Data Structure for a Time-Based Bandwidth Reservations Problem

We discuss a problem of handling resource reservations. The resource can be reserved for some time, it can be freed or it can be queried what is the largest amount of reserved resource during a time interval. We show that the problem has a lower bound of $\Omega(\log n)$ per operation on average and we give a matching upper bound algorithm. Our solution also solves a dynamic version of the related problems of a prefix sum and a partial sum.