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The KLEE'S MESURE of $n$ axis-parallel boxes in $\mathbb{R}^d$ is the volumeof their union. It can be computed in time within $O(n^{d/2})$ in the worstcase. We describe three techniques to boost its computation: one based on sometype of "degeneracy'' of the input, and two ones on the inherent "easiness'' ofthe structure of the input. The first technique benefits from instances wherethe MAXIMA of the input is of small size $h$, and yields a solution running intime within $O(n\log^{2d-2}{h}+ h^{d/2}) \subseteq O(n^{d/2}$). The secondtechnique takes advantage of instances where no $d$-dimensional axis-alignedhyperplane intersects more than $k$ boxes in some dimension, and yields asolution running in time within $O(n \log n + n k^{(d-2)/2}) \subseteqO(n^{d/2})$. The third technique takes advantage of instances where the\emph{intersection graph} of the input has small treewidth $\omega$. It yieldsan algorithm running in time within $O(n^4\omega \log \omega + n (\omega \log\omega)^{d/2})$ in general, and in time within $O(n \log n + n \omega ^{d/2})$if an optimal tree decomposition of the intersection graph is given. We showhow to combine these techniques in an algorithm which takes advantage of allthree configurations.

Adaptive Computation of the Klee's Measure in High Dimensions