Product Range Search Problem

Abstract

Given a metric space, a standard metric range search, given a query $(q,r)$, finds all points within distance $r$ of the point $q$. Suppose now we have two different metrics $d_1$ and $d_2$. A product range query $(q, r_1, r_2)$ is a point $q$ and two radii $r_1$ and $r_2$. The output is all points within distance $r_1$ of $q$ with respect to $d_1$ and all points within $r_2$ of $q$ with respect to $d_2$. In other words, it is the intersection of two searches. We present two data structures for approximate product range search in doubling metrics. Both data structures use a net-tree variant, the greedy tree. The greedy tree is a data structure that can efficiently answer approximate range searches in doubling metrics. The first data structure is a generalization of the range tree from computational geometry using greedy trees rather than binary trees. The second data structure is a single greedy tree constructed on the product induced by the two metrics.

Figures (3)

• Figure 1: Illustration of how a spatial query is processed through the tree structure.
• Figure 2: Each small square represents a pair of covering balls—one from $X$, one from $Y$, under the $\ell_\infty$ product metric. Balls have radius $(1 + \varepsilon)r$, allowing slight overlap beyond the exact query region (dashed), ensuring full approximate coverage.
• Figure :

Theorems & Definitions (9)

• Lemma 2.1: Subadditivity of Dimension
• Lemma 2.2: Standard Packing Lemma krauthgamer2004navigating
• Lemma 2.3
• Theorem 3.1
• Theorem 3.2
• Theorem 4.1
• proof
• Theorem 4.2
• proof