The Derivation of a Tighter Bound for Top-Down Skew Heaps

In this paper we present and analyze functional programs for a number of priority queue operations. These programs are based upon the top-down skew heaps|a truly elegant data structure|designed by D.D. Sleator and R.E. Tarjan. We show how their potential technique can be used to determine the time complexity of functional programs. This functional approach enables us to derive a potential function leading to tighter bounds for the amortized costs of the priority queue operations. From the improved bounds it follows, for instance, that Skewsort, a simple sorting program using these operations, requires only about 1:44Nlog 2 N comparisons to sort N numbers (in the worst case). 1 Amortized complexity in a functional setting By means of a simple example we explain how the potential technique of Sleator and Tarjan 7] can be used to determine the time complexity of functional programs. In this example lists of zeros and ones are used as binary representations of natural numbers. We denote the empty list by ], and the list with head b and tail s is denoted by b] + + s. We abbreviate b] + + ] to b]. Binary list s represents natural number s] ], where ] ] is the abstraction function deened by ]] ] = 0 and b] + + s] ] = b+22 s] ]. The successor function on the natural numbers is in this representation implemented by program suc: suc: ] = 1] suc:((0] + + s) = 1] + + s suc:((1] + + s) = 0] + + suc:s; where the dot \." denotes function application. Note that suc:s] ] = s] ] + 1. As a suitable cost measure for the evaluation of expressions involving function