Deformable image registration is an important tool in medical image analysis. In the case of lung computed tomography (CT) registration there are three major challenges: large motion of small features, sliding motions between organs, and changing image contrast due to compression. Recently, Markov random field (MRF)-based discrete optimization strategies have been proposed to overcome problems involved with continuous optimization for registration, in particular its susceptibility to local minima. However, to date the simplifications made to obtain tractable computational complexity reduced the registration accuracy. We address these challenges and preserve the potentially higher quality of discrete approaches with three novel contributions. First, we use an image-derived minimum spanning tree as a simplified graph structure, which copes well with the complex sliding motion and allows us to find the global optimum very efficiently. Second, a stochastic sampling approach for the similarity cost between images is introduced within a symmetric, diffeomorphic B-spline transformation model with diffusion regularization. The complexity is reduced by orders of magnitude and enables the minimization of much larger label spaces. In addition to the geometric transform labels, hyper-labels are introduced, which represent local intensity variations in this task, and allow for the direct estimation of lung ventilation. We validate the improvements in accuracy and performance on exhale-inhale CT volume pairs using a large number of expert landmarks.