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.