A Direct-Sum Theorem for Read-Once Branching Programs

We study a direct-sum question for read-once branching programs. If M(f) denotes the minimum average memory required to compute a function f(x1, x2, . . . , xn) how much memory is required to compute f on k independent inputs that arrive in parallel? We show that when the inputs are sampled independently from some domain X and M(f) = Ω(n), then computing the value of f on k streams requires average memory at least Ω ( k · M(f) n ) . Our results are obtained by defining new ways to measure the information complexity of read-once branching programs. We define two such measures: the transitional and cumulative information content. We prove that any read-once branching program with transitional information content I can be simulated using average memory O(n(I + 1)). On the other hand, if every read-once branching program with cumulative information content I can be simulated with average memory O(I+ 1), then computing f on k inputs requires average memory at least Ω(k · (M(f)− 1)).