A Deterministic Algorithm for Testing the Equivalence of Read-Once Branching Programs with Small Discrepancy

Abstract

The problem to test the equivalence of two given read-once branching programs is a well-known problem in the class BPP that is not known to be solvable in deterministic polynomial time. The standard probabilistic algorithm to solve the problem reduces it to an instance of Polynomial Identity Testing and then applies the Schwartz-Zippel Lemma to test the equivalence. This method needs \(O(n\log n)\) random bits, where n is the number of variables in the branching programs. We provide a new method for testing the equivalence of read-once branching programs that uses \(O(\log n +\log |D|)\) random bits, where D is the set of assignments for which the two branching programs compute different results. This means O(n) random bits in the worst case and a deterministic polynomial time algorithm when the discrepancy set D is at most polynomial.