On Constant Factors in Comparison-Based Geometric Algorithms and Data Structures

Abstract

Many standard problems in computational geometry have been solved asymptotically optimally as far as comparison-based algorithms are concerned, but there has been little work focusing on improving the constant factors hidden in big-Oh bounds on the number of comparisons needed. In this paper, we consider orthogonal-type problems and present a number of results that achieve optimality in the constant factors of the leading terms, including

• an algorithm for the 2D maxima problem that uses $n\lg h + O(n\sqrt{\lg h})$ comparisons, where $h$ denotes the output size;

• a randomized algorithm for the 3D maxima problem that uses $n\lg h + O(n\lg ^{2/3} h)$ expected number of comparisons;

• a randomized algorithm for detecting intersections among a set of orthogonal line segments that uses $n\lg n + O(n\sqrt{\lg n})$ expected number of comparisons;

• a data structure for point location among 3D disjoint axis-parallel boxes that can answer queries in $(3/2)\lg n + O(\lg \lg n)$ comparisons;

• a data structure for point location in a 3D box subdivision that can answer queries in $(4/3)\lg n + O(\sqrt{\lg n})$ comparisons.

Some of the results can be adapted to solve nonorthogonal problems, such as 2D convex hulls and general line segment intersection. Our algorithms and data structures use a variety of techniques, including Seidel and Adamy’s planar point location method, weighted binary search, and height-optimal BSP trees.

Appendix: Lower Bounds

Kirkpatrick and Seidel [33, 34] proved an $\varOmega (n\lg h)$ lower bound for the 2D convex hull and 2D maxima problem (the version in which the output need not be in sorted order) in the algebraic computation tree model using Ben-Or’s technique [3]. Ben-Or’s technique also implies an $\varOmega (n\lg n)$ lower bound for the orthogonal line segment intersection detection problem. The technique has hidden constant factors, however.

In this section, we note that for 2D maxima and orthogonal segment intersection detection, the hidden constant factor in the lower bounds can be made equal to 1 in the linear decision tree model, where a comparison refers to testing the sign of a linear combination of the input values. Although the model is not as powerful as higher-degree algebraic decision trees, it permits ordinary comparisons between two input coordinates, which are what this paper focuses on anyway. For linear decision trees, a simpler technique [20, 48] suffices. The main observation is that cells in a linear decision tree are intersections of halfspaces and are thus convex.

Maxima in 2D. Let $F$ consist of all functions $f:\{1,\ldots ,n\}\rightarrow \{1,\ldots ,h\}$ such that $f(1),\ldots ,f(h)$ is a permutation of $\{1,\ldots ,h\}$. Given $f\in F$, define $S_f$ to be the sequence of $n$ points where the $i$th point is $(f(i)-\varepsilon i, -f(i)-\varepsilon i)$. Then $S_f$ has exactly $h$ maximal points, namely the first $h$ points.

Suppose that the two inputs $S_f$ and $S_g$ $(f,g\in F)$ end up in the same leaf cell of the decision tree. Imagine moving the point positions from $S_f$ to $S_g$ linearly. At any moment in time, the input point sequence belongs to the same leaf cell, by convexity. The $i$th point can only move along the line $\{(t-\varepsilon i, -t-\varepsilon i) : t\in \mathbb {R}\}$.

For $i,j\in \{1,\ldots ,h\}$, the $i$th and $j$th points cannot change order, for otherwise one of these two points would not be maximal at some moment in time. Thus, $f(1)=g(1),\ldots , f(h)=g(h)$. In particular, the staircase of the first $h$ points stays unmoved.

For $i\in \{h+1,\ldots ,n\}$, the $i$th point cannot cross this staircase, for otherwise the set of maximal points would change (assuming a sufficiently small $\varepsilon >0$). Thus, $f(i)=g(i)$.

We conclude that the inputs $S_f$ over all $f\in F$ must lie in distinct leaf cells. Consequently, the number of leaf cells is at least $|F|=h!h^{n-h}$, and the height of the decision tree is at least $\lg |F| = n\lg h - O(h)$.

In fact, the average depth over $|F|$ distinct leaves in the decision tree is at least $\lg |F|$. It follows that the expected number of comparisons on the input $S_f$ for a random $f\in F$ is at least $n\lg h-O(h)$. By Yao’s principle, this gives the same expected lower bound for randomized algorithms.

Theorem 9

Any algorithm that computes the $h$ maxima of a set of $n$ points in two dimensions requires at least $n\lg h - O(h)$ worst-case expected number of comparisons under the linear decision tree model.

Orthogonal Line Segment Intersection Detection. Let $F$ consist of all pairs $f=(f_v,f_h)$ where $f_v,f_h:\{1,\ldots n/2\}\rightarrow \{1,\ldots , n/2\}$ are bijections. Given $f\in F$, define $S_f$ to be a sequence of $n/2$ vertical and $n/2$ horizontal line segments where the $i$th vertical segment has endpoints $(f(i),-1)$ and $(f(i),1)$ and the $i$th horizontal segment has endpoints $(f(i)+1/3,0)$ and $(f(i)+2/3,0)$. Then $S_f$ has no intersections.

Suppose that the two inputs $S_f$ and $S_g$ $(f,g\in F)$ end up in the same leaf cell of the decision tree. Imagine moving the coordinates of the segments from $S_f$ to $S_g$ linearly. At any moment in time, the input point sequence belongs to the same leaf cell, by convexity. Each segment can only move horizontally. The segments cannot change $x$-order, for otherwise the input would have to change from no to yes at some moment in time. We thus have $f=g$.

We conclude that the inputs $S_f$ over all $f\in F$ must lie in distinct leaf cells. Consequently, the number of leaf cells is at least $|F|=(n/2)!^2$, and the height of the decision tree is at least $\lg |F| = n\lg n - O(n)$.

As before, we also obtain the same expected lower bound for randomized algorithms.

Theorem 10

Any algorithm that detects the existence of an intersection for a two-dimensional set of $n$ line segments, each vertical or horizontal, requires at least $n\lg n - O(n)$ worst-case expected number of comparisons under the linear decision tree model.