The Probabilistic Method Week 10: Applications Joshua Brody - - PowerPoint PPT Presentation

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The Probabilistic Method Week 10: Applications Joshua Brody - - PowerPoint PPT Presentation

Jan 30, 2023 242 likes •449 views

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)

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The Probabilistic Method

Joshua Brody CS49/Math59 Fall 2015

Week 10: Applications

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Reading Quiz

(A) dist(x,y) = #i such that xi != yi (B) dist(x,y) = sqrt(∑i |xi - yi|) (C) dist(x,y) = maxi |xi - yi| (D) multiple answers correct (E) none of the above

What is the Hamming Distance between two n-bit strings x,y?

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Reading Quiz

(A) dist(x,y) = #i such that xi != yi (B) dist(x,y) = sqrt(∑i |xi - yi|) (C) dist(x,y) = maxi |xi - yi| (D) multiple answers correct (E) none of the above

What is the Hamming Distance between two n-bit strings x,y?

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Coding Theory

[Shannon 48]

00010101

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Coding Theory

[Shannon 48]

Problem:

  • p-fraction of bits are corrupted
  • Bob doesn’t know which bits get corrupted.

01001101

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Coding Theory

[Shannon 48]

Problem:

  • p-fraction of bits are corrupted
  • Bob doesn’t know which bits get corrupted.

How can Alice and Bob reliably communicate over an unreliable channel?

01001101

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SLIDE 7

Error-Correcting Codes

10011

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Error-Correcting Codes

0001010101 10011

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Error-Correcting Codes

0100110100 10011

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SLIDE 10

Error-Correcting Codes

0100110100 10011 10011

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Clicker Question

(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)

What is the rate and distance of a 5-repetition code?

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Clicker Question

(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)

What is the rate and distance of a 5-repetition code?

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SLIDE 13

Adjacency Matrix of Magic Graphs

a11 a12 a13 a14 a15 a16 a17 a18 a21 a22 a23 a24 a25 a26 a27 a28 a31 a32 a33 a34 a35 a36 a37 a38 a41 a42 a43 a44 a45 a46 a47 a48 a51 a52 a53 a54 a55 a56 a57 a58 a61 a62 a63 a64 a65 a66 a67 a68

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SLIDE 14

Adjacency Matrix of Magic Graphs

a11 a12 a13 a14 a15 a16 a17 a18 a21 a22 a23 a24 a25 a26 a27 a28 a31 a32 a33 a34 a35 a36 a37 a38 a41 a42 a43 a44 a45 a46 a47 a48 a51 a52 a53 a54 a55 a56 a57 a58 a61 a62 a63 a64 a65 a66 a67 a68

columns indexed by vertices in L rows indexed by vertices in R

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Adjacency Matrix of Magic Graphs

a11 a12 a13 a14 a15 a16 a17 a18 a21 a22 a23 a24 a25 a26 a27 a28 a31 a32 a33 a34 a35 a36 a37 a38 a41 a42 a43 a44 a45 a46 a47 a48 a51 a52 a53 a54 a55 a56 a57 a58 a61 a62 a63 a64 a65 a66 a67 a68

columns indexed by vertices in L rows indexed by vertices in R aij = 1 iff (vj, ri) is edge in magic graph

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Magic Graph Matrix Multiplication

a11 a12 a13 a14 a15 a16 a17 a18 a21 a22 a23 a24 a25 a26 a27 a28 a31 a32 a33 a34 a35 a36 a37 a38 a41 a42 a43 a44 a45 a46 a47 a48 a51 a52 a53 a54 a55 a56 a57 a58 a61 a62 a63 a64 a65 a66 a67 a68

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Magic Graph Matrix Multiplication

a11 a12 a13 a14 a15 a16 a17 a18 a21 a22 a23 a24 a25 a26 a27 a28 a31 a32 a33 a34 a35 a36 a37 a38 a41 a42 a43 a44 a45 a46 a47 a48 a51 a52 a53 a54 a55 a56 a57 a58 a61 a62 a63 a64 a65 a66 a67 a68 x1 x2 x3 x4 x5 x6 x7 x8

= *

y1 y2 y3 y4 y5 y6

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Magic Graph Matrix Multiplication

a11 a12 a13 a14 a15 a16 a17 a18 a21 a22 a23 a24 a25 a26 a27 a28 a31 a32 a33 a34 a35 a36 a37 a38 a41 a42 a43 a44 a45 a46 a47 a48 a51 a52 a53 a54 a55 a56 a57 a58 a61 a62 a63 a64 a65 a66 a67 a68 x1 x2 x3 x4 x5 x6 x7 x8

= *

y1 y2 y3 y4 y5 y6

yi = ∑ aik * xk

k

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SLIDE 19

The Probabilistic Method