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

2. 58131385514113711531054 Sorting (Integers) 00001010 0000110100110101001101100011011100111010100010011000101010001101 10111353545558137138141 00111010000011011000101000110111100011011000100100001010001101010000101000110110 Input Output 12345678910 123456789 w bits n integers Trie 12345678910 Patricia trie (branching charcters only) permutation 0 0 0 1 1 1 1 0 1 0 1 0 1 0 1 0 0 0 1 1 1 1 0 0 0 1 1 1 0 0 0 1 10 110 0 0 0 - - 1 - 1 - 1 - - - - - 1 0 - 0 1 - 1 - 0 - - - 1 - - - 0 - 10 --0 0

3. Comparison Based Sorting 1113535558137138141 58 13 138 55 141 54 137 53 10 11 1358 55138 137141 1153 1054 135558138 1153137141 10111353545558137138141 next 135558138 1153137141 1113535558137138141 scanning

4. Sorting Networks O(n·log 2 n) size : Batcher 1968 O(n·log n) size : Ajtai, Komlós, Szemerédi 1983; Paterson 1990; Goodrich 2014 138 13 58 55 13 138 55 58 13 55 58 138 55 58 138 13 58 55 13 55 58 138 InputOutput  Original motivated by hardware implementations  Renewed interest since data-oblivious ̶applications in privacy preserving computations ̶allows for parallel computations on bit-level and GPU

5. Results Bucket sortO(n+2 w ) Radix sort; Hollerith 1887 van Emde Boas 1975 Willard 1983 superlinear space expected Kirkpatrick and Reicsh 1983 Merge sort: von Neumann 1945 comparison based optimal Thorup and Han 2002expected Andersson et al. 1998O(n) NEWO(n) Time per element w log 2 n·loglog nlog 2+ε n O(1) log n 1 2 3 1 2 3 NEW

6. Open Question Sorting in linear time for all w? w log 2 n·loglog nlog 2+ε n O(1) log n NEW Time per element