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 [b₀ , b₁) ∪ [b₁ , b₂) ∪ … ∪ [b_{B − 1} , b_B), where b₀ = 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...