Effect of number representation on the achievable minimum number of operations in multiple constant multiplications

In this work, we analyze the effect of representing constants under binary, CSD, and MSD representations on the minimum number of operations required in a multiple constant multiplications problem. To this end, we resort to a recently proposed algorithm that computes the exact minimum solution. To extend the applicability of this algorithm to much larger instances, we propose problem reduction and model simplification techniques that significantly reduce the search space. We have conducted experiments on a rich set of instances including randomly generated and FIR filter instances. The results show that, contrary to common belief, the binary representation clearly yields better solutions than CSD, and even provides slightly better solutions than MSD. Moreover, the superiority of the binary solutions increases as the number and bit-width of the constants increase.