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Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Union Bound, Geometric Variables, Coupon collector’s problem and Minimum Cut

Maria-Eirini Pegia

Study Group

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Context

1

Section 1: Union Bound

2

Section 2: Geometric variables

3

Section 3: Coupon collector’s problem

4

Section 4: Minimum Cut Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Union Bound

Boole’s inequality known as the union bound

P ✣ ❇ P P❼ ➁

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Union Bound

Boole’s inequality known as the union bound for any finite or countable set of events, the probability that at least one of the events happens is

no greater

than the sum of the probabilities of the individual events

P ✣ ❇ P P❼ ➁

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Union Bound

Boole’s inequality known as the union bound for any finite or countable set of events, the probability that at least one of the events happens is

no greater

than the sum of the probabilities of the individual events

Formally

for a countable set of events A1, A2, A3, ..., we have

P( ✣i Ai) ❇ Pi P❼Ai➁

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

George Boole ( 1815 - 1864 )

• English mathematician,

philosopher and logician

• differential equations, algebraic

logic

• author of The Laws of Thought

(1854)

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

George Boole

”No general method for the solution of questions in the theory of probabilities can be established which does not explicity recognize, not only the special numerical bases of the science, but also those universal laws of thought which are the basis of all reasoning, and which, whatever they may be as to their essence, are at least mathematical as to their form”

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Countable and uncountable sets

Let E set. Finite Set If E ① ❣ or E ✂ ➌1,2,...,n➑ then E is called finite ✂

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Countable and uncountable sets

Let E set. Finite Set If E ① ❣ or E ✂ ➌1,2,...,n➑ then E is called finite Countably infinity If E ✂ N then E countably infinite

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Countable and uncountable sets

Let E set. Finite Set If E ① ❣ or E ✂ ➌1,2,...,n➑ then E is called finite Countably infinity If E ✂ N then E countably infinite Countable Set If E is either a finite set or a countably infinite set, then E is called countable set.

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Countable and uncountable sets

Let E set. Finite Set If E ① ❣ or E ✂ ➌1,2,...,n➑ then E is called finite Countably infinity If E ✂ N then E countably infinite Countable Set If E is either a finite set or a countably infinite set, then E is called countable set. Uncountable Set If E is neither a finite set nor a countably infinite set, then E is called uncountable set (uncountably infinite set).

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Examples: countable set

Examples: ❺ Z is a countable (infinity) set. f : Z Ð N f ❼n➁ ➣ ➝ ➝ ➛ ➝ ➝ ↕ 2n if n ❈ 0 2❼✏n➁ ✔ 1 if n ❅ 0 ❺ ❃ Ô ✟

✏

❃ Ð

• ✏

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Examples: countable set

Examples: ❺ Z is a countable (infinity) set. f : Z Ð N f ❼n➁ ➣ ➝ ➝ ➛ ➝ ➝ ↕ 2n if n ❈ 0 2❼✏n➁ ✔ 1 if n ❅ 0 ❺ N x N is a countable (infinity) set. Let k ❃ N. fundamental theorem of arithmetic Ô ✟ k = 2m✏1 (2n - 1), m,n ❃ N f : N x N Ð N f(n,m) = 2m✏1 (2n - 1)

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Example: uncountable set

The set [0,1] is uncountable Proof: ❺ [0,1] is not finite. ❺ [0,1] is not countable infinity. ➌ ➑

• ✆

✆ ✆ ➯

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Example: uncountable set

The set [0,1] is uncountable Proof: ❺ [0,1] is not finite. ❺ [0,1] is not countable infinity. Assume [0,1] is countable infinity. Then [0,1] = ➌x1,x2,x3,...➑

• Divide [0,1] into 3 equal length intervals. 0, 1

3✆,1 3, 2 3✆,2 3,1✆

• Let x1 ➯ I1.

