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lines changed Original file line number Diff line number Diff line change @@ -341,11 +341,15 @@ def hypervolume(
341341 Compute the hypervolume metric with respect to a given reference point
342342 assuming minimization of all objectives. For 2D and 3D, the algorithm used
343343 :footcite:p:`FonPaqLop06:hypervolume,BeuFonLopPaqVah09:tec` has :math:`O(n
344- \log n)` complexity. For 4D or higher, the algorithm
345- :footcite:p:`FonPaqLop06:hypervolume` has :math:`O(n^{d-2} \log n)` time
346- and linear space complexity in the worst-case, but experimental results
347- show that the pruning techniques used may reduce the time complexity even
348- further.
344+ \log n)` complexity. For 4D or higher, it uses a recursive algorithm that
345+ has the 3D algorithm as a base case :footcite:p:`FonPaqLop06:hypervolume`,
346+ which has :math:`O(n^{d-2} \log n)` time and linear space complexity in the
347+ worst-case, but experimental results show that the pruning techniques used
348+ may reduce the time complexity even further. Andreia P. Guerreiro improved
349+ the integration of the 3D case with the recursive algorithm, which leads to
350+ significant reduction of computation time. She has also enhanced the
351+ numerical stability of the algorithm by avoiding floating-point comparisons
352+ of partial hypervolumes.
349353
350354 Parameters
351355 ----------
Original file line number Diff line number Diff line change 33# ' Compute the hypervolume metric with respect to a given reference point
44# ' assuming minimization of all objectives. For 2D and 3D, the algorithm used
55# ' \citep{FonPaqLop06:hypervolume,BeuFonLopPaqVah09:tec} has \eqn{O(n \log n)}
6- # ' complexity. For 4D or higher, the algorithm \citep{FonPaqLop06:hypervolume}
6+ # ' complexity. For 4D or higher, it uses a recursive algorithm that has the 3D
7+ # ' algorithm as a base case algorithm \citep{FonPaqLop06:hypervolume}, which
78# ' has \eqn{O(n^{d-2} \log n)} time and linear space complexity in the
89# ' worst-case, but experimental results show that the pruning techniques used
9- # ' may reduce the time complexity even further.
10+ # ' may reduce the time complexity even further. Andreia P. Guerreiro improved
11+ # ' the integration of the 3D case with the recursive algorithm, which leads to
12+ # ' significant reduction of computation time. She has also enhanced the
13+ # ' numerical stability of the algorithm by avoiding floating-point comparisons
14+ # ' of partial hypervolumes.
1015# '
1116# ' @inherit epsilon params return
1217# '
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