Nearly-2-SAT Solutions for Segmented-Channel Routing

Abstract

This paper proposes a nearly-2-satisfiability (SAT) routing solution for segmented-channels in row-based field programmable gate array architectures. Both dogleg-free routing and routing with doglegging are considered here. The constraints for these problems have been expressed as Boolean conjunctive norm form (CNF) clauses and the conjunction of these clauses creates a Boolean function. The formulation ensures that the maximum number of clauses created consist of 2 literals. Hence, it converts the routing problem, which is an optimization problem, to a nearly-2-SAT problem. Since 2-SAT, i.e., checking the SAT of a Boolean CNF function with clauses having maximum 2 literals, is polynomial time computable, a nearly-2-SAT problem will also compute faster than other SAT-based solutions. We have used statistical distributions to experimentally test the efficiency of our technique and compared the performance with existing SAT-based technique for segmented-channel routing. The results show that more than 90% clauses are 2-literal for both routing options in the proposed methodology and around $20 {\times }$ speedup is observed with respect to existing technique.