Strictly Implicit Priority Queues: On the Number of Moves and Worst-Case Time

TL;DR

This paper introduces two new strictly implicit priority queues with improved move and time complexities, achieving constant amortized or worst-case Insert times while maintaining efficient ExtractMin operations, advancing the theoretical understanding of implicit heaps.

Contribution

The paper presents two novel strictly implicit priority queues with optimal move and time complexities, including the first with worst-case constant Insert time and minimal state information.

Findings

First structure supports amortized O(1) Insert and O(log n) ExtractMin with O(1) moves.

Second structure achieves worst-case O(1) Insert and O(log n) ExtractMin with minimal state.

Advances theoretical bounds for implicit priority queues with move-efficient operations.

Abstract

The binary heap of Williams (1964) is a simple priority queue characterized by only storing an array containing the elements and the number of elements $n$ - here denoted a strictly implicit priority queue. We introduce two new strictly implicit priority queues. The first structure supports amortized $O(1)$ time Insert and $O(\log n)$ time ExtractMin operations, where both operations require amortized $O(1)$ element moves. No previous implicit heap with $O(1)$ time Insert supports both operations with $O(1)$ moves. The second structure supports worst-case $O(1)$ time Insert and $O(\log n)$ time (and moves) ExtractMin operations. Previous results were either amortized or needed $O(\log n)$ bits of additional state information between operations.