Frequency Moments

DEFINITION Consider a stream (i.e., an ordered list) S = a1, a2, . . . , an of elements ai ∈ [m] def = {1, 2, . . . ,m}. For i ∈ [m], its frequency fi is the number of times it occurs in S. The k-th frequency moment Fk of S, for real k > 0, is defined to be Fk(S) = ∑ i∈[m] f k i . Interpreting 0 0 as 0, we also define F0 this way, so that it equals the number of distinct elements in S. Observe that F1 = n is the length of S. In the database community, F2 is known as the repeat rate or Gini’s index of homogeneity. It is also natural to define F∞ = max1≤i≤m fi. It is usually assumed that n is very large and that algorithms which compute the frequency moments do not have enough storage to keep the entire stream in memory. It is also common to assume that they are only given a constant (usually one) number of passes over the data. It is also assumed that the stream is presented in an arbitrary, possibly worst-case order. This necessitates the use of extremely efficient randomized approximation algorithms. An algorithm A ( , δ)-approximates the kth frequency moment Fk if for any input stream S, Pr[|A(S)−Fk(S)| ≤ Fk(S)] ≥ 1− δ, where the probability is over the coin tosses of A. Here, by A(S) we mean that A is presented items in S one-by-one. Efficiency is measured in terms of the amount of memory and update time of the algorithm.