A Tight Lower Bound for Decrease-Key in the Pure Heap Model

TL;DR

This paper establishes a new lower bound on the decrease-key operation's cost in the pure heap model, showing it must be at least logarithm of logarithm of n, which matches or surpasses previous bounds for various heap types.

Contribution

It improves the lower bound for decrease-key in the pure heap model, demonstrating asymptotic optimality of certain heap variants and surpassing previous bounds for others.

Findings

Lower bound of Ω(log log n) for decrease-key in pure heap model.

Pure-heap variants of heaps are asymptotically optimal for decrease-key.

The bound matches or exceeds previous bounds for specific heap algorithms.

Abstract

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.