Search Results

We bound the minimum number w of wires needed to compute any (asymptotically good) error-correcting code C:{0,1}^Ω(n) -> {0,1}ⁿ with minimum distance Ω(n), using unbounded fan-in circuits of depth d with arbitrary gates. Our main results are: (1) If d=2 ...

We prove superlinear lower bounds for the number of edges in constant depth circuits with n inputs and up to n outputs. Our lower bounds are proved for all types of constant depth circuits, e.g., constant depth arithmetic circuits and constant depth ...

We show that the size of the smallest depth-two $N$-superconcentrator is $$ \Theta(N\log^2 N/\log\log N). $$ Before this work, optimal bounds were known for all depths except two. For the upper bound, we build superconcentrators by putting together a ...

The chaining problem is defined as follows. Given values $a_{1},\ldots,a_{n},\,a_{i} = 0$ or 1, $1 \leq i \leq n$, compute $b_{1},\ldots,b_{n}$ such that $b_{i} = \max\{j \mid a_{j} = 1,\, j < i \}$. (Define $\max\{\} = 0.$) The chaining problem ...