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-2s2N-e2N, where s2N is the sum of all digits of the binary representation of N and e2N 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$\lceil$lgN$\rceil$-2$\lceil$lgN$\rceil$+1-2s2N-e2N + 5.
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