1 7 A ug 1 99 8 A Comparison Of Two Methods For Random Labelling of Balls by Vectors of Integers

1 Greg Kirk[Ki] raised the question of comparing the following two ways for labelling balls. Given r predetermined positive integers n i , (1 ≤ i ≤ r), and given N balls (N large), consider two ways to randomly assign r− component vectors of integers (a 1 ,. .. , a r) to them, such that 1 ≤ a i ≤ n i. We will call these vectors 'labels'. Of course altogether there are r i=1 n i possible labels. smaller pots, each with capacity N/n i balls, and labeled with labels 1 through n i , and, uniformly at random, distribute them into these smaller pots. Assign the i th component of the vector-label of each ball, a i , to be the label of the pot in which it was dropped. Having done that, you dump all the balls back into the big pot, and go on to the next iteration. Second Way: Do the same as above for i = 1, except that at the end of the first iteration you do not dump back the balls into the large ball but proceed as follows. For i = 2,. .. , r, assuming that the balls have already received their first i − 1 components, leaving the balls in their pots from the (i − 1) th iteration, you line-up n i new pots, each with a capacity of N/n i balls, and labeled with labels 1 through n i. For each of the n i−1 pots from the previous iteration, individually, we uniformly at random, distribute their contents into the new pots, each of the n i new pots getting exactly N/(n i−1 n i) balls from each of the n i−1 pots from the previous, (i − 1) th iteration. Note that in the First Way, assuming that we can reuse the pots, we need 1 + max(n 1 ,. .. , n r) pots, one of which should have a capacity of N balls, while in the Second Way, we need max(1 + The goal is to maximize the 'equal representation' of all the possible r i=1 n i vector-labels. It is obvious, with either way, that the probability of a ball to be assigned any given label is r i=1 n −1 i , and hence that the expected number of balls to be given label v, for each of v ∈ r …