The multi-log₂N network architecture is attractive for constructing optical switches, and the related routing algorithms are critical for the operation and efficiency of such switches. Although several routing algorithms have been proposed for multi-log₂N networks, a full performance comparison among them has not been published up to now. Thus, we rectify this omission by providing such a comparison in terms of blocking probability, time complexity, hardware cost and load balancing capability. Notice that the load balance is important for reducing the peak power requirement of a switch, so we also propose in this paper a new routing algorithm for optical multi-log₂N networks to achieve a better load balance.