Cache Oblivious Distribution Sweeping

Abstract

We adapt the distribution sweeping method to the cache oblivious model. Distribution sweeping is the name used for a general approach for divide-and-conquer algorithms where the combination of solved subproblems can be viewed as a merging process of streams. We demonstrate by a series of algorithms for specific problems the feasibility of the method in a cache oblivious setting. The problems all come from computational geometry, and are: orthogonal line segment intersection reporting, the all nearest neighbors problem, the 3D maxima problem, computing the measure of a set of axis-parallel rectangles, computing the visibility of a set of line segments from a point, batched orthogonal range queries, and reporting pairwise intersections of axis-parallel rectangles. Our basic building block is a simplified version of the cache oblivious sorting algorithm Funnelsort of Frigo et al., which is of independent interest.

Cache-Oblivious Red-Blue Line Segment Intersection

“…Subsequently, algorithms and data structures for a range of problems have been developed [3]. Relevant to this paper, Bender et al [5] developed a cache-oblivious algorithm that solves the offline planar point location problem using O(Sort(N )) memory transfers; Brodal and Fagerberg [6] developed a cache-oblivious version of distribution sweeping and showed how to use it to solve the orthogonal line segment intersection problem, as well as several other problems involving axis-parallel objects, cache-obliviously using O(Sort(N ) + T /B) memory transfers. To the best of our knowledge, no cache-oblivious algorithm was previously known for any intersection problem involving non-axis-parallel objects.…”

“…In the case of axis-parallel objects, such an ordering is equivalent to the y-ordering of the vertices of the objects; in the non-axis-parallel case, this ordering is more difficult to obtain [4]. Similar to the cache-oblivious orthogonal line-segment intersection algorithm [6], we employ the cache-oblivious distribution sweeping paradigm, which uses two-way merging rather than M/B-way distribution. While this eliminates the need for multislabs, which do not seem to have an efficient cache-oblivious counterpart, it also results in a recursion depth of Θ(log 2 N ) rather than Θ(log M/B N ).…”

“…Our new algorithm borrows ideas from both the external-memory algorithm for the redblue line segment intersection problem [4] and the cache-oblivious algorithm for the orthogonal line-segment intersection problem [6]. In order to obtain a useful notion of sweeping the plane top-down or bottom-up, we utilize the same total ordering as in [4] on a set of non-intersecting segments, which arranges the segments intersected by any vertical line in the same order as the y-coordinates of their intersections with the line.…”

“…This creates a new challenge, as a segment may have intersections with all segments in the output stream of a given merge process and, thus, needs access to the entire output stream to report these intersections. To overcome this problem, Brodal and Fagerberg [6] provided a technique to detect, count, and collect intersected segments at each level of recursion that ensures that the number of additional accesses needed to report intersections is proportional to the output size.…”

“…This implies that one cannot afford to spend even 1/B memory transfers per line segment at each level of the recursion. For axis-parallel objects, Brodal and Fagerberg [6] addressed this problem using the so-called k-merger technique, which was introduced as the central idea in Funnel Sort (ie., cache-oblivious Merge Sort) [8]. This technique allows N elements to be passed through a log 2 N -level merge process using only O(Sort(N )) memory transfers, but generates the output of each merge process in bursts, each of which has to be consumed by the next merge process before the next burst is produced.…”