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Example: uncountable set

• Divide I1 into 3 equal length intervals.
• Let x2 ➯ I2 (I2 ❵ I1).
• I1 ❛ I2 ❛ I3 ❛ ...

such that xm ➯ Im and Length(In) = 1 3n Ð n➟ 0. Heine–Borel theorem Ô ✟ ✜✔➟

n1 In = ➌x➑ ❵ [0,1]

So x = xk for some k. ➯

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Example: uncountable set

• Divide I1 into 3 equal length intervals.
• Let x2 ➯ I2 (I2 ❵ I1).
• I1 ❛ I2 ❛ I3 ❛ ...

such that xm ➯ Im and Length(In) = 1 3n Ð n➟ 0. Heine–Borel theorem Ô ✟ ✜✔➟

n1 In = ➌x➑ ❵ [0,1]

So x = xk for some k. Contradiction ( xk ➯ Ik ). So [0,1] is uncountable.

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Georg Cantor ( 1845 - 1918 )

• German mathematician
• introduced the term countable set
• invented set theory
• Discrete Mathematics
• proved that R are more numerous

than the N Ô ✟ ➜ ”infinity of infinities”

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Cantor paradox

Input: line segment AB, length 1, which is washed by rain completely vertically. We place shelters in AB with the following procedure.

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Cantor paradox

Assume that we select randomly a point of AB and let as symbolized it by C = ➌ a point of AB that is washed ➑ C ❵ [0,1] Cantor set C is uncountable, compact (closed + bounded in Euclidean space) and has length 0. C’ = [0,1] ❷ C has length 1 1 3 ✔ 2 9 ✔ 4 27 ✔ ... 1 3 P➟

k0❼2

3➁k = 1

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Cantor paradox

❺ Set Theory Ð C is huge (uncountable)

_\_ ( " ) ) _/_

❺ Probability Theory Ð C is nothing

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Geometric distribution

Definition 1 The probability distribution of the number X of failures until the first success, supported on the set ➌0,1,2,3,...➑. Definition 2 The probability distribution of the number Y = X + 1 of Bernoulli trials needed to get one success, supported on the set ➌1,2,3,...➑.

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Geometric distribution

Definition 1 The probability distribution of the number X of failures until the first success, supported on the set ➌0,1,2,3,...➑. Definition 2 The probability distribution of the number Y = X + 1 of Bernoulli trials needed to get one success, supported on the set ➌1,2,3,...➑. Which of these one calls ”the” geometric distribution?

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

• p(x) = θ❼1 ✏ θ➁x , x = 0, 1, 2, ...
• P➟

x0 p❼x➁ = P➟ x0 θ❼1 ✏ θ➁x =

θ 1 ✏ ❼1 ✏ θ➁ = 1 Ô ✟ ❼ ✏ ➁ ❼ ✏ ➁ ✏

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

• p(x) = θ❼1 ✏ θ➁x , x = 0, 1, 2, ...
• P➟

x0 p❼x➁ = P➟ x0 θ❼1 ✏ θ➁x =

θ 1 ✏ ❼1 ✏ θ➁ = 1

• Let X the number of failures until the first success.

θ = const success probability S = Success F = Failure F F ... F S, x failures Ô ✟ p = ❼1 ✏ θ➁xθ ❼ ✏ ➁ ✏

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

• p(x) = θ❼1 ✏ θ➁x , x = 0, 1, 2, ...
• P➟

x0 p❼x➁ = P➟ x0 θ❼1 ✏ θ➁x =

θ 1 ✏ ❼1 ✏ θ➁ = 1

• Let X the number of failures until the first success.

θ = const success probability S = Success F = Failure F F ... F S, x failures Ô ✟ p = ❼1 ✏ θ➁xθ

• Let Y the number of trials until the first success then Y = X + 1.

p(y) = θ❼1 ✏ θ➁y✏1, y = 1, 2, ... (Pascal distribution)

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Nice Problem

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Nice Problem

A drunk man tries to open the door of his house randomly selected

• ne of the five keys in his pocket and of which only one opens the

door. a) What is the probability the k-the key that will try to be the first to open the door, for k = 1, 2, ..., when the key does not open the door by putting the other pocket? b) What is the same possibility if the key does not open the door to put it in the same pocket? c) if any test takes two seconds of time, how long it is expected to

• pen the door in both previous cases?

