A SWAR Approach to Counting Ones

We investigate the complexity of algorithms counting ones in different sets of operations. With addition and logical operations (but no shift) $O({\log }^2(n))$ steps suffice to count ones. Parity can be computed with complexity $O(\log (n))$, which is the same bound as for methods using shift-operations. If multiplication is available, a solution of time complexity $O({\log }^{\ast }(n))$ is possible improving the known bound $O(\log \log (n))$.