A Tight Amortized Bound for Path Reversal

Let T be a rooted tree. A path reversal at a node x in T is performed by traversing the path from x to the tree root r and making x the parent of each node on the path other than X. Thus x becomes the new tree root. (See Fig. 1). The cost of the reversal is the number of edges on the path reversed. Path reversal is a variant of the standard path compression algorithm for maintaining disjoint sets under union [5]. It has also been used in a novel mutual exclusion algorithm [2,6]. Suppose that a sequence of m reversals is performed on an arbitrary initial n-node tree. What is the total cost of the sequence? Let T(n, m) be the