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A lower bound is presented which shows that a class of heap algorithms in thepointer model with only heap pointers must spend Omega(log log n / log log logn) amortized time on the decrease-key operation (given O(log n) amortized-timeextract-min). Intuitively, this bound shows the key to having O(1)-timedecrease-key is the ability to sort O(log n) items in O(log n) time; Fibonacciheaps [M.L. Fredman and R. E. Tarjan. J. ACM 34(3):596-615 (1987)] do thisthrough the use of bucket sort. Our lower bound also holds no matter how muchdata is augmented; this is in contrast to the lower bound of Fredman [J. ACM46(4):473-501 (1999)] who showed a tradeoff between the number of augmentedbits and the amortized cost of decrease-key. A new heap data structure, thesort 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-timedecrease-key, but with a straightforward implementation such that our lowerbound holds. Thus a natural model is presented for a pointer-based heap suchthat the amortized runtime of a self-adjusting structure and amortized lowerasymptotic bounds for decrease-key differ by but a O(log log log n) factor.

Why some heaps support constant-amortized-time decrease-key operations, and others do not