Soft Heaps Simplified

In 1998, Chazelle [J. ACM, 47 (2000), pp. 1012--1027] introduced a new kind of meldable heap (priority queue) called the soft heap. Soft heaps trade accuracy for speed: the heap operations are allowed to increase the keys of certain items, thereby making these items bad, as long as the number of bad items in the data structure is at most $\varepsilon m$, where $m$ is the total number of insertions performed so far, and $\varepsilon$ is an error parameter. The amortized time per heap operation is $O(\lg \frac{1}{\varepsilon})$, reduced from $O(\lg n)$, where $n$ is the number of items in the heap. Chazelle used soft heaps in several applications, including a faster deterministic minimum-spanning-tree algorithm and a new deterministic linear-time selection algorithm. We give a simplified implementation of soft heaps that uses less space and avoids Chazelle's dismantling operations. We also give a simpler, improved analysis that yields an amortized time bound of $O(\lg \frac{1}{\varepsilon})$ for each deleti...