A Non-Oblivious Reduction of Counting Ones to Multiplication

An algorithm counting the number of ones in a binary word is presented running in time $O(\log\log b)$ where $b$ is the number of ones. The operations available include bit-wise logical operations and multiplication.

💡 Research Summary

The paper addresses the classic problem of counting the number of set bits (population count) in a binary word x of length n. While the naïve approach scans all n bits, more sophisticated methods have been developed. Wegner’s technique removes the least‑significant 1 in each iteration using the expression x & (x‑1), achieving O(b) time where b = ν(x) is the number of ones. This method is “non‑oblivious” because its runtime depends on the actual input. In contrast, the Gillies‑Miller algorithm and related techniques achieve O(log log n) time but are oblivious: the number of steps is independent of b, so a sparse input does not enjoy any speed‑up.

The contribution of the paper is a hybrid algorithm that retains the non‑oblivious property while matching the asymptotic speed of the Gillies‑Miller method. Specifically, it shows that counting ones can be performed in O(log log b) steps under a unit‑cost model that includes bitwise logical operations and integer multiplication. The algorithm is a modification of the Gillies‑Miller reduction, using a family of “magic masks” m