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Solution

Let us symbolized: Ak the fact the k-th key which he will test opens the door and Bk the fact the k-th key which he will test be the first to will open the door. (a) P(Bk), k = 1, 2, 3, 4, 5 (P(Bk) = 1

5 for all k)

k = 1. B1 = A1, P(B1) = P(A1) = 1

5

k = 2. B2 = A

➐

1 A2

P(B2) = P(A

➐

1 A2) = P(A

➐

1) P(A2❙A

➐

1) = ( 4 5) ( 1 4) = 1 5

k = 3. B3 = A

➐

1 A

➐

2 A3

P(B3) = P(A

➐

1 A

➐

2 A3) = P(A

➐

1) P(A

➐

2❙A

➐

1) P(A3❙A

➐

1A

➐

2) =( 4 5)( 3 4)( 1 3)

= 1

5

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Solution

(b) P(Bk) for k = 1, 2, 3, ..., n, ... k = 1. B1 = A1, P(B1) = P(A1) = 1

5

k ❃ N (random physical number) P(Bk) = P(A

➐

1 A

➐

2 A

➐

2 ... A

➐

k✏1 Ak) = ❼4 5➁k✏1 ( 1 5), k = 1, 2, ...

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Solution

(c) Case (a) X := random variable that has the value k when the k-the key that is tested it is the first to open the door Ô ✟ X discrete uniform distribution with values 1, 2, 3, 4, 5 and probability 1

5 for each value.

So EX = ❼1✔2✔3✔4✔5➁

5

= 3 6 sec (on the average) P➟

• ❼ ➁ ✏ ❼

P➟

• ❼ ➁ ✏

P➟

• ❙

✏

❙

❼ ✏ ➁

❙

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Solution

(c) Case (a) X := random variable that has the value k when the k-the key that is tested it is the first to open the door Ô ✟ X discrete uniform distribution with values 1, 2, 3, 4, 5 and probability 1

5 for each value.

So EX = ❼1✔2✔3✔4✔5➁

5

= 3 6 sec (on the average) Case (b) EX = P➟

k1 k❼4 5➁k✏1❼1 5) = ( 1 5) P➟ k1 k❼4 5➁k✏1 = 1 5 d dt (P➟ k1 tk)❙t 4

5 = 1

5 d dt ( t 1✏t )❙t 4

5 = 1

5( 1 ❼1✏t➁2 )❙t 4

5 = 5

10 sec (on the average)

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Coupon collector’s problem

Given n coupons, how many coupons do you expect you need to draw with replacement before having drawn each coupon at least

• nce?

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Claim: The expected number of steps is Θ(n logn)

✏❼ ✏ ➁

✂ Ô ✟ P ❼ ➁ P P

✏❼ ✏ ➁

P ✏ P ✏

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Claim: The expected number of steps is Θ(n logn) Proof: T : the time to collect all n coupons ti : the time to collect the i-th coupon after i-1 coupons have been collected

✏❼ ✏ ➁

✂ Ô ✟ P ❼ ➁ P P

✏❼ ✏ ➁

P ✏ P ✏

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Claim: The expected number of steps is Θ(n logn) Proof: T : the time to collect all n coupons ti : the time to collect the i-th coupon after i-1 coupons have been collected pi = n✏❼i✏1➁

n

: the probability of collecting a new coupon ti ✂ geometric distribution Ô ✟ E(ti) = 1

pi

E(T) = Pn

i1 E❼ti➁ = Pn i1 1 pi = n Pn i1 1 n✏❼i✏1➁ = n P1 jn✏1 1 j =

n Pjn✏1

1 1 j = n Hn

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

E(T) = n Hn = n logn + γ n + 1

2 + o(1) ,

as n ➟ where γ = limn➟❼Pn

i1 1 i ✏ ln♥➁ ☎ 0.5772156649 is the

Euler - Mascheroni constant.

