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.

翻译：给定一个度量空间，标准度量范围搜索在给定查询$(q,r)$时，找出所有与点$q$距离在$r$以内的点。现在假设我们有两个不同的度量$d_1$和$d_2$。一个积范围查询$(q, r_1, r_2)$由一个点$q$和两个半径$r_1$与$r_2$构成。输出结果是所有在$d_1$下与$q$距离不超过$r_1$的点，以及在$d_2$下与$q$距离不超过$r_2$的点。换言之，这是两个搜索结果的交集。我们提出了两种用于加倍度量下近似积范围搜索的数据结构。这两种数据结构都使用了净树变体——贪心树。贪心树是一种能够高效回答加倍度量下近似范围搜索的数据结构。第一种数据结构是计算几何中范围树的推广，使用贪心树而非二叉树。第二种数据结构则是在由两个度量诱导的积度量上构建的单个贪心树。