Optimal Resilient Dynamic Dictionaries⋆

Brodal, Gerth Stølting; Fagerberg, Rolf; Jørgensen, Allan Grønlund; Moruz, Gabriel; Mølhave, Thomas

doi:10.7146/dpb.v36i585.7222

“…We show that Multiway Merge-Sort achieves this tradeoff by adopting a multiway merger with a low number of branch mispredictions. For adaptive sorting algorithms we similarly obtain that an algorithm performing O(dn(1 + log(1 + Inv /n))) comparisons must perform Ω(n log d (1 + Inv /n)) branch mispredictions, where Inv is the number of 1 The paper Optimal Resilient Dynamic Dictionaries by Gerth S. Brodal, Rolf Fagerberg, Irene Finocchi, Fabrizio Grandoni, Giuseppe F. Italiano, Allan G. Jørgensen, Gabriel Moruz, and Thomas Mølhave [24], has been accepted for publication as a merged paper. The original contributions were Resilient Search Trees: Randomization and Prejudice, by Irene Finocchi, Fabrizio Grandoni, and Giuseppe F. Italiano, and Optimal Resilient Dynamic Dictionaries, by Gerth S. Brodal, Rolf Fagerberg, Allan G. Jørgensen, Gabriel Moruz, and Thomas Mølhave.…”

Section: Contributionsmentioning

confidence: 80%

“…The original contributions were Resilient Search Trees: Randomization and Prejudice, by Irene Finocchi, Fabrizio Grandoni, and Giuseppe F. Italiano, and Optimal Resilient Dynamic Dictionaries, by Gerth S. Brodal, Rolf Fagerberg, Allan G. Jørgensen, Gabriel Moruz, and Thomas Mølhave. The contributions of the latter will appear as a technical report in [27].…”

Section: Contributionsmentioning

confidence: 99%

“…Optimal resilient dictionaries (Chapter 10, [24]). We introduce in [24] two optimal resilient static dictionaries, a randomized one and a deterministic one.…”

Section: Contributionsmentioning

confidence: 99%

“…Optimal resilient dictionaries (Chapter 10, [24]). We introduce in [24] two optimal resilient static dictionaries, a randomized one and a deterministic one. The randomized dictionary supports searches in O(log n+δ) expected time using O(log δ) random bits in the worst case under the assumption of an oblivious adversary.…”

Section: Contributionsmentioning

confidence: 99%

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Various computer hardware components are affecting the running time of algorithms in different proportions, or may have severe implications on the accuracy of algorithms. In this dissertation we propose algorithms and data structures that are efficient and robust with respect to different hardware factors. The hardware factors affecting the running time that we consider include branch mispredictions occurring in the processor, memory transfers occurring between consecutive levels of the memory hierarchy in modern computers, as well as the high rate sequential access on modern hard-disks which is a major motivation for developing streaming algorithms. Regarding the factors affecting the accuracy of algorithms, we consider soft memory errors which determine corruptions in RAM memories.Branch mispredictions incur significant performance losses for algorithms. First, we show that the running time of randomized Quicksort is adaptive, i.e. it depends on the presortedness of the input. We prove theoretically that the number of element swaps performed by Quicksort depends on the number Inv of inversions in the input and show experimentally that the number of element swaps is closely correlated to the number of branch mispredictions. We then give lower bound trade-offs between comparisons and branch mispredictions for sorting and adaptive sorting, and propose sorting and adaptive sorting algorithms matching these bounds. Finally, we show experimentally that for random queries perfectly balanced binary search trees can be outperformed by skewed binary search trees, i.e. trees for which at any given node there is a fixed ratio between the nodes in the left and right subtrees. This happens because the skewed binary search trees perform more comparisons, but fewer branch mispredictions, compared to perfectly binary search trees for random queries.Memory transfers occurring between consecutive levels of the memory hierarchy in modern computers are modeled in the I/O model and cache oblivious model, and the complexity of algorithms is given by the number of memory transfers performed. We introduce I/O lower bounds for adaptive sorting algorithms and give two algorithms that are optimal and I/O optimal with respect to Inv .In a streaming setting, algorithms are restricted to access data sequentially, while having at their disposal a working memory that can be accessed for free, but which is usually much smaller than the problem size. We give general reductions from parallel algorithms to streaming algorithms, and show that we can obtain optimal algorithms (up to poly-log factors) for several combinatorial problems, such as sorting, connected components, minimum spanning tree, v biconnected components, or maximal independent set.To analyze memory corruptions, we use the faulty-memory RAM model proposed by Finocchi and Italiano. In this model, any memory cell can get corrupted at any time during the execution of an algorithm, with no possibility of distinguishing between corrupted and uncorrupted cells. The number of corr...

