[PPT] - The Probabilistic Method Week 7: Alterations Joshua Brody PowerPoint Presentation

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The Probabilistic Method

Joshua Brody CS49/Math59 Fall 2015

Week 7: Alterations

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Reading Quiz

(A) Generate GOOD object by choosing random object, then removing any badness “by hand” (B) Generate GOOD object by choosing random object, then showing BAD probability < 1 (C) Generate GOOD object by choosing random object, then using Chernoff Bound (D) Choose edges of graph by alternating between red edges and blue edges (E) None of the above

What is the alteration technique?

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Reading Quiz

(A) Generate GOOD object by choosing random object, then removing any badness “by hand” (B) Generate GOOD object by choosing random object, then showing BAD probability < 1 (C) Generate GOOD object by choosing random object, then using Chernoff Bound (D) Choose edges of graph by alternating between red edges and blue edges (E) None of the above

What is the alteration technique?

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Ramsey Theory

R(k,k) := smallest n such that for every two-coloring

f Kn, there is red Kk subgraph or a blue Kk subgraph.

Basic Method: If then R(k,k) > n.

( )

n k

1-( )

k 2

2 < 1 Alterations: R(k,k) > n -

( )

n k

1-( )

k 2

2

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Ramsey Theory

R(k,k) := smallest n such that for every two-coloring

f Kn, there is red Kk subgraph or a blue Kk subgraph.

Basic Method: R(k,k) > (1+o(1)) k2k/e√2 Alterations: R(k,k) > (1+o(1)) k2k/e

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Clicker Question

(A) Yes (B) No (C) Maybe (D)None of the above

Will alterations always give improvement over basic probabilistic method approach?

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Independent Sets

Independent Set: set of

vertices which share no edges

α(G): size of largest

independent set

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Independent Sets

Independent Set: set of

vertices which share no edges

α(G): size of largest

independent set Theorem: If G = (V,E) is a graph with n vertices and nd/2 edges, then α(G) ≥ n/2d

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Theorem: If G = (V,E) is a graph with n vertices and nd/2 edges, then α(G) ≥ n/2d

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Theorem: If G = (V,E) is a graph with n vertices and nd/2 edges, then α(G) ≥ n/2d

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Theorem: If G = (V,E) is a graph with n vertices and nd/2 edges, then α(G) ≥ n/2d

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Theorem: If G = (V,E) is a graph with n vertices and nd/2 edges, then α(G) ≥ n/2d

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Clicker Question

(A) E[Y] = (nd/2)*p (B) E[Y] = nd*p (C) E[Y] = nd*p2 (D) E[Y] = 2nd*p2 (E) None of the above

What is E[Y]?

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Clicker Question

(A) E[Y] = (nd/2)*p (B) E[Y] = nd*p (C) E[Y] = nd*p2 (D) E[Y] = 2nd*p2 (E) None of the above

What is E[Y]?

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Minimizing Area

Let S be a set of n points on unit square [0,1]x[0,1] T(S) := minP

,Q,R ∈ S area(PQR)

T(n) := maxS T(S) Theorem: T(n) = Ω(1/n2)

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Clicker Question

(A) E[#small triangles] ≤ (n choose 3)*16*∏/100n2 (B) E[#small triangles] ≤ (n choose 3)/100n2 (C) E[#small triangles] ≤ (2n choose 3)*16*∏/100n2 (D) E[#small triangles] ≤ 2nd*p2 (E) None of the above

What is E[#triangles w/area ≤ 1/100n2]?

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Clicker Question

(A) E[#small triangles] ≤ (n choose 3)*16*∏/100n2 (B) E[#small triangles] ≤ (n choose 3)/100n2 (C) E[#small triangles] ≤ (2n choose 3)*16*∏/100n2 (D) E[#small triangles] ≤ 2nd*p2 (E) None of the above

What is E[#triangles w/area ≤ 1/100n2]?

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Minimizing Area

Let S be a set of n points on unit square [0,1]x[0,1] T(S) := minP

,Q,R ∈ S area(PQR)

T(n) := maxS T(S) Theorem: T(n) = Ω(1/n2)

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