The Probabilistic Method Week 10: Applications Joshua Brody CS49/Math59 Fall 2015
Reading Quiz What is the Hamming Distance between two n-bit strings x,y? (A) dist(x,y) = #i such that x i != y i (B) dist(x,y) = sqrt( ∑ i |x i - y i |) (C) dist(x,y) = max i |x i - y i | (D) multiple answers correct (E) none of the above
Reading Quiz What is the Hamming Distance between two n-bit strings x,y? (A) dist(x,y) = #i such that x i != y i (B) dist(x,y) = sqrt( ∑ i |x i - y i |) (C) dist(x,y) = max i |x i - y i | (D) multiple answers correct (E) none of the above
Coding Theory [Shannon 48] 00010101
Coding Theory [Shannon 48] 01001101 Problem: • p -fraction of bits are corrupted • Bob doesn’t know which bits get corrupted.
Coding Theory [Shannon 48] 01001101 How can Alice and Bob reliably Problem: communicate over an unreliable channel? • p -fraction of bits are corrupted • Bob doesn’t know which bits get corrupted.
Error-Correcting Codes 10011
Error-Correcting Codes 10011 0001010101
Error-Correcting Codes 10011 0100110100
Error-Correcting Codes 10011 10011 0100110100
Clicker Question What is the rate and distance of a 5-repetition code? (A) (R, 휹 ) = (1/3, 1/3) (B) (R, 휹 ) = (1/5, 1/5) (C) (R, 휹 ) = (1/5, 1) (D) (R, 휹 ) = (2/5, 1/5) (E) (R, 휹 ) = (1/5, 2/5)
Clicker Question What is the rate and distance of a 5-repetition code? (A) (R, 휹 ) = (1/3, 1/3) (B) (R, 휹 ) = (1/5, 1/5) (C) (R, 휹 ) = (1/5, 1) (D) (R, 휹 ) = (2/5, 1/5) (E) (R, 휹 ) = (1/5, 2/5)
Adjacency Matrix of Magic Graphs a 11 a 12 a 13 a 14 a 15 a 16 a 17 a 18 a 21 a 22 a 23 a 24 a 25 a 26 a 27 a 28 a 31 a 32 a 33 a 34 a 35 a 36 a 37 a 38 a 41 a 42 a 43 a 44 a 45 a 46 a 47 a 48 a 51 a 52 a 53 a 54 a 55 a 56 a 57 a 58 a 61 a 62 a 63 a 64 a 65 a 66 a 67 a 68
Adjacency Matrix of Magic Graphs columns indexed by vertices in L a 11 a 12 a 13 a 14 a 15 a 16 a 17 a 18 a 21 a 22 a 23 a 24 a 25 a 26 a 27 a 28 rows indexed by a 31 a 32 a 33 a 34 a 35 a 36 a 37 a 38 vertices in R a 41 a 42 a 43 a 44 a 45 a 46 a 47 a 48 a 51 a 52 a 53 a 54 a 55 a 56 a 57 a 58 a 61 a 62 a 63 a 64 a 65 a 66 a 67 a 68
Adjacency Matrix of Magic Graphs columns indexed a ij = 1 iff (v j , r i ) is by vertices in L edge in magic graph a 11 a 12 a 13 a 14 a 15 a 16 a 17 a 18 a 21 a 22 a 23 a 24 a 25 a 26 a 27 a 28 rows indexed by a 31 a 32 a 33 a 34 a 35 a 36 a 37 a 38 vertices in R a 41 a 42 a 43 a 44 a 45 a 46 a 47 a 48 a 51 a 52 a 53 a 54 a 55 a 56 a 57 a 58 a 61 a 62 a 63 a 64 a 65 a 66 a 67 a 68
Magic Graph Matrix Multiplication a 11 a 12 a 13 a 14 a 15 a 16 a 17 a 18 a 21 a 22 a 23 a 24 a 25 a 26 a 27 a 28 a 31 a 32 a 33 a 34 a 35 a 36 a 37 a 38 a 41 a 42 a 43 a 44 a 45 a 46 a 47 a 48 a 51 a 52 a 53 a 54 a 55 a 56 a 57 a 58 a 61 a 62 a 63 a 64 a 65 a 66 a 67 a 68
Magic Graph Matrix Multiplication a 11 a 12 a 13 a 14 a 15 a 16 a 17 a 18 x 1 y 1 a 21 a 22 a 23 a 24 a 25 a 26 a 27 a 28 x 2 y 2 = a 31 a 32 a 33 a 34 a 35 a 36 a 37 a 38 x 3 y 3 * a 41 a 42 a 43 a 44 a 45 a 46 a 47 a 48 x 4 y 4 a 51 a 52 a 53 a 54 a 55 a 56 a 57 a 58 x 5 y 5 a 61 a 62 a 63 a 64 a 65 a 66 a 67 a 68 x 6 y 6 x 7 x 8
Magic Graph Matrix Multiplication a 11 a 12 a 13 a 14 a 15 a 16 a 17 a 18 x 1 y 1 a 21 a 22 a 23 a 24 a 25 a 26 a 27 a 28 x 2 y 2 = a 31 a 32 a 33 a 34 a 35 a 36 a 37 a 38 x 3 y 3 * a 41 a 42 a 43 a 44 a 45 a 46 a 47 a 48 x 4 y 4 a 51 a 52 a 53 a 54 a 55 a 56 a 57 a 58 x 5 y 5 a 61 a 62 a 63 a 64 a 65 a 66 a 67 a 68 x 6 y 6 x 7 y i = ∑ a ik * x k x 8 k
The Probabilistic Method
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