An In-Place Priority Queue with O(1) Time for Push and $\lg n + O(1)$ Comparisons for Pop

Abstract

An in-place priority queue is a data structure that is stored in an array, uses constant extra space in addition to the array elements, and supports the operations $top$ ($find$-$min$), $push$ ($insert$), and $pop$ ($delete$-$min$). In this paper we introduce an in-place priority queue, for which $top$ and $push$ take O(1) worst-case time, and $pop$ takes $O(\lg {} n)$ worst-case time and involves at most $\lg {} n + O(1)$ element comparisons, where n denotes the number of elements currently in the data structure. The achieved bounds are optimal to within additive constant terms for the number of element comparisons, hereby solving a long-standing open problem. Compared to binary heaps, we surpass the comparison bound for $pop$ and the time bound for $push$. Our data structure is similar to a binary heap with two crucial differences:

• (1) To improve the comparison bound for $pop$, we reinforce a stronger heap order at the bottom levels of the heap such that the element at any right child is not smaller than that at its left sibling.

• (2) To speed up $push$, we buffer insertions and allow $O(\lg ^2 n)$ nodes to violate heap order in relation to their parents.