Optimal dislocation with persistent errors in subquadratic time. ETH Zurich

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1 Optimal dislocation with persistent errors in subquadratic time Barbara Geissmann, Chih-Hung Liu, Stefano Leucci, Paolo Penna ETH Zurich

2 The Problem Sorting with erroneous comparisons

3 The Problem Sorting with erroneous comparisons

4 The Problem Sorting with erroneous comparisons

5 The Problem

6 The Problem 4>

7 The Problem 4<

8 The Problem 5<

9 The Problem 5< error probability p constant 1 2 6

10 The Problem 5< error probability p constant independent for each pair 1 2 6

11 The Problem 5< error probability p constant independent for each pair persistent errors 1 2 6

12 The Problem 5< error probability p constant independent for each pair persistent errors %

13 The Problem Repeating does not help 5< error probability p constant independent for each pair persistent errors 1 2 6

14 The Problem Repeating does not help 5<

15 Can you sort? Algorithm errors

16 Can you sort? Algorithm errors

17 Can you sort? Algorithm errors Appro Sorted

18 Can you sort? Algorithm errors Appro Sorted

19 Can you sort? Algorithm errors Dislocation

20 What can be done?

21 Prior Results

22 Prior Results MAX O(log n) Dislocation TOTAL O(n) Braverman & Mossel (SODA 08)

23 Prior Results Dislocation Time: MAX TOTAL O(n 3+C ) O(log n) O(n) Braverman & Mossel (SODA 08)

24 Prior Results Dislocation Time: MAX TOTAL O(n 3+C ) O(log n) O(n) (1/2 p) 4 Braverman & Mossel (SODA 08)

25 Prior Results Dislocation Time: MAX TOTAL O(n 3+C ) O(log n) O(n) O(n 2 ) O(log n) Braverman & Mossel (SODA 08) Klein, Penninger, Sohler, Woodruff (ESA 11)

26 Prior Results Dislocation Time: MAX TOTAL O(n 3+C ) O(log n) O(n) O(n 2 ) O(n 2 ) O(log n) O(log n) O(n) Braverman & Mossel (SODA 08) Klein, Penninger, Sohler, Woodruff (ESA 11) Geissmann, Leucci, Liu, Penna (ISAAC 17)

27 Prior Results Time: MAX Dislocation TOTAL O(n 3+C ) O(log n) O(n) O(n 2 ) O(log n) O(n 2 ) O(log n) O(n) Ω(log n) Ω(n) Braverman & Mossel (SODA 08) Klein, Penninger, Sohler, Woodruff (ESA 11) Geissmann, Leucci, Liu, Penna (ISAAC 17)

28 Prior Results Time: MAX Dislocation TOTAL O(n 3+C ) O(log n) O(n) O(n 2 ) O(log n) O(n 2 ) O(log n) O(n) Ω(log n) Ω(n) Braverman Subquadratic & Mossel (SODA 08) time? Klein, Penninger, Sohler, Woodruff (ESA 11) Geissmann, Leucci, Liu, Penna (ISAAC 17)

29 Our Contribution YES

30 Our Contribution YES Dislocation Time: MAX TOTAL O(n 3/2 ) O(log n) O(n)

31 Our Contribution YES Dislocation Time: MAX TOTAL O(n 3/2 ) O(log n) O(n) randomized algorithm derandomized algorithm

32 O(n 2 )-Time Algorithm Geissmann, Leucci, Liu, Penna (ISAAC 17)

33 O(n 2 )-Time Algorithm errors well-spread success Geissmann, Leucci, Liu, Penna (ISAAC 17)

34 O(n 2 )-Time Algorithm errors well-spread success initial dislocation D time O(Dn) Geissmann, Leucci, Liu, Penna (ISAAC 17)

35 O(n 2 )-Time Algorithm D d errors well-spread success initial dislocation D time O(Dn) Geissmann, Leucci, Liu, Penna (ISAAC 17)

36 O(n 2 )-Time Algorithm D d with high prob d = O(log n) errors well-spread success initial dislocation D time O(Dn) Geissmann, Leucci, Liu, Penna (ISAAC 17)

37 O(n 2 )-Time Algorithm D n d errors well-spread success initial dislocation D time O(Dn) Geissmann, Leucci, Liu, Penna (ISAAC 17)

38 O(n 2 )-Time Algorithm New Algo D d errors well-spread success initial dislocation D time O(Dn) Geissmann, Leucci, Liu, Penna (ISAAC 17)

39 O(n 2 )-Time Algorithm New Algo D d errors well-spread success initial dislocation D time O(Dn) tricky part Geissmann, Leucci, Liu, Penna (ISAAC 17)

