The problem of random sampling without replacement (RS) calls for the selection of m distinct random items out of a population of size n. If all items have the same probability to be selected, the problem is known as uniform RS. Uniform random sampling in one pass is discussed in [1,6,11]. Reservoir-type uniform sampling algorithms over data streams are discussed in [12]. A parallel uniform random sampling algorithm is given in [10]. In weighted random sampling (WRS) the items are weighted and the probability of each item to be selected is determined by its relative weight. WRS can be defined with the following algorithm D:
Algorithm D, a definition of WRS
A population V of n weighted items
A set S with a WRS of size m
For \( < k=1 >\) to m do
Let \( < p_i(k) = / w_j> > \) be the probability of item v i to be selected in round k
Randomly select an item \(
