S-Metric calculation by considering dominated hypervolume as Klee's measure problem.

The dominated hypervolume (or S-metric) is a commonly accepted quality measure for comparing approximations of Pareto fronts generated by multi-objective optimizers. Since optimizers exist, namely evolutionary algorithms, that use the S-metric internally several times per iteration, a fast determination of the S-metric value is of essential importance. This work describes how to consider the S-metric as a special case of a more general geometric problem called Klee's measure problem (KMP). For KMP, an algorithm exists with runtime O(n log n + n(d/2) log n), for n points of d >or= 3 dimensions. This complex algorithm is adapted to the special case of calculating the S-metric. Conceptual simplifications realize the algorithm without complex data structures and establish an upper bound of O(n(d/2) log n) for the S-metric calculation for d >or= 3. The performance of the new algorithm is studied in comparison to another state of the art algorithm on a set of academic test functions.