Histograms on Streams

Synonyms

Piecewise-constant approximations

Definition

A B-bucket histogram of length N is a partition of the set [0 , N) of N integers into intervals [b ₀ , b ₁) ∪ [b ₁ , b ₂) ∪  …  ∪ [b _B − 1 , b _B ), where b ₀ = 0 and b _B  = N, together with a collection of B heights h _j , for 0 ≤ j < B, one for each bucket. On point query i, the histogram answer is h _j , where j is the index of the interval (or “bucket”) containing i; that is, the unique j with b _j  ≤ i < b _j + 1. In vector notation, χ _S is the vector that is 1 on the set S and zero elsewhere and the answer vector of a histogram is $\overrightarrow{H}={\displaystyle {\sum}_{0\le j<B^h_j}{\chi}_{\left[{b}_j,{b}_{j+1}\right).}}$

A histogram, $\overrightarrow{H}$, is often used to approximate some other function, $\overrightarrow{A}$, on [0 , N). In building a B-bucket histogram, it is desirable to choose B − 1 boundaries b _j and B heights h _j that tend to minimize some distance, e.g., the sum square error \( {\left\Vert...