Dynamic data structures for approximate Hausdorff distance in the word RAM

We give a fully dynamic data structure for maintaining an approximation of the Hausdorff distance between two point sets in a constant dimension d, a standard problem in computational geometry. Our solution has an approximation factor of 1+ε for any constant ε>0 and expected update time O(log⁡Ulog⁡log⁡n), where U is the universe size, and n is the number of the points. The result of the paper greatly improves over the previous exact method, which required O(n5/6polylogn) time and worked only in a semi-online setting. The model of computation is the word RAM model.