A linear time algorithm for the feasibility of pebble motion on trees

Abstract

We consider the following generalization of the popular “15 puzzle.” Let T be a tree with n vertices and with k < n distinct pebbles numbered 1, ..., k on distinct vertices. A move consists in transferring a pebble from its current position to an adjacent unoccupied vertex. We ask the following question: Is a given arrangement of pebbles reachable from another?

We present an algorithm that on input two arrangements of k pebbles on a tree with n vertices decides in time O(n) whether the two arrangements are reachable from one another. We also give an algorithm that, on input two reachable configurations, return a sequence of moves that transforms one configuration into the other.

The pebble motion problem on trees has various applications including memory management in distributed systems, robot motion planning, and deflection routing.

Partially supported by Progetto MURST 40%, Algoritmi, Modelli di Calcolo e Strutture Informative. Part of this work was done while the last author was at International Computer Science Institute, Berkeley, CA, USA.