Mackworth and Freuder have analyzed the time complexity of several constraint satisfaction algorithms [5]. We present here new algorithms for arc and path consistency and show that the arc consistency algorithm is optimal in time complexity and of the same-order space complexity as the earlier algorithms. A refined solution for the path consistency problem is proposed. However, the space complexity of the path consistency algorithm makes it practicable only for small problems. These algorithms are the result of the synthesis techniques used in Image (a general constraint satisfaction system) and local consistency methods [3].