Figure: (1707 - 1783) Figure: (1750 - 1800)

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Minimum Cut

What is a cut of a graph?

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Minimum Cut

What is a cut of a graph?

A cut is a partition of the vertices of a graph into two disjoint subsets that are joined by at least one edge.

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Minimum Cut

What is a cut of a graph?

A cut is a partition of the vertices of a graph into two disjoint subsets that are joined by at least one edge.

What is a minimum cut of a graph?

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Minimum Cut

What is a cut of a graph?

A cut is a partition of the vertices of a graph into two disjoint subsets that are joined by at least one edge.

What is a minimum cut of a graph?

In graph theory a minimum cut of a graph is a cut that is minimal in some sense.

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Global Minimum Cut

Given a connected, non-directed graph G = (V, E) find a cut (A, B) of minimum cardinality

Figure: red line: a cut with three crossing edges, green line: a min-cut of this graph

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

David Ron Karger (1967)

❮ is a professor of computer science and a member of the Computer Science and Artificial Intelligence Laboratory at the Massachusetts Institute of Technology ❮ Bachelor of Arts (Harvard University) ❮ PhD in computer science (Stanford University) ❮ Doctoral advisor:

• Rajeev Motwani

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Karger’s (contraction) algorithm [1995]

❮ Pick an edge e = (u,v) uniformly at random. ❮ Contract edge e.

• replace u and v by single new node w.
• preserve edges, updating endpointsof u and v to w.
• keep parallel edges, but delete self-loops.

❮ Repeat until graph has just 2 nodes v1 and v2. ❮ Return the cut (all nodes that were contracted to form v1)

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Contraction algorithm

Claim: The contraction algorithm returns a min cut with probability ❈

2 n2 ❻ ❻ ❻ ❻ ❻

❙

❻❙

❮

❻ ❙ ❙

❮ ❈

❻ ❻

Ô ✟ ❙ ❙ ❈ ❮

❻

❇

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Contraction algorithm

Claim: The contraction algorithm returns a min cut with probability ❈

2 n2

Proof: Consider a global min cut (A❻,B❻) of G. Let F ❻ be edges with one endpoint in A❻ and the other in B❻. Let k = ❙F ❻❙ size of min cut. ❮ In first step, algorithm contracts an edge in F ❻ probablity

k ❙E❙

❮ Every node has degree ❈ k since otherwise (A❻,B❻) would not be min cut Ô ✟ ❙E❙ ❈ 1

2 kn

❮ Thus, algorithm contracts an edge in F ❻ with probability ❇ n

2

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Contraction algorithm

❮ Let G’ be graph after j iterations. There are n’ = n - j nodes. ❮ Suppose no edge in F ❻ has been contracted. The min cut in G’ is still k. ❮ Since value of min cut is k, ❙E ➐❙ ❈ 1

2 kn’

❮ Thus algorithm contracts an edge in F ❻ with probability ❇ 2

n➐

❮ Let Ej := event that an edge in F ❻ is not contracted in iteration j. ✾ ✾ ✾

✏ ✾ ✏

❈

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Contraction algorithm

❮ Let G’ be graph after j iterations. There are n’ = n - j nodes. ❮ Suppose no edge in F ❻ has been contracted. The min cut in G’ is still k. ❮ Since value of min cut is k, ❙E ➐❙ ❈ 1

2 kn’

❮ Thus algorithm contracts an edge in F ❻ with probability ❇ 2

n➐

❮ Let Ej := event that an edge in F ❻ is not contracted in iteration j. P(E1 ✾ E2 ✾ ... ✾ En✏3 ✾ En✏2) ❈

