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
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