Dynamic Algorithms for Interval Scheduling on a Single Machine

TL;DR

This paper develops and analyzes dynamic algorithms for interval scheduling on a single machine, achieving efficient query and update times, especially for monotonic interval sets, with experimental comparison.

Contribution

Introduces new dynamic algorithms with optimal amortised times for interval scheduling, including specialized solutions for monotonic interval sets.

Findings

Amortised query time: O(log n)

Amortised update time: O(d log^2 n

Algorithms outperform existing methods in experiments

Abstract

We investigate dynamic algorithms for the interval scheduling problem. Our algorithm runs in amortised time $O(\log n)$ for query operation and $O(d\log^2 n)$ for insertion and removal operations, where $n$ and $d$ are the maximal numbers of intervals and pairwise overlapping intervals respectively. We also show that for a monotonic set, that is when no interval properly contains another interval, the amortised complexity is $O(\log n)$ for both query and update operations. We compare the two algorithms for the monotonic interval sets using experiments.