Priority Queues with Multiple Time Fingers

TL;DR

This paper introduces a new priority queue data structure supporting various operations with worst-case optimal or near-optimal time complexities, and establishes theoretical equivalences among key distribution-sensitive bounds.

Contribution

It presents a priority queue with worst-case guarantees for all operations, introduces the time-finger property, and proves the equivalence between the working-set property and the unified bound.

Findings

Supports insert and find-min in constant time.

Supports delete and delete-min in logarithmic time based on element order.

Establishes the equivalence between the working-set property and the unified bound.

Abstract

A priority queue is presented that supports the operations insert and find-min in worst-case constant time, and delete and delete-min on element x in worst-case O(lg(min{w_x, q_x}+2)) time, where w_x (respectively q_x) is the number of elements inserted after x (respectively before x) and are still present at the time of the deletion of x. Our priority queue then has both the working-set and the queueish properties, and more strongly it satisfies these properties in the worst-case sense. We also define a new distribution-sensitive property---the time-finger property, which encapsulates and generalizes both the working-set and queueish properties, and present a priority queue that satisfies this property. In addition, we prove a strong implication that the working-set property is equivalent to the unified bound (which is the minimum per operation among the static finger, static optimality, and the working-set bounds). This latter result is of tremendous interest by itself as it had gone unnoticed since the introduction of such bounds by Sleater and Tarjan [JACM 1985]. Accordingly, our priority queue satisfies other distribution-sensitive properties as the static finger, static optimality, and the unified bound.