# The Logarithmic Funnel Heap: An Efficient Priority Queue For Extremely Large Sets

## TL;DR

This paper introduces the Logarithmic Funnel Heap, an efficient indexed priority queue with near-logarithmic expected operation times suitable for handling extremely large data sets in computational tasks.

## Contribution

It presents a novel heap data structure with expected $O( ext{log}^* n)$ and $O( ext{log} ext{log} n)$ operation times, improving efficiency for large-scale applications.

## Key findings

- Insert, delete, decrease run in expected O(log* n) time.
- Increase operation is conjectured to run in expected O(log log n) time.
- Space and construction time are linear, O(n).

## Abstract

For many data-processing applications, a comprehensive set of efficient operations for the management of priority values is required. Indexed priority queues are particularly promising to satisfy this requirement by design. In this work, we report the design and analysis of an efficient indexed priority queue with a comprehensive set of operations. In particular, $\mathtt{insert}$, $\mathtt{delete}$ and $\mathtt{decrease}$ all run in expected $O(\log^{*}{n})$ time, while $\mathtt{increase}$ is conjectured by means of Monte Carlo simulations to run in expected $O(\log\log{n})$ time. The space complexity as well as the time complexity for the construction of the empty heap data structure is $O(n)$. For certain massive computational problems, such as specific analyses of extremely large graphs and (chemical) simulations, this heap system may exhibit considerable utility.