨ y, one path would be: root node (x) → child node for x=1 → terminal 1. This path represents all input assignments where x=1 (no matter what y is) that make the function evaluate to 1.

二、Why does the number of paths matter? #

Your paper notes that FDDs have fewer paths than BDDs (even with similar node counts), and links this to the number of prime implicants (pi terms) in a canonical two-level RME (Reduced Multiple-Output Expression) form. Here’s why that’s a big deal:

• Direct link to boolean function structure: Every path ending at the 1 terminal corresponds to a unique set of variable assignments that satisfy the boolean function (i.e., make it return 1). When converting the diagram to a two-level sum-of-products (SOP) form, each of these paths maps directly to a prime implicant. So the total number of paths equals the number of pi terms in that canonical expression.
• Impact on logic synthesis efficiency: Fewer paths mean fewer pi terms, which translates to simpler, more efficient logic circuits. Less hardware is needed, power consumption drops, and the circuit runs faster. This is the key advantage FDDs bring to multilevel logic synthesis: they can represent some boolean functions more compactly in terms of path count, even when node counts are comparable to BDDs.
• Computational complexity: The number of paths also affects how efficiently we can perform operations on BDD/FDDs (like function minimization or composition). Fewer paths mean these operations take less time and computational resources, which is critical when working with large-scale logic designs.

To connect this directly to your paper’s quote: Since the number of paths equals the number of pi terms in the canonical two-level RME, FDDs’ reduced path count means the corresponding two-level logic expression is more concise—leading to better, more optimized circuits in multilevel synthesis.

内容的提问来源于stack exchange，提问作者sirius