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Efficient Integer Sorting in the RAM Model: Techniques and Challenges

This work investigates efficient sorting algorithms for integers within the RAM model framework, focusing on techniques such as merging networks and Patricia tries. Presented at the 14th Scandinavian Workshop on Algorithm Theory, the research delves into comparison-based sorting efficiencies and the quest for linear-time sorting methods for various input sizes. The challenges posed by hardware implementations and the implications for privacy-preserving computations underscore the importance of these algorithms in contemporary computing environments.

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Efficient Integer Sorting in the RAM Model: Techniques and Challenges

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  1. Sorting Integers in the RAM Model Gerth Stølting Brodal Joint work with Djamal Belazzougui and Jesper Sindal Nielsen (to bepresented at the 14th Scandinavian Workshop on Algorithm Theory) MADALGO Review Meeting, April 28, 2014

  2. Sorting (Integers) n integers Input w bits 0 0 1 1 0 - - 0 0 - 0 0 1 1 0 - 0 - 1 - 1 - - 1 0 0 1 1 Output 1 1 0 0 - 1 0 - 1 1 1 1 - 0 - 1 0 0 1 1 1 - 0 0 - 0 - 1 0 0 1 1 - 1 1 - 0 0 - 1 0 - 1 - 0 - Trie Patricia trie (branchingcharctersonly) 0 0 permutation 9 9 7 7 2 2 8 8 10 10 4 4 1 1 6 6 3 3 5 5

  3. ComparisonBasedSorting • Merge sort: von Neumann 1945 • Ford et al. 1959: Lowerbound≥ comparions next scanning next scanning next scanning comparions

  4. Sorting Networks O(n·log2n) size : Batcher 1968 O(n·logn) size : Ajtai, Komlós, Szemerédi 1983; Paterson 1990; Goodrich 2014 Input Output 138 13 13 138 13 13 138 58 55 13 55 58 55 55 58 58 58 55 58 138 55 138 • Original motivated by hardware implementations • Renewedinterestsincedata-oblivious • applications in privacypreservingcomputations • allows for parallel computations on bit-leveland GPU

  5. Time per element Results 2 NEW 1 3 O(1) w log n log2n·loglogn log2+εn 1 2 3

  6. Open Question Time per element Sortingin lineartime for all w? NEW O(1) w log n log2n·loglogn log2+εn

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