2 n2

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Chain rule for conditional probability P(E1 ✾ E2 ✾ ... ✾ En✏3 ✾ En✏2) = P(E1)P(E2❙E1)...P(En✏3❙E1 ✾ E2 ...En✏4)P(En✏2❙E1 ✾ E2 ...En✏3)= (1- 2

n) (1- 2 n✏1) ... (1- 2 4) (1- 2 3) =

( n✏2

n ) ( n✏3 n✏1) ... ( 2 4) ( 1 3) = 2 n❼n✏1➁ ❈ 2 n2

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Amplification: To amplify the probability of success, run the contraction algorithm many times. ♥ ❼ ✏ ➁

♥

☎❼ ✏

❼ ➁➁

✠

♥

❇ ❼ ✏ ➁

♥

❮

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Amplification: To amplify the probability of success, run the contraction algorithm many times. Claim: If we repeat the contraction algorithm n2 ln♥ times with independent random choices, the probability of failing to find the global min cut is at most 1

n2 .

❼ ✏ ➁

♥

☎❼ ✏

❼ ➁➁

✠

♥

❇ ❼ ✏ ➁

♥

❮

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Amplification: To amplify the probability of success, run the contraction algorithm many times. Claim: If we repeat the contraction algorithm n2 ln♥ times with independent random choices, the probability of failing to find the global min cut is at most 1

n2 .

Proof: By independence the probability of failure is at most ❼1 ✏ 2

n2 ➁n2ln♥ = ☎❼1 ✏ 1 ❼ n2

2 ➁➁ n2 2 ✠

2ln♥

❇ ❼e✏1➁2ln♥ =

1 n2

❮ Deathbed formula

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Deathbed formulas

❨ limn➟❼1 ✔ 1

n➁n = e

❮ 1

4 ❅ ❼1 ✏ 1 n➁n ❅ 1 e

• limn➟❼1 ✏ 1

n➁n = 1 e

• n = 2 : ❼1 ✏ 1

n➁n❙n2 = 1 4

❮ 1

e ❅ ❼1 ✏ 1 n➁n✏1 ❅ 1 2

• limn➟❼1 ✏ 1

n➁n✏1 = 1 e

• n = 2: ❼1 ✏ 1

n➁n✏1❙n2 = 1 2

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Global Min Cut

Remark: Overall running time is slow since we perform Θ❼n2logn➁ iterations and each takes Ω(m) time. Improvement: Karger - Stein (1996) O (n2 log3n) Best known: Karger (2000) O(m log3n)

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

References

Motwani, Rajeev; Raghavan, Prabhakar (1995), ”3.6. The Coupon Collector’s Problem”, Randomized algorithms, Cambridge: Cambridge University Press, pp. 57–63, MR 1344451. J.Kleinberg, E.Tardos. Algorithm Design. Boston, ”13.2 Minimum Cut”, Mass.: Pearson/Addison-Wesley, cop. 2006. ❉✳▼♣❡ts❼❦♦❝✳ ❊✐s❛❣✇❣➔ st❤♥ ♣r❛❣♠❛t✐❦➔ ❛♥❼❧✉s❤✳ ❏❡ss❛❧♦♥Ð❦❤✿ ❆❢♦Ð ❑✉r✐❛❦Ð❞❤✱ ✷✵✶✻✳ ❙✳❑♦✉♥✐❼❝✱ P✳▼✇ôs✐❼❞❤❝✳ ❏❡✇rÐ❛ ♣✐❥❛♥♦t➔t✇♥✳ ❏❡ss❛❧♦♥Ð❦❤✿ ❩➔t❤✱ ✶✾✾✺✳

http://delab.csd.auth.gr/✂tsichlas/Algo Compl/7 random algorithms.pdf

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and

Section 1: Union Bound Section 2: Geometric variables Section 3: Coupon collector’s problem Section 4: Minimum Cut

Thank you!!!

Maria-Eirini Pegia Union Bound, Geometric Variables, Coupon collector’s problem and