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Cacheoblivious Btrees
, 2000
"... Abstract. This paper presents two dynamic search trees attaining nearoptimal performance on any hierarchical memory. The data structures are independent of the parameters of the memory hierarchy, e.g., the number of memory levels, the blocktransfer size at each level, and the relative speeds of me ..."
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Cited by 145 (23 self)
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Abstract. This paper presents two dynamic search trees attaining nearoptimal performance on any hierarchical memory. The data structures are independent of the parameters of the memory hierarchy, e.g., the number of memory levels, the blocktransfer size at each level, and the relative speeds of memory levels. The performance is analyzed in terms of the number of memory transfers between two memory levels with an arbitrary blocktransfer size of B; this analysis can then be applied to every adjacent pair of levels in a multilevel memory hierarchy. Both search trees match the optimal search bound of Θ(1+logB+1 N) memory transfers. This bound is also achieved by the classic Btree data structure on a twolevel memory hierarchy with a known blocktransfer size B. The first search tree supports insertions and deletions in Θ(1 + logB+1 N) amortized memory transfers, which matches the Btree’s worstcase bounds. The second search tree supports scanning S consecutive elements optimally in Θ(1 + S/B) memory transfers and supports insertions and deletions in Θ(1 + logB+1 N + log2 N) amortized memory transfers, matching the performance of the Btree for B = B Ω(log N log log N).
Engineering a cacheoblivious sorting algorithm
 In Proc. 6th Workshop on Algorithm Engineering and Experiments
, 2004
"... The cacheoblivious model of computation is a twolevel memory model with the assumption that the parameters of the model are unknown to the algorithms. A consequence of this assumption is that an algorithm efficient in the cache oblivious model is automatically efficient in a multilevel memory mod ..."
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Cited by 25 (1 self)
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The cacheoblivious model of computation is a twolevel memory model with the assumption that the parameters of the model are unknown to the algorithms. A consequence of this assumption is that an algorithm efficient in the cache oblivious model is automatically efficient in a multilevel memory model. Since the introduction of the cacheoblivious model by Frigo et al. in 1999, a number of algorithms and data structures in the model has been proposed and analyzed. However, less attention has been given to whether the nice theoretical proporities of cacheoblivious algorithms carry over into practice. This paper is an algorithmic engineering study of cacheoblivious sorting. We investigate a number of implementation issues and parameters choices for the cacheoblivious sorting algorithm Lazy Funnelsort by empirical methods, and compare the final algorithm with Quicksort, the established standard for comparison based sorting, as well as with recent cacheaware proposals. The main result is a carefully implemented cacheoblivious sorting algorithm, which we compare to the best implementation of Quicksort we can find, and find that it competes very well for input residing in RAM, and outperforms Quicksort for input on disk. 1
Cacheoblivious algorithms and data structures
 IN SWAT
, 2004
"... Frigo, Leiserson, Prokop and Ramachandran in 1999 introduced the idealcache model as a formal model of computation for developing algorithms in environments with multiple levels of caching, and coined the terminology of cacheoblivious algorithms. Cacheoblivious algorithms are described as stand ..."
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Cited by 10 (2 self)
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Frigo, Leiserson, Prokop and Ramachandran in 1999 introduced the idealcache model as a formal model of computation for developing algorithms in environments with multiple levels of caching, and coined the terminology of cacheoblivious algorithms. Cacheoblivious 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 twolevel I/O model of Aggarwal and Vitter for an arbitrary memory and block size and an optimal offline cache replacement strategy. The result are algorithms that automatically apply to multilevel memory hierarchies. This paper gives an overview of the results achieved on cacheoblivious algorithms and data structures since the seminal paper by Frigo et al.
An InPlace Sorting Algorithm Performing O(n log n) Comparisons and . . .
, 2003
"... In this paper we give a positive answer to the longstanding problem of finding an inplace sorting algorithm performing O(n log n) comparisons and O(n) data moves in the worst case. So far, the better inplace sorting algorithm with O(n) moves was proposed by Munro and V. Raman. Their algorithm ..."
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Cited by 6 (2 self)
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In this paper we give a positive answer to the longstanding problem of finding an inplace sorting algorithm performing O(n log n) comparisons and O(n) data moves in the worst case. So far, the better inplace sorting algorithm with O(n) moves was proposed by Munro and V. Raman. Their algorithm requires O(n ) comparisons in the worst case, for any # > 0. Later,
Abstract
, 2005
"... Questions about order versus disorder in systems and models have been fascinating scientists over the years. In Computer Science, order is intimately related to sorting, commonly meant as the task of arranging keys in increasing or decreasing order with respect to an underlying total order relation. ..."
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Questions about order versus disorder in systems and models have been fascinating scientists over the years. In Computer Science, order is intimately related to sorting, commonly meant as the task of arranging keys in increasing or decreasing order with respect to an underlying total order relation. The sorted organization is amenable for searching a set of n keys, since each search requires Θ(log n) comparisons in the worst case, which is optimal if the cost of a single comparison can be considered a constant. Nevertheless, we prove that disorder implicitly provides more information than order does. For the general case of searching an array of multidimensional keys, whose comparison cost is proportional to their length (and hence cannot be considered a constant), we demonstrate that “suitable ” disorder gives better bounds than those derivable by using the natural lexicographic order. We start out from previous work done by Andersson, Hagerup, H˚astad and Petersson [SIAM Journal on Computing, 30(2), 2001], who proved that k log log n
Implicit Btrees: A New Data Structure for the Dictionary Problem*
, 2004
"... Abstract An implicit data structure for the dictionary problem maintains n data values in the first n locations of an array in such a way that it efficiently supports the operations insert, delete and search. No information other than that in O(1) memory cells and in the input data is to beretained ..."
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Abstract An implicit data structure for the dictionary problem maintains n data values in the first n locations of an array in such a way that it efficiently supports the operations insert, delete and search. No information other than that in O(1) memory cells and in the input data is to beretained; and the only operations performed on the data values (other than reads and writes) are comparisons. This paper describes the Implicit Btree, a new data structure supporting theseoperations in O(logB n) block transfers like in regular Btrees, under the realistic assumptionthat a block stores B = \Omega (log n) keys, so that reporting r consecutive keys in sorted order hasa cost of O(logB n + r/B) block transfers. En route a number of space efficient techniques forhandling segments of a large array in a memory hierarchy are developed. Being implicit, the proposed data structure occupies exactly dn/Be blocks of memory after each update, where nis the number of keys after each update and B is the number of keys contained in a memoryblock. In main memory, the time complexity of the operations is
Optimal implicit dictionaries over . . .
, 2005
"... An array of n distinct keys can be sorted for logarithmic searching or can be organized as a heap for logarithmic updating, but it is unclear how to attain logarithmic time for both searching and updating. This natural question dates back to the heap of Williams and Floyd in the sixties and relates ..."
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An array of n distinct keys can be sorted for logarithmic searching or can be organized as a heap for logarithmic updating, but it is unclear how to attain logarithmic time for both searching and updating. This natural question dates back to the heap of Williams and Floyd in the sixties and relates to the fundamental issue whether additional space besides those for the keys gives more computational power in dictionaries and how data ordering helps. Implicit data structures have been introduced in the eighties with this goal, providing the best bound of O(log² n) time, until a recent result showing O(log² n / log log n) time. In this paper we describe the flat implicit tree, which is the first data structure obtaining O(log n) time for search and (amortized) update using an array of n cells.