Optimal Pebble Motion on a Tree

In this paper we consider the following pebble coordination problem. Consider a tree with n vertices and k pebbles located at distinct vertices of the tree. Each pebble can be moved from its current position to an adjacent unoccupied vertex. Among the k pebbles, one distinguished pebbles has been assigned a destination. We give an O(n5) algorithm for the problem of designing the shortest sequence of moves that takes the distinguished pebble from its original position to its destination. Our algorithm improves the running time of the best previously presented algorithm that needed to solve O(n6) min-cost flow problems on graphs of size O(n). Our algorithm does not resort to reduction to flow but is instead based on a novel dynamic programming approach.