In this paper, we propose an efficient, non-iterative method for estimating optical flow. We develop a probabilistic framework that is appropriate for describing the inherent uncertainty in the brightness constraint due to errors in image derivative computation. We separate the flow into two one-dimensional representations and pose the problem of flow estimation as one of solving for the most probable configuration of one-dimensional labels in a Markov Random Fields (MRF) with linear clique potentials. The global optimum for this problem can be efficiently solved for using the max-flow computation in a graph. We develop this formulation and describe how the use of the probabilistic framework, the parametrization and the MRF formulation together enables us to capture the desirable properties for flow estimation, especially preserving motion discontinuities. We demonstrate the performance of our algorithm and compare our results with that of other algorithms described in the performance evaluation paper of Barron et. al [2].