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 $\vec{H} = \mathop{{\sum} }\nolimits_{0 \le j < B} h_j \chi_{\left[ \left.b_j ,b_{j+1}\right) \right.}$.

A histogram, $\vec{H}$, is often used to approximate some other function, $\vec{A}$, 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\|\vec{A} -\vec{H}\right\|^2 =...