Pub Date : 1900-01-01DOI: 10.1109/SFFCS.1999.814588
A. Amir, A. Efrat, P. Indyk, H. Samet
Investigates data structures obtained by a recursive partitioning of the input domain into regions of equal size. One of the most well-known examples of such a structure is the quadtree, which is used in this paper as a basis for more complex data structures; we also provide multidimensional versions of the stratified tree of P. van Emde Boas (1997). We show that, under the assumption that the input points have limited precision (i.e. are drawn from an integer grid of size u), these data structures yield efficient solutions to many important problems. In particular, they allow us to achieve O(log log u) time per operation for finding the dynamic approximate nearest neighbor (under insertions and deletions) and the exact online closest pair (under insertions only) in any constant dimension. They allow O(log log u) point location in a given planar shape or in its expansion (dilation by a ball of a given radius). Finally, we provide a linear-time (optimal) algorithm for computing the expansion of a shape represented by a quadtree. This result shows that the spatial order imposed by this regular data structure is sufficient to optimize the dilation by a ball operation.
研究通过将输入域递归划分为大小相等的区域而获得的数据结构。这种结构最著名的例子之一是四叉树，本文将其用作更复杂数据结构的基础;我们还提供了P. van Emde Boas(1997)分层树的多维版本。我们表明，假设输入点具有有限的精度(即从大小为u的整数网格中绘制)，这些数据结构为许多重要问题提供了有效的解决方案。特别是，它们允许我们在每次操作中获得O(log log u)的时间来查找动态近似最近邻(在插入和删除情况下)和准确的在线最近邻对(仅在插入情况下)。它们允许O(log log u)点的位置在给定的平面形状或在其膨胀(膨胀由一个给定半径的球)。最后，我们提供了一个线性时间(最优)算法来计算由四叉树表示的形状的展开。结果表明，该规则数据结构所施加的空间顺序足以优化球操作的膨胀。
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