Retroactive Monotonic Priority Queues via Range Searching

The best-known fully retroactive priority queue costs $O(\log^2 m \log \log m)$ time per operation, where $m$ is the number of operations performed on the data structure. In contrast, standard (non-retroactive) and partially retroactive priority queues can cost $O(\log m)$ time per operation. So far, it is unknown whether this $O(\log m)$ bound can be achieved for fully retroactive priority queues. In this work, we study a restricted variant of priority queues known as monotonic priority queues. First, we show that finding the minimum in a retroactive monotonic priority queue is a special case of the range-searching problem. Then, we design a fully retroactive monotonic priority queue with a cost of $O(\log m + T(m))$ time per operation, where $T(m)$ is the maximum between the query and the update time of a specific range-searching data structure with $m$ elements. Finally, we design a fully retroactive monotonic priority queue that costs $O(\log m \log \log m)$ time per operation.