Fig 8. Long-/Heavy-Tails.

This figure shows various plots of the unimodal instance 6g_5color_164_100_01 (Hartigans’ dip test value 0.003). This is an example of an instance with a long-tailed runtime distribution. (a) The plot shows the histogram of runtimes (in gray) and the fitted pdf (in red). Both are shifted to the left by the minimal time T₀ required to solve any extended instance. The obtained shape parameter of the fit is k = 0.884 < 1. Thus, the distribution is long-tailed. (b) We have plotted the logarithm of the tail of the distribution, i. e., logS(x). By visual inspection, one can see that it decays sub-linearly. In this case, ${\text{lim}\text{inf}}_{x\to \infty }-\text{log}{\mathcal{P}}_{}[X>x]/x=0$. This property characterizes the class of so-called heavy-tailed distributions (a superset of the class of long-tailed distributions) [55]. Intuitively, this means that the algorithm has a non-vanishing probability of requiring very long runtimes. For comparison, we have plotted the logarithmic survival function of an exponentially distributed random variable with the same expectation in blue. The logarithm of the tail of such an exponential distribution decays linearly. (c) Zoomed in version of (b). This clearly shows the sublinear decay by focusing on the curvature.