Dynamic Generation of Discrete Random Variates

We present and analyze efficient new algorithms for generating a random variate distributed according to a dynamically changing set of N weights. The base version of each algorithm generates the discrete random variate in O(log ^* N) expected time and updates a weight in O(2^{log^* N}) expected time in the worst case. We then show how to reduce the update time to O(log ^* N) amortized expected time. We finally show how to apply our techniques to a lookup-table technique in order to obtain expected constant time in the worst case for generation and update. We give parallel algorithms for parallel generation and update having optimal processor—time product. Besides the usual application in computer simulation, our method can be used to perform constant-time prediction in prefetching applications. We also apply our techniques to obtain an efficient dynamic algorithm for maintaining an approximate heap of N elements, in which each query is required to return an element whose value is within an ɛ multiplicative factor of the maximal element value. For ɛ= 1/polylog (N), each query, insertion, or deletion takes O(log log log N) time.