Faster 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 faster determination of the S-metric value is of essential importance. This paper describes how to consider the S-metric as a special case of a more general geometrical problem called Klee s measure problem (KMP). For KMP, an algorithm exists with run time O (n log n + n^ (d/2) log n), for n points of d>= 3 dimensions. This complex algorithmis adapted to the special case of calculating the S-metric. Conceptual simplifications of the implementation are concerned that save on a factor of O (log n) and establish an upper bound of O (n log n + n^ (d/2) ) for the S-metric calculation, improving the previously known bound of O (n^ (d-1) ).