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

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.