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“…We show that Multiway Merge-Sort achieves this tradeoff by adopting a multiway merger with a low number of branch mispredictions. For adaptive sorting algorithms we similarly obtain that an algorithm performing O(dn(1 + log(1 + Inv /n))) comparisons must perform Ω(n log d (1 + Inv /n)) branch mispredictions, where Inv is the number of 1 The paper Optimal Resilient Dynamic Dictionaries by Gerth S. Brodal, Rolf Fagerberg, Irene Finocchi, Fabrizio Grandoni, Giuseppe F. Italiano, Allan G. Jørgensen, Gabriel Moruz, and Thomas Mølhave [24], has been accepted for publication as a merged paper. The original contributions were Resilient Search Trees: Randomization and Prejudice, by Irene Finocchi, Fabrizio Grandoni, and Giuseppe F. Italiano, and Optimal Resilient Dynamic Dictionaries, by Gerth S. Brodal, Rolf Fagerberg, Allan G. Jørgensen, Gabriel Moruz, and Thomas Mølhave.…”

Section: Contributionsmentioning

confidence: 88%

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Various computer hardware components are affecting the running time of algorithms in different proportions, or may have severe implications on the accuracy of algorithms. In this dissertation we propose algorithms and data structures that are efficient and robust with respect to different hardware factors. The hardware factors affecting the running time that we consider include branch mispredictions occurring in the processor, memory transfers occurring between consecutive levels of the memory hierarchy in modern computers, as well as the high rate sequential access on modern hard-disks which is a major motivation for developing streaming algorithms. Regarding the factors affecting the accuracy of algorithms, we consider soft memory errors which determine corruptions in RAM memories.Branch mispredictions incur significant performance losses for algorithms. First, we show that the running time of randomized Quicksort is adaptive, i.e. it depends on the presortedness of the input. We prove theoretically that the number of element swaps performed by Quicksort depends on the number Inv of inversions in the input and show experimentally that the number of element swaps is closely correlated to the number of branch mispredictions. We then give lower bound trade-offs between comparisons and branch mispredictions for sorting and adaptive sorting, and propose sorting and adaptive sorting algorithms matching these bounds. Finally, we show experimentally that for random queries perfectly balanced binary search trees can be outperformed by skewed binary search trees, i.e. trees for which at any given node there is a fixed ratio between the nodes in the left and right subtrees. This happens because the skewed binary search trees perform more comparisons, but fewer branch mispredictions, compared to perfectly binary search trees for random queries.Memory transfers occurring between consecutive levels of the memory hierarchy in modern computers are modeled in the I/O model and cache oblivious model, and the complexity of algorithms is given by the number of memory transfers performed. We introduce I/O lower bounds for adaptive sorting algorithms and give two algorithms that are optimal and I/O optimal with respect to Inv .In a streaming setting, algorithms are restricted to access data sequentially, while having at their disposal a working memory that can be accessed for free, but which is usually much smaller than the problem size. We give general reductions from parallel algorithms to streaming algorithms, and show that we can obtain optimal algorithms (up to poly-log factors) for several combinatorial problems, such as sorting, connected components, minimum spanning tree, v biconnected components, or maximal independent set.To analyze memory corruptions, we use the faulty-memory RAM model proposed by Finocchi and Italiano. In this model, any memory cell can get corrupted at any time during the execution of an algorithm, with no possibility of distinguishing between corrupted and uncorrupted cells. The number of corr...

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Jørgensen

¹

,

Phillips

²

,

Löffler³

2011

Lecture Notes in Computer Science

1

0

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Various computer hardware components are affecting the running time of algorithms in different proportions, or may have severe implications on the accuracy of algorithms. In this dissertation we propose algorithms and data structures that are efficient and robust with respect to different hardware factors. The hardware factors affecting the running time that we consider include branch mispredictions occurring in the processor, memory transfers occurring between consecutive levels of the memory hierarchy in modern computers, as well as the high rate sequential access on modern hard-disks which is a major motivation for developing streaming algorithms. Regarding the factors affecting the accuracy of algorithms, we consider soft memory errors which determine corruptions in RAM memories.Branch mispredictions incur significant performance losses for algorithms. First, we show that the running time of randomized Quicksort is adaptive, i.e. it depends on the presortedness of the input. We prove theoretically that the number of element swaps performed by Quicksort depends on the number Inv of inversions in the input and show experimentally that the number of element swaps is closely correlated to the number of branch mispredictions. We then give lower bound trade-offs between comparisons and branch mispredictions for sorting and adaptive sorting, and propose sorting and adaptive sorting algorithms matching these bounds. Finally, we show experimentally that for random queries perfectly balanced binary search trees can be outperformed by skewed binary search trees, i.e. trees for which at any given node there is a fixed ratio between the nodes in the left and right subtrees. This happens because the skewed binary search trees perform more comparisons, but fewer branch mispredictions, compared to perfectly binary search trees for random queries.Memory transfers occurring between consecutive levels of the memory hierarchy in modern computers are modeled in the I/O model and cache oblivious model, and the complexity of algorithms is given by the number of memory transfers performed. We introduce I/O lower bounds for adaptive sorting algorithms and give two algorithms that are optimal and I/O optimal with respect to Inv .In a streaming setting, algorithms are restricted to access data sequentially, while having at their disposal a working memory that can be accessed for free, but which is usually much smaller than the problem size. We give general reductions from parallel algorithms to streaming algorithms, and show that we can obtain optimal algorithms (up to poly-log factors) for several combinatorial problems, such as sorting, connected components, minimum spanning tree, v biconnected components, or maximal independent set.To analyze memory corruptions, we use the faulty-memory RAM model proposed by Finocchi and Italiano. In this model, any memory cell can get corrupted at any time during the execution of an algorithm, with no possibility of distinguishing between corrupted and uncorrupted cells. The number of corr...

show abstract

Optimal Resilient Dynamic Dictionaries⋆

Cited by 5 publications

References 3 publications

Algorithms and Data Structures

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