Evaluating Rational Functions: Infinite Precision Is Finite Cost And Tractable On Average

We consider the following generalization of the familiar ´15-puzzle´ which arises from issues in memory manngrment in distributed systems: Iet G be a graph with n vertices with k < n pebbles numbered 1,...,k on distinct verticcs. A move consistes of transferring a pebble to an adjacent unocccupied vertex. Is one arrangement of the pebbles reachable from another? We present a P-time decision algorithm, and prove matching O(n³) upper and lower bounds on the number of moves required. We have the following subexponential bound for certain unbounded cycles, one of which has prime length p ⩽ 2n/3, and G is primitive, then G = A_n or S_n and has diameter ⩽ 2^{6 [(√(p+4))}n⁸.