A Fast Exact Algorithm for Computing the Hypervolume Contributions in 4-D Space

Jingda Deng, Qingfu Zhang, Jianyong Sun, Hui Li

Abstract: The hypervolume (HV) contribution is widely used in indicator-based multiobjective algorithms. We propose an algorithm to compute exact 4-D HV contributions for a set of $n$ points in $O(n^{[{3}/{2}]}\log n)$ time. Our algorithm improves the currently best time complexity $O(n^{2})$ by $O({\sqrt {n}}/{\log n})$ , and it is the first algorithm of subquadratic time for this problem. Our algorithm is built upon a space partition method in computational geometry and a geometric structure called the anchored gradient. We also propose a new space partition strategy to reduce the practical running time and the space overhead of this algorithm. Experimental results on a variety of test instances show that our proposed algorithm performs better than the existing state-of-the-art algorithm, especially, on point sets with cliff or other irregular properties.