[Paper Review] Efficient Enumeration of Unidirectional Cuts for Technology Mapping of Boolean Networks

Niranjan Kulkarni, Sarma Vrudhula|arXiv (Cornell University)|Mar 23, 2016

VLSI and FPGA Design Techniques20 references3 citations

TL;DR

This paper proposes an efficient enumeration method for unidirectional (strong line) cuts in Boolean networks, leveraging network flow-based graph pruning to accelerate minimal independent set (MIS) enumeration. The approach enables practical, exhaustive enumeration of k-feasible strong line cuts—equivalent to unidirectional node cuts—without degrading solution quality, significantly expanding the match space beyond minimal node cuts while maintaining low computational overhead.

ABSTRACT

In technology mapping, enumeration of subcircuits or cuts to be replaced by a standard cell is an important step that decides both the quality of the solution and execution speed. In this work, we view cuts as set of edges instead of as set of nodes and based on it, provide a classification of cuts. It is shown that if enumeration is restricted to a subclass of cuts called unidirectional cuts, the quality of solution does not degrade. We also show that such cuts are equivalent to a known class of cuts called strong line cuts first proposed in [14]. We propose an efficient enumeration method based on a novel graph pruning algorithm that utilizes network flow to approximate minimum strong line cut. The runtimes for the proposed enumeration method are shown to be quite practical for enumeration of a large number of cuts.

Motivation & Objective

• To address the structural bias in conventional node-cut-based technology mapping that limits the discovery of high-quality matches.
• To explore the potential of non-minimal, unidirectional cuts (strong line cuts) as an alternative to minimal node cuts in technology mapping.
• To develop a practical, scalable method for enumerating k-feasible strong line cuts in large Boolean networks.
• To demonstrate that strong line cuts preserve solution quality while expanding the search space for optimal cell replacements.
• To enable efficient enumeration of large numbers of strong line cuts through a novel graph pruning technique based on approximate minimum MIS computation.

Proposed method

• Represents cuts as edge sets (line cuts) rather than node sets, enabling a duality between node cuts and line cuts in DAGs.
• Introduces unidirectional cuts (strong line cuts) as a distinct class of cuts that are equivalent to strong line cuts and avoid structural bias of minimal node cuts.
• Applies a network flow-based pruning transformation to the original DAG, reducing the graph size while preserving relevant k-feasible strong line cuts.
• Performs MIS enumeration on the pruned graph G′ instead of the original graph G, significantly reducing enumeration time.
• Uses a recursive MIS enumeration algorithm with on-the-fly size filtering (discarding sets larger than k) to focus only on relevant cuts.
• Employs an approximate minimum MIS computation to guide pruning, ensuring the transformation remains computationally efficient and accurate for practical use.

Experimental results

Research questions

• RQ1 Can unidirectional (strong line) cuts be efficiently enumerated in large Boolean networks while preserving solution quality?
• RQ2 Does restricting enumeration to strong line cuts degrade the quality of technology mapping compared to minimal node cuts?
• RQ3 Can a network flow-based pruning technique significantly reduce the computational cost of enumerating k-feasible strong line cuts?
• RQ4 How does the combination of pruning and MIS enumeration scale across standard benchmark circuits?
• RQ5 To what extent do strong line cuts expand the match space beyond what is reachable via minimal node cuts?

Key findings

• The proposed method enables practical, exhaustive enumeration of strong line cuts across ISCAS benchmarks, with runtimes remaining feasible even for large circuits.
• For k=6 and cone size limited to 300, c1908 circuit produced 2,099,375 cuts after pruning, with enumeration time of 5.02 seconds.
• At k=10 and cone size 100, c1908 generated over 1.3 billion cuts, with enumeration time of 6,510.5 seconds, demonstrating scalability for larger k.
• The pruning step dominates runtime for small k, indicating that the method is effectively polynomial in practice for small k values.
• The method outperforms naive MIS enumeration by orders of magnitude due to effective pruning, making large-scale enumeration feasible.
• Strong line cuts are shown to be equivalent to strong line cuts from prior work, confirming theoretical equivalence and practical utility.