Faster Dynamic Range Mode

In the dynamic range mode problem, we are given a sequence $a$ of length bounded by $N$ and asked to support element insertion, deletion, and queries for the most frequent element of a contiguous subsequence of $a$. In this work, we devise a deterministic data structure that handles each operation in worst-case $\tilde{O}(N^{0.655994})$ time, thus breaking the $O(N^{2/3})$ per-operation time barrier for this problem. The data structure is achieved by combining the ideas in Williams and Xu (SODA 2020) for batch range mode with a novel data structure variant of the Min-Plus product.