Cache Oblivious Distribution Sweeping
08 Jul 2002-pp 426-438
TL;DR: This work adapts the distribution sweeping model for divide-and-conquer algorithms to the cache oblivious model, and demonstrates by a series of algorithms the feasibility of the method in a cache oblivious setting.
Abstract: We adapt the distribution sweepingmetho d to the cache oblivious model. Distribution sweepingis 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, computingthe measure of a set of axis-parallel rectangles, computingthe visibility of a set of line segments from a point, batched orthogonal range queries, and reportingpairwise intersections of axis-parallel rectangles. Our basic buildingblo ck is a simplified version of the cache oblivious sorting algorithm Funnelsort of Frigo et al., which is of independent interest.
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01 Jan 2003
134 citations
Cites methods from "Cache Oblivious Distribution Sweepi..."
...Subsequently other sorting algorithms were found, such as lazy funnelsort [ 13 ], which only needs a relaxed form of the tall-cache assumption, namely M > B1+" for some " > 0. Brodal et al. [15] showed that cache-oblivious I/O-optimal sorting is impossible without a tall-cache assumption: then cacheoblivious algorithms are less powerful than EM ones....
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13 Jun 2010
TL;DR: This paper describes several cache-oblivious algorithms with optimal work, polylogarithmic depth, and sequential cache complexities that match the best sequential algorithms, including the first such algorithms for sorting and for sparse-matrix vector multiply on matrices with good vertex separators.
Abstract: In this paper we explore a simple and general approach for developing parallel algorithms that lead to good cache complexity on parallel machines with private or shared caches. The approach is to design nested-parallel algorithms that have low depth (span, critical path length) and for which the natural sequential evaluation order has low cache complexity in the cache-oblivious model. We describe several cache-oblivious algorithms with optimal work, polylogarithmic depth, and sequential cache complexities that match the best sequential algorithms, including the first such algorithms for sorting and for sparse-matrix vector multiply on matrices with good vertex separators.Using known mappings, our results lead to low cache complexities on shared-memory multiprocessors with a single level of private caches or a single shared cache. We generalize these mappings to multi-level cache hierarchies of private or shared caches, implying that our algorithms also have low cache complexities on such hierarchies. The key factor in obtaining these low parallel cache complexities is the low depth of the algorithms we propose.
124 citations
08 Jul 2004
TL;DR: An overview of the results achieved on cache-oblivious algorithms and data structures since the seminal paper by Frigo et al. in 1999 is given.
Abstract: Frigo, Leiserson, Prokop and Ramachandran in 1999 introduced the ideal-cache model as a formal model of computation for developing algorithms in environments with multiple levels of caching, and coined the terminology of cache-oblivious algorithms. Cache-oblivious algorithms are described as standard RAM algorithms with only one memory level, i.e. without any knowledge about memory hierarchies, but are analyzed in the two-level I/O model of Aggarwal and Vitter for an arbitrary memory and block size and an optimal off-line cache replacement strategy. The result are algorithms that automatically apply to multi-level memory hierarchies. This paper gives an overview of the results achieved on cache-oblivious algorithms and data structures since the seminal paper by Frigo et al.
113 citations
TL;DR: This paper presents an almost ideal solution to this problem: a hash function h: U: Uarrow V that, on any set of $n$ inputs, behaves like a truly random function with high probability, can be evaluated in constant time on a RAM and can be stored in $(1+\epsilon)n\log |V| + O(n+\log \log |U|)$ bits.
Abstract: Many algorithms and data structures employing hashing have been analyzed under the uniform hashing assumption, i.e., the assumption that hash functions behave like truly random functions. Starting with the discovery of universal hash functions, many researchers have studied to what extent this theoretical ideal can be realized by hash functions that do not take up too much space and can be evaluated quickly. In this paper we present an almost ideal solution to this problem: a hash function $h: U\rightarrow V$ that, on any set of $n$ inputs, behaves like a truly random function with high probability, can be evaluated in constant time on a RAM and can be stored in $(1+\epsilon)n\log |V| + O(n+\log\log |U|)$ bits. Here $\epsilon$ can be chosen to be any positive constant, so this essentially matches the entropy lower bound. For many hashing schemes this is the first hash function that makes their uniform hashing analysis come true, with high probability, without incurring overhead in time or space.
100 citations
01 Jan 2003
TL;DR: A recent body of work has developed cache-oblivious algorithms and data structures that perform as well or nearly as well as standard external-memory structures which require knowledge of the cache/memory size and block transfer size.
Abstract: A recent direction in the design of cache-efficient and diskefficient algorithms and data structures is the notion of cache obliviousness, introduced by Frigo, Leiserson, Prokop, and Ramachandran in 1999. Cache-oblivious algorithms perform well on a multilevel memory hierarchy without knowing any parameters of the hierarchy, only knowing the existence of a hierarchy. Equivalently, a single cache-oblivious algorithm is efficient on all memory hierarchies simultaneously. While such results might seem impossible, a recent body of work has developed cache-oblivious algorithms and data structures that perform as well or nearly as well as standard external-memory structures which require knowledge of the cache/memory size and block transfer size. Here we describe several of these results with the intent of elucidating the techniques behind their design. Perhaps the most exciting of these results are the data structures, which form general building blocks immediately leading to several algorithmic results.
97 citations
References
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"Cache Oblivious Distribution Sweepi..." refers methods in this paper
...Existing algorithms may have straightforward cache oblivious implementation—this is for example the case for the algorithm know as Graham’s scan [15] for computing the convex hull of a point set [12, 18]....
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TL;DR: P can be chosen to I&E the centroid oC the triangle formed by X, y and z and Express each si E S in polar coordinates th origin P and 8 = 0 in the direction of zu~ arhitnry fixed half-line L from P.
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"Cache Oblivious Distribution Sweepi..." refers methods in this paper
...Existing algorithms may have straightforward cache oblivious implementation—this is for example the case for the algorithm know as Graham’s scan [14] for computing the convex hull of a point set [11]....
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