Two new parallel algorithms are presented for the problem of labeling the connected components of a binary image, which is also known as the connected ones problem. The machine model is an SIMD two-dimensional mesh connected computer consisting of an N x N array of processing elements, each containing a single pixel of an N x N image. Both new algorithms use a shrinking operation defined by Levialdi and have time complexities of O(N log N) bit operations, which makes them the fastest local algorithms for the problem. Compared with other approaches having similar or better time complexities, this local approach dramatically simplifies the algorithms and reduces the constants of proportionality by nearly two orders of magnitude, thus making them the first practical algorithms for the problem. The two algorithms differ in the amount of memory required per processing element; the first uses O(N) bits while the second employs a novel compression scheme to reduce the requirement to O(log N) bits.