40 Simple Faster Algo New Algo D d

41 Simple Faster Algo D d

42 n Simple Faster Algo D d

43 n Simple Faster Algo D d

44 n Simple Faster Algo D d

45 n Simple Faster Algo O(n) O(n) O(n) D d

46 n Simple Faster Algo O(n) O(n) O(n) O(n 3/2 ) D d

47 Simple Faster Algo n O(n 3/2 ) NOT ENOUGH! D d

48 Simple Faster Algo n O(n 3/2 ) 1 NOT ENOUGH! D d

49 Simple Faster Algo n O(n 3/2 ) 1 NOT ENOUGH! D d 1

50 Simple Faster Algo n O(n 3/2 ) 1 NOT ENOUGH! 1 D 1 d

51 Simple Faster Algo O(n 3/2 ) D d

52 Simple Faster Algo O(n 3/2 ) D d

53 Simple Faster Algo O(n 3/2 ) D d

54 Simple Faster Algo O(n 3/2 ) STILL NOT ENOUGH D d

55 Simple Faster Algo 1 2 n O(n 3/2 ) D d

56 Simple Faster Algo 1 2 n O(n 3/2 ) 1 2 D d

57 Simple Faster Algo 1 2 n O(n 3/2 ) 1 2 D d

58 Simple Faster Algo 1 2 n O(n 3/2 ) 1 2 n D d

59 Simple Faster Algo 1 2 n O(n 3/2 ) 1 2 n 1 D d

60 Simple Faster Algo 1 2 n O(n 3/2 ) 1 2 n 1 2 D d

61 Simple Faster Algo 1 2 n O(n 3/2 ) 1 2 n 1 2 D n d

62 Simple Faster Algo random permutation O(n 3/2 ) D d

63 Simple Faster Algo random permutation O(n 3/2 ) D = n 3/4 d

64 Simple Faster Algo random permutation O(n 3/2 ) D = n 3/4 O(n 7/4 ) d

65 That was the simple version...

66 O(n 2 ) O(n 2 δ )

67 O(n 2 ) O(n 2 δ ) O(n 2 δ )

68 O(n 2 ) O(n 2 δ ) O(n 2 δ )

69 O(n 2 ) O(n 2 δ ) O(n 2 δ ) O(n 3/2 )

70 Part II: Derandomization

71 Comparisons Randomness < > p 1 p > < p 1 p? XOR?? < > 1 c log n 2 ± 1 1 n 4 2 ± 1 n 4 Comparisons One random bit

72 Naive Approach k??? k(n k) log n

73 Naive Approach k??? SORT k(n k) log n

74 Naive Approach k??? SORT k(n k) log n REINSERT

75 Open Questions Time: MAX Dislocation TOTAL O(n 3+C ) O(log n) O(n 2 ) O(log n) O(n) O(n 2 ) O(n 3/2 ) O(log n) O(log n) O(n) O(n) Braverman & Mossel (SODA 08) Klein, Penninger, Sohler, Woodruff (ESA 11) Geissmann, Leucci, Liu, Penna (ISAAC 17)

76 Open Questions Time: MAX Dislocation TOTAL O(n 3+C ) O(log n) O(n 2 ) O(log n) O(n) O(n 2 ) O(n 3/2 ) O(log n) O(log n) O(n) O(n) Faster? O(n log n) comparisons Braverman & Mossel (SODA 08) Klein, Penninger, Sohler, Woodruff (ESA 11) Geissmann, Leucci, Liu, Penna (ISAAC 17)

77 Open Questions Time: MAX Dislocation TOTAL O(n 3+C ) O(log n) O(n) p < 1/16 O(n 2 ) O(log n) O(n 2 ) O(n 3/2 ) O(log n) O(log n) O(n) O(n) Braverman & Mossel (SODA 08) Klein, Penninger, Sohler, Woodruff (ESA 11) Geissmann, Leucci, Liu, Penna (ISAAC 17)

78 Open Questions Time: MAX Dislocation TOTAL O(n 3+C ) O(log n) O(n) p < 1/16 O(n 2 ) O(log n) O(n 2 ) O(n 3/2 ) O(log n) O(log n) O(n) O(n) Any p < 1/2? Braverman & Mossel (SODA 08) Klein, Penninger, Sohler, Woodruff (ESA 11) Geissmann, Leucci, Liu, Penna (ISAAC 17)

79 Open Questions Time: MAX Dislocation TOTAL O(n 3+C ) O(log n) O(n) p < 1/16 O(n 2 ) O(log n) O(n 2 ) O(n 3/2 ) O(log n) O(log n) O(n) O(n) Any p < 1/2? p < 1/2 Braverman & Mossel (SODA 08) Klein, Penninger, Sohler, Woodruff (ESA 11) Geissmann, Leucci, Liu, Penna (ISAAC 17)

80 Open Questions Time: MAX Dislocation TOTAL O(n 3+C ) O(log n) O(n 2 ) O(log n) O(n) O(n 2 ) O(n 3/2 ) O(log n) O(log n) O(n) O(n) Other error models? Braverman & Mossel (SODA 08) Klein, Penninger, Sohler, Woodruff (ESA 11) Geissmann, Leucci, Liu, Penna (ISAAC 17)

81 Tahnk You