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(n
³
) 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))}
n
⁸
.