A Tight Lower Bound for Decrease-Key in the Pure Heap Model
  We improve the lower bound on the amortized cost of the decrease-key operation in the pure heap model and show that any pure-heap-model heap (that has a \bigoh{\log n} amortized-time extract-min operation) must spend \bigom{\log\log n} amortized time on the decrease-key operation. Our result shows that sort heaps as well as pure-heap variants of numerous other heaps have asymptotically optimal decrease-key operations in the pure heap model. In addition, our improved lower bound matches the lower bound of Fredman [J. ACM 46(4):473-501 (1999)] for pairing heaps [M.L. Fredman, R. Sedgewick, D.D. Sleator, and R.E. Tarjan. Algorithmica 1(1):111-129 (1986)] and surpasses it for pure-heap variants of numerous other heaps with augmented data such as pointer rank-pairing heaps. 
John Iacono, \"Ozg\"ur \"Ozkan
cs.DS