Dynamic Products of Ranks

TL;DR

This paper introduces a data structure that efficiently maintains a set of points and dynamically tracks the point with the minimum or maximum product of ranks in two coordinate orderings, with updates in sublinear time.

Contribution

It presents a novel data structure capable of maintaining the extremal point based on product of ranks under dynamic updates efficiently.

Findings

Supports dynamic updates in O(√n log n) time

Maintains the point with extremal product of ranks efficiently

Applicable to problems involving rank-based point selection

Abstract

We describe a data structure that can maintain a dynamic set of points given by their Cartesian coordinates, and maintain the point whose product of ranks within the two coordinate orderings is minimum or maximum, in time $O(\sqrt{n\log n})$ per update.