XBDD: A Highly Optimized ROBDD with Per-Edge Variable-Flip Maps

2026-09-29

AI Summary

This study addresses the node explosion problem in traditional Reduced Ordered Binary Decision Diagrams (ROBDDs) caused by neglecting local polarity differences, and proposes a novel structure termed XBDD. Its core innovation lies in introducing the first per-edge variable flipping mapping mechanism, which enables polarity-aware node sharing to merge redundant nodes that differ solely in polarity. From an engineering perspective, bitmaps, mapping pools, normalized operators, and complemented edges are integrated to optimize space and time overhead. Experimental results demonstrate that XBDD achieves exponential compression in specific scenarios, trading controllable computational cost for substantial memory savings, thereby effectively validating the superiority of the proposed mechanism.

Abstract

The Reduced Ordered Binary Decision Diagram (ROBDD) is a canonical representation of Boolean functions and is widely used in tasks such as equivalence checking and satisfiability checking of combinational circuits. Classical ROBDD packages greatly improve the efficiency of building ROBDDs through a series of optimization techniques, and compress the node scale of the ROBDD through complement edges. However, existing implementations do not take into account the local polarity differences of isomorphic Boolean functions, and still produce a distinct node for each polarity combination, thereby causing an explosion in the number of nodes. This paper proposes XBDD, a highly optimized ROBDD that, on the basis of fully implementing complement edges and their accompanying engineering techniques, introduces a per-edge variable-flip map. XBDD attaches a flip map to each edge to indicate which input variables must be negated when that edge is followed. This allows nodes that differ only in local input polarities to be merged, further reducing the node count. For certain function families, this sharing even yields exponential compression. We also propose methods that use a bitmap and a map pool to substantially reduce the extra overhead brought by the map, and propose normalization and cofactor operators for the map. In addition, XBDD implements several other engineering optimizations to further improve both time and space efficiency. Experiments show that XBDD trades a controllable time cost for a significant space gain, validating the effectiveness of the per-edge variable-flip map.

Problem

• ROBDD
• Boolean functions
• node explosion
• local polarity
• isomorphic subgraphs

Innovation

• ROBDD
• Variable-Flip Map
• Boolean Function Compression
• Complement Edges
• Bitmap Optimization