Elementary Yet Precise Worst-Case Analysis of Floyd's Heap-Construction Program

Abstract

The worst-case behavior of the heap-construction phase of Heapsort escaped mathematically precise characterization by a closed-form formula for almost five decades. This paper offers a proof that the exact number of comparisons of keys performed in the worst case during construction of a heap of size N is: 2N − 2s2(N)− e2(N), where s2(N) is the sum of all digits of the binary representation of N and e2(N) is the exponent of 2 in the prime factorization of N . It allows for derivation of this best-known upper bound on the number of comparisons of Heapsort: (2N − 1)dlgNe − 2dlgNe+1 − 2s2(N)− e2(N) + 5.

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