Separating Overlapped Intervals on a Line

TL;DR

This paper presents an optimal algorithm for repositioning overlapping intervals on a line to eliminate overlaps with minimal maximum movement, achieving $O(n ext{log} n)$ time complexity.

Contribution

It introduces a novel approach that reduces the problem to considering at most n candidate interval orderings, enabling an efficient optimal solution.

Findings

Algorithm computes optimal repositioning in $O(n ext{log} n)$ time.

Proves an $ ext{Omega}(n ext{log} n)$ lower bound, confirming optimality.

Uses pruning techniques to handle candidate lists efficiently.

Abstract

Given $n$ intervals on a line $\ell$, we consider the problem of moving these intervals on $\ell$ such that no two intervals overlap and the maximum moving distance of the intervals is minimized. The difficulty for solving the problem lies in determining the order of the intervals in an optimal solution. By interesting observations, we show that it is sufficient to consider at most $n$ "candidate" lists of ordered intervals. Further, although explicitly maintaining these lists takes $\Omega(n^2)$ time and space, by more observations and a pruning technique, we present an algorithm that can compute an optimal solution in $O(n\log n)$ time and $O(n)$ space. We also prove an $\Omega(n\log n)$ time lower bound for solving the problem, which implies the optimality of our algorithm.