Number of Laser Beams in a Bank

Reduce rows to device counts

Scan each binary-string row and count its '1' characters. Empty rows do not participate and do not break the relationship between nonempty rows. For each nonempty row with current devices, add previous * current to the answer, then set previous = current. This works because every device in one nonempty row connects to every device in the next nonempty row, and any device row in between would block a beam.

The invariant is that previous is the count in the latest nonempty row above the current row. Each character is read once, so time is O(total characters) and auxiliary space is O(1). Use a wide enough numeric type if row sizes make the product large. Test all empty rows, one nonempty row, consecutive nonempty rows, multiple empty rows between device rows, and rows with one device. Do not multiply every pair of rows; only adjacent nonempty rows are relevant.