Optimal-Cost Construction of Shallow Cuttings for 3-D Dominance Ranges in the I/O-Model

Abstract

Shallow cuttings are a fundamental tool in computational geometry and spatial databases for solving offline and online range searching problems. For a set $P$ of $N$ points in 3-D, at SODA'14, Afshani and Tsakalidis designed an optimal $O(N\log_2N)$ time algorithm that constructs shallow cuttings for 3-D dominance ranges in internal memory. Even though shallow cuttings are used in the I/O-model to design space and query efficient range searching data structures, an efficient construction of them is not known till now. In this paper, we design an optimal-cost algorithm to construct shallow cuttings for 3-D dominance ranges. The number of I/Os performed by the algorithm is $O\left(\frac{N}{B}\log_{M/B}\left(\frac{N}{B}\right) \right)$, where $B$ is the block size and $M$ is the memory size. As two applications of the optimal-cost construction algorithm, we design fast algorithms for offline 3-D dominance reporting and offline 3-D approximate dominance counting. We believe that our algorithm will find further applications in offline 3-D range searching problems and in improving construction cost of data structures for 3-D range searching problems.

Figures (5)

• Figure 1: (a) A schematic representation of a $k$-level 3-D shallow cutting, (b) 3-D dominance reporting and approximate counting query, (c) 3-D orthogonal range reporting query, (d) 3-D rectangle stabbing query.
• Figure 2: An example of a $k$-level 2-D shallow cutting. Here $k=3$, so each inner corner $d_i$ dominates three points and each outer corner $c_i$ dominates six points.
• Figure 3: (a) Outer boundary (in bold) before the patching procedure. The dotted boundary is the new curve created during the patching procedure. (b) Outer boundary (in bold) after the patching procedure. The dotted curve was part of the outer boundary before patching. Each outer corner created during the patching procedure corresponds to the apex point of a 3-D cell in the $k$-level cutting of $P$.
• Figure 4: An example of $x$-selection query in Lemma \ref{['lem:significance']}. The point $p$ is an outer corner of a 2-D shallow cutting of the $k_i$-level. The corner $c$ is the outer corner of the parent staircase corresponding to the parent cell $C$ of $p$, since $c$ has the smallest $y$-coordinate among all apex points of the parent staircase dominating $p$.
• Figure 5: For simplicity, the root node is shown in 2-D. (a) An example with $24$ points and $\alpha=5$. If the apex point of a query dominance range lies in cell $C$, then the cells shaded pink lie completely inside the query region. Therefore, for cell $C$ we have $n(C)=10$. (b) The blue portion of the query region is handled at the root itself. We will recurse on the pink portion.

Theorems & Definitions (43)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 1
• Lemma 2
• Lemma 3
• proof
• Lemma 4