Maximizing Range Sum in Spatial Databases

Definition

Let O be a set of objects (a.k.a. points) in 2D space $\mathbb{R}^{2}$, where $\mathbb{R}$ represents the real domain. Each object o ∈ O is associated with a positive value w(o) as its weight. Given non-negative values d₁ and d₂, the goal of the maximizing range sum (MaxRS) problem is to place a d₁ × d₂ rectangle r in $\mathbb{R}^{2}$ to maximize the covered weight of r, defined as:

$$\displaystyle{ covered - weight\left (r\right ) =\sum \limits _{o\in O\, \cap \,r}w\left (p\right ). }$$

In plain words, covered-weight(r) equals the total weight of the objects of O that are covered by r. As a special case, if every object in O has weight 1, then covered-weight(r) simply indicates how many objects of O fall in r. Note that the position of rcan be anywhere in the data space, namely, there are infinitely many possible rectangles that could have been chosen. To...