Applying fast matrix multiplication to Klee’s measure problem

Determine how to apply fast matrix multiplication techniques to Klee’s measure problem—computing the volume of the union of n axis-aligned boxes in R^d—to obtain an exact algorithm with running time strictly faster than O(n^{d/2} + n log n) for constant dimension d.

Background

Klee’s measure problem asks for the volume of the union of n axis-aligned boxes in R^d. The best known exact algorithm for constant d runs in time O(n^{d/2} + n log n). A conditional lower bound suggests that any improvement over this bound would likely require using fast matrix multiplication (FMM) techniques.

Despite this indication, the applicability of FMM to Klee’s measure problem is not understood. The paper notes that it is unclear how to leverage FMM to achieve a faster exact algorithm, leaving a concrete methodological gap even though such techniques might be necessary for progress.