MPI-I-91-104

Bottom-Up-Heapsort is a variant of Heapsort. Its worst case complexity for the number of comparisons is known to be bounded from above by ${3\over2}n\log n+O(n)$, where $n$ is the number of elements to be sorted. There is also an example of a heap which needs ${5\over4}n\log n- O(n\log\log n)$ comparisons. We show in this paper that the upper bound is asymptotical tight, i.e.~we prove for large $n$ the existence of heaps which need at least $c_n\cdot({3\over2}n\log n-O(n\log\log n))$ comparisons where $c_n=1-{1\over\log^2n}$ converges to 1. This result also proves the old conjecture that the best case for classical Heapsort needs only asymptotical $n\log n+O(n\log\log n)$ comparisons.