Smallest circuit for square of sparse symmetric matrix

Question

I have an $n \times n$ symmetric matrix, and I would like to compute its square in as small a circuit complexity as possible. It's sparse: there are $\sqrt{n}$ nonzero entries in each row/column, so the input will be sparse by the output will be nearly a density of $1$. A normal sparse multiplication algorithm would take $O (n^2)$ time which is clearly optimal, but the circuit depth for any sparse representation I can think of is at least $O (n^3)$. (The act of "accessing a given location in memory" is not efficient in circuits.)

Using a fast matrix multiplication method like Coppersmith-Winograd (C-W) would bring me to $O \left( n^{\sim 2.4} \right)$. I know I can use the symmetry to reduce by a constant factor, but I figure not more than that. I'm hoping that the fact that it's squaring (and not multiplying two different matrices) can reduce the exponent, but I don't know how — I know that it can cut my work down a bit because the first iteration of C-W will have some identical results. I can't find any way to use the sparsity at all.

Numerical precision isn't an issue for me; the input is a 0-1 matrix.

In terms of practical purposes, I particularly care about $n \sim 3000$, but am generally curious about asymptotically good answers too. (Just saying this to explain that very huge constant factors are a big negative for me.) Any references very much appreciated!

Answer 1

If it's really depth that you care about, I think you should be able to do it with a circuit of depth $O(\log^2 n)$ using sorting networks. I'll assume we want to compute the product $AB$, and that $A$ is represented as an array of rows, where each row is a sorted list of non-zero entries; and $B$ is an array of columns, where each column is a sorted list of non-zero entries. Then the $ik$ cell of the product $AB$ (namely, $(AB)_{ik}$) can be computed as follows: copy the entire row $A_{i\cdot}$; copy the entire column $B_{\cdot k}$; concatenate these two lists; sort them based on their indices, using a sorting network, to bring together entries with a common index; at each position where both matrices have an entry, multiply those entries; and sum up those products. You can build a sorting network with depth $O(\log^2 n)$, and the multiply-and-sum step can be done with depth $O(\log n)$ using an addition tree.