In this paper we study sweep coverage path planning, in which a robot must cover all points in a workspace with its footprint. In many coverage applications, including cleaning and monitoring, it is beneficial to use coverage paths with minimal robot turns. To address this, we provide an efficient method to compute the minimum altitude of a non-convex polygonal region, which captures the number of parallel line segments, and thus turns, needed to cover the region. Then, given a non-convex polygon, we provide a method to cut the polygon into two pieces that minimizes the sum of their altitudes. Given an initial convex decomposition of a workspace, we apply this method to iteratively re-optimize and delete cuts of the decomposition. Finally, we compute a coverage path of the workspace by placing parallel line segments in each region, and then computing a tour of the segments of minimum cost. We present simulation results on several workspaces with obstacles, which demonstrate improvements in both the number of turns in the final coverage path and runtime.