# Hybridizing Non-dominated Sorting Algorithms: Divide-and-Conquer Meets Best Order Sort

## TL;DR

This paper introduces a hybrid non-dominated sorting algorithm that combines divide-and-conquer with Best Order Sort, achieving improved practical performance especially on large datasets with many points and objectives.

## Contribution

The paper proposes a novel hybrid algorithm that merges asymptotically efficient divide-and-conquer with a practical quadratic method for non-dominated sorting.

## Key findings

- Hybrid algorithm outperforms original algorithms by at least 20% on large datasets.
- Speedup reaches up to four times for small numbers of objectives.
- Hybrid maintains comparable performance on small problem instances.

## Abstract

Many production-grade algorithms benefit from combining an asymptotically efficient algorithm for solving big problem instances, by splitting them into smaller ones, and an asymptotically inefficient algorithm with a very small implementation constant for solving small subproblems. A well-known example is stable sorting, where mergesort is often combined with insertion sort to achieve a constant but noticeable speed-up. We apply this idea to non-dominated sorting. Namely, we combine the divide-and-conquer algorithm, which has the currently best known asymptotic runtime of $O(N (\log N)^{M - 1})$, with the Best Order Sort algorithm, which has the runtime of $O(N^2 M)$ but demonstrates the best practical performance out of quadratic algorithms. Empirical evaluation shows that the hybrid's running time is typically not worse than of both original algorithms, while for large numbers of points it outperforms them by at least 20%. For smaller numbers of objectives, the speedup can be as large as four times.