A Linear-Time Algorithm for the Feasibility of Pebble Motion on Trees

Abstract

We consider the following pebble motion problem. We are given a tree T with n vertices and two arrangements R and S of k < n distinct pebbles numbered 1, . . . , k on distinct vertices of the tree. Pebbles can move along edges of T provided that at any given time at most one pebble is traveling along an edge and each vertex of T contains at most one pebble. We are asked the following question:Is arrangement S reachable from R?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, returns 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.

Graph-Based Multi Robot Motion Planning: Feasibility and Structural Properties

“…To date, the notion of graph solvability has been essentially dependent to the initial and final configurations (situations) of the moving objects. For instance, the question whether a tree-like graph is solvable for a given initial and final configuration of pebbles is solvable or not is addressed in (Auletta et al, 1999). However, no work exists in the literature for all types of graphs, and never has the problem of deciding if a graph is always solvable for a specific number of robots for any initial and final configuration been mentioned or addressed.…”

Motion Planning and Reconfiguration for Systems of Multiple Objects

“…Ratner and Warmuth have shown that finding a solution with minimum number of moves for the (N x N) extension of the 15-puzzle is intractable [30], so the reconfiguration problem in graphs with labeled chips is NP-hard. Auletta et al gave a linear time algorithm for the pebble motion on a tree [3]. This problem is the labeled variant of the same reconfiguration problem studied in [13], however each move is along one edge only.…”

Reconfigurations in Graphs and Grids

“…Auletta et al gave a linear time algorithm for the pebble motion on a tree [5]. This problem is the labeled variant of the same reconfiguration problem we study here, however each move is along one edge only.…”

“…In another application, the chips are indivisible packets (copies) of the same data that need to be moved from one site to another of a wide-area communication network without ever exceeding the capacities of the communication buffers at each site [5,16].…”