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A lower bound is presented which shows that a class of heap algorithms in the pointer model with only heap pointers must spend Omega(log log n / log log log n) amortized time on the decrease-key operation (given O(log n) amortized-time extract-min). Intuitively, this bound shows the key to having O(1)-time decrease-key is the ability to sort O(log n) items in O(log n) time; Fibonacci heaps [M.L. Fredman and R. E. Tarjan. J. ACM 34(3):596-615 (1987)] do this through the use of bucket sort. Our<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1302.6641v3">arXiv:1302.6641v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/f63whiaw3fe2fjt6b4gu54mduu">fatcat:f63whiaw3fe2fjt6b4gu54mduu</a> </span>
more »... wer bound also holds no matter how much data is augmented; this is in contrast to the lower bound of Fredman [J. ACM 46(4):473-501 (1999)] who showed a tradeoff between the number of augmented bits and the amortized cost of decrease-key. A new heap data structure, the sort heap, is presented. This heap is a simplification of the heap of Elmasry [SODA 2009: 471-476] and shares with it a O(log log n) amortized-time decrease-key, but with a straightforward implementation such that our lower bound holds. Thus a natural model is presented for a pointer-based heap such that the amortized runtime of a self-adjusting structure and amortized lower asymptotic bounds for decrease-key differ by but a O(log log log n) factor.
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