# Depth Distribution in High Dimensions

## TL;DR

This paper introduces an algorithm to compute the Depth Distribution of axis-aligned boxes in high dimensions, generalizing existing measures like Klee's Measure and Maximum Depth, with improved complexity bounds.

## Contribution

It presents a new algorithm for depth distribution computation in high-dimensional spaces, extending previous methods for Klee's Measure and Maximum Depth.

## Key findings

- Algorithm complexity within O(n^{(d+1)/2} log n)
- Space complexity within O(n log n)
- Refined results based on input difficulty measures

## Abstract

Motivated by the analysis of range queries in databases, we introduce the computation of the Depth Distribution of a set $\mathcal{B}$ of axis aligned boxes, whose computation generalizes that of the Klee's Measure and of the Maximum Depth. In the worst case over instances of fixed input size $n$, we describe an algorithm of complexity within $O({n^\frac{d+1}{2}\log n})$, using space within $O({n\log n})$, mixing two techniques previously used to compute the Klee's Measure. We refine this result and previous results on the Klee's Measure and the Maximum Depth for various measures of difficulty of the input, such as the profile of the input and the degeneracy of the intersection graph formed by the boxes.