Distributed scheduling algorithms for cell-based Virtual Output Queuing (VOQ) switches have led to high-performance schedulers. However, existing algorithms suffer from low service guarantees and a need for several iterations to provide high throughput. In this paper, we introduce the mutual priority concept, a new mechanism that addresses these issues. We prove that mutual priority algorithms achieve optimal service guarantee (N in an N × N switch) and lead to packet delays that outperform all existing algorithms with significant improvements: the delay achieved by the most efficient mutual priority algorithm is 4.5 times lower for high loads (i.e., reduced by 78%) than the delay achieved by the best existing scheduler for a single iteration execution, while for log N iterations the improved delay is 2.5 times lower (i.e., reduced by 60%). Importantly, mutual priority schedulers provide significantly high performance with a single iteration, outperforming all alternatives and achieving performance comparable to that with multiple Iterations.