Computing discrete geodesic distance over triangle meshes is one of the fundamental problems in computational geometry and computer graphics. In this problem, an effective window pruning strategy can significantly affect the actual running time. Due to its importance, we conduct an in-depth study of window pruning operations in this paper, and produce an exhaustive list of scenarios where one window can make another window partially or completely redundant. To identify a maximal number of redundant windows using such pairwise cross checking, we propose a set of procedures to synchronize local window propagation within the same triangle by simultaneously propagating a collection of windows from one triangle edge to its two opposite edges. On the basis of such synchronized window propagation, we design a new geodesic computation algorithm based on a triangle-oriented region growing scheme. Our geodesic algorithm can remove most of the redundant windows at
the earliest possible stage, thus significantly reducing computational cost and memory usage at later stages. In addition, by adopting triangles instead of windows as the primitive in propagation management, our algorithm significantly cuts down the data management overhead. As a result, it runs 4-15 times faster than MMP and ICH algorithms, 2-4 times faster than FWP-MMP and FWP-CH algorithms, and also incurs the least memory usage.