Scale to N×N and count completion edges

Separate spatial size from reduction length

An N×N PE grid produces one N×N output tile. K is the reduction dimension and need not equal N. With N=3 and K=5, A is 3×5, B is 5×3 and C is 3×3. Nine PEs each consume five terms, performing forty-five MACs. N determines wiring and skew; K determines accumulation width, term counting and injection duration.

Use the established convention: clear finishes before E0, inputs are stable before E_t, MAC and registered forwarding occur at that edge, and there is no extra multiplier pipeline. A(i,k) is injected on k+i and travels j hops. B(k,j) is injected on k+j and travels i hops. Both reach PE(i,j) on k+i+j.

The final PE completes after E_(K+2N−3). Counting E0 inclusively gives K+2N−2 advance edges. For N=K=2, the last edge number is three and the count is four. For N=3, K=5, the last number is eight and the count is nine.

Parameterize the links and counters

At column zero, a PE's A input comes from the boundary; elsewhere it comes from its left neighbor's registered A output. The B rule is analogous for the top boundary and neighbor above. Connect valid and job/k tags with the same links, and share clear and advance across the grid.

Require N≥1, K≥1 and AW≥2W. A count storing zero through K needs max(1,ceil(log2(K+1))) bits. A k tag storing zero through K−1 needs max(1,ceil(log2 K)) bits. These widths differ when K is a power of two; avoid zero-width declarations for K=1.

The last boundary injection occurs on E_(K+N−2), followed by N−1 drain edges. Computation counts exclude initialization, memory loading and output transfer. With P pause edges, elapsed physical edges become K+2N−2+P. Extra pipeline stages require a new timing analysis.