Dynamic sampling from a discrete probability distribution with a known distribution of rates

In this paper, we consider several efficient data structures for the problem of sampling from a dynamically changing discrete probability distribution, where some prior information is known on distribution rates, in particular maximum and minimum rate, number possible outcomes N large. We three basic structures, Acceptance-Rejection method, Complete Binary Tree Alias method. These can be used as building blocks multi-level structure, at each levels, one used, with top level selecting group events, bottom an element group. Depending assumptions rates outcomes, different combinations used. prove that expected time update constant when rate follows certain conditions. show any combining tree structure have $O\left(\log\log{r_{max}}/{r_{min}}\right)$ possible, $r_{max}$ $r_{min}$ rate. also discuss implementation Two Levels allows sampling, amortized updates, assuming are events sufficiently present experimental verification, highlighting limits given by constraints real-life setting.