Tradeoffs in Depth-Two Superconcentrators

摘要

An N-superconcentrator is a directed graph with N input vertices and iV output vertices and some intermediate vertices, such that for k = 1,2,... ,N, between any set of k input vertices and any set of k output vertices, there are k vertex disjoint paths. In a depth-two N-superconcentrator each edge either connects an input vertex to an intermediate vertex or an intermediate vertex to an output vertex. We consider tradeoffs between the number of edges incident on the input vertices and the number of edges incident on the output vertices in a depth-two N-superconcentrator. For an N-superconcentrator G, let a(G) be the average degree of the input vertices and b(G) be the average degree of the output vertices. Assume that b(G) ≥ a(G). We show that there is a constant k_1 > 0 such that a(G) log ((2b(G))/(a(G))) log b(G) ≥ k_1·log~2 N. We further show a complementary sufficient condition: there is a constant k_2 > 0, such that if some a and b (a ≤ b) satisfy the above inequality with k_1 replaced by k_2, then there is an N-superconcentrator G with a(G) ≤ a and b(G) ≤ b. In particular, these results imply that the minimum size of a depth-two N-superconcentrator is θ (N (log~2N)/(log log N)), which was already known. Our results are motivated by the connection between the size of depth-two superconcentrators and the problem of maintaining the Discrete Fourier Transform (DFT) in the straight-line program model. Our necessary condition implies that in this model, for any solution to the problem of maintaining the DFT of a vector of length N over an algebraically closed field of characteristic 0, if each update is processed using at most d atomic operations (for d ≤ ( (log N)/(log log N ))~2),then at least N~(Ω (1/d~(1/2))) atomic operations are required to process a query, in the worst case. In particular, if each update is to be processed in constant time, then some query takes Ω (N~ε) worst-case time (for some constant ε > 0). Before this work, it was only known that one of these operations requires Ω ((log~2N)/(log log N))time.