Frequency Moments

Synonyms

L_p distances; L_p norms

Definition

Consider a stream (i.e., an ordered list) = a₁ , a₂ … , a_n of elements a_i ∈ [m]^def = {1, 2, … , m}. For i ∈ [m], its frequency f_i is the number of times it occurs in . The kth frequency moment F_k of , for real k > 0, is defined to be ${F}_k({\cal S})={\displaystyle {\sum}_{i\in \left[m\right]}{f}_i^k}$. Interpreting 0⁰ as 0, one can also define F₀ this way, so that it equals the number of distinct elements in . 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_∞ = max_{1 ≤ i ≤ m}f_i.

It is usually assumed that nis 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 further assumed that the stream is presented in an arbitrary, possibly...