Randomized AlgorithmsCS648

Lecture 13Expected duration of a randomized experiment

Part I

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COUPON COLLECTOR PROBLEM

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Coupon Collector Problem

• There is a bag containing distinct coupons. • Each coupon has a unique label from [].

Experiment:Repeat 1. Select a coupon randomly uniformly from the bag 2. Note down its label 3. Place the coupon back into the bagUntil every coupon has appeared at least once

: the number of iterations of the loop (number of coupons drawn).

Question: What is E[] ?

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Example

=5

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3 1 3 3 1 2 2 1 5 1 3 5 3 4

5 5 1 2 1 2 5 4 4 1 2 3

3 2 2 3 3 3 1 1 5 1 3 5 3 2 2 5

1 42

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Done in 14 samplings

Done in 12 samplings

Coupon Collector Problem

: the number of iterations of the loop (number of coupons drawn).Question: What is E[] ?

Standard method:

E[] =

5

No easy way !!

?

Coupon Collector Problem

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no coupon

seen

all coupons

seen

This transition is not sudden. In fact it is a gradual transition through various discrete stages. Can you

see these discrete stages ?

Coupon Collector Problem

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no coupon

seen

all coupons

seen1 2 3 4

This transition is not sudden. In fact it is a gradual transition through various discrete stages. Can you

see these discrete stages ?

Reviewing Example

=5

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3 1 3 3 1 2 2 1 5 1 3 5 3 4

1 42

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Reviewing Example

=5

9

3 2 2 3 3 3 1 1 5 1 3 5 3 2 2 5

1 42

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1 2 3 4 50

5 5 1 2 1 2 5 4 4 1 2 3

1 2 3 4 50

1 32 40

3 1 3 3 1 2 2 1 5 1 3 5 3 4

Each instance of coupon collector problem has to

pass through these stages. Does it give you

some inspiration to calculate E[X] ?

Coupon Collector Problem

: the number of iterations of the loop (number of coupons drawn).Question: What is E[] ?

no. of coupons sampled from the moment ?? to the moment ??

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th distinct coupon was selected

th distinct coupon was selected

Reviewing Example

=5

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1 42

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1 2 3 450

3 1 3 3 1 2 2 1 5 1 3 5 3 4

=1 =1 =4 =3 =5

This picture validates the equality

Coupon Collector Problem

: the number of iterations of the loop (number of coupons drawn).

Question: What is ] ?

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Calculating E[]

Experiment (in th stage):Repeat 1. Select a coupon randomly uniformly from the bag 2. Note down its label 3. Place the coupon back into the bagUntil th distinct coupon appears.

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𝑛

Calculating E[]

Experiment (in th stage):Repeat 1. Select a coupon randomly uniformly from the bag 2. Note down its label 3. Place the coupon back into the bagUntil th distinct coupon appears.

=Probability an iteration is successfulQuestion: What is ?E[] = = = =

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𝑛

𝑖(𝑛−𝑖)/𝑛

Coupon Collector Problem

: the number of iterations of the loop (number of coupons drawn).

Theorem: Expected duration of coupon collector experiment is .

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DISCRETE RANDOM WALK ON A LINE

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Discrete Random Walk

• Particle starts from origin• In each second, particle moves 1 unit to the left or to the right with equal

probability.• While at origin, the particle moves to 1 always.

Question: What is the expected number of steps of the random walk to reach milestone n ?

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0 1 2 3 4 5 6 7 8 … n n+1

An example

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0 1 2 3 4 5 6 7 8 …

I, and perhaps you too,

could not notice the walk. So let

us trace the walk slowly.

Formalism

: No. of steps of a random walk which starts at and terminates on reaching for the first time.

Aim: To calculate E[]

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Careful look at the example

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0 1 2 3 4 5 6 7 8 …

Can you break the walk 08 into stages ? Think

carefully …

Careful look at the example

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0 1 2 3 4 5 6 7 8 …

Walk starting from 0 and terminating at 5

Careful look at the example

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0 1 2 3 4 5 6 7 8 …

Walk starting from 5 and terminating at 8

Walk starting from 0 and terminating at 5

Relation among ’s

For any = +

Breaking down to the limits, we get =

Hence using linearity of expectation E[] = ]

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Relation among ’s

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0 1 2 3 4 5 6 7 8 …

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1

1

3

1

5

1

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HOW TO CALCULATE E[] ?

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Conditional Expectation

Given any event and a random variable defined over a probability space (,P).

E[] = E[| ] P() + E[| ] P()

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ε Ωε

A useful tool to calculate expected value of a random variable

E[| ]

E[| ]

Calculating E[]

E[] = ??

= ½ E[| first move is L] + ½ . 1

= ½ E[| first move is L] + ½

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0 1 … 𝑖 +1

½ E[| first move is L] + ½ E[| first move is R]

?

Calculating E[| first move is L]

E[| first move is L] = ?? = 1 + E[] + E[] //by linearity of expectation

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0 1 … 𝑖 +1

1 + E[]

Calculating E[]

E[] = ½ E[| first move is L] + ½ . 1 = ½ + ½ . 1 = 1 + ½ 2 E[] = 2 + ] E[] = 2 + Question: What is E[] ?Question: What is E[] ?Question: What is E[] ?Answer: ??

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0 1 … 𝑖 +1

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2 𝑖+1

CALCULATING E[]

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Calculating E[]

Lemma (just proved): E[] =

E[] = = = = =

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0 1 2 3 4 5 6 7 … 𝑛 +1

Theorem: Expected number of steps of a random walk starting from 0 and terminating on reaching is .

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Expected duration of a random experiment

Let X denote the random variable for the duration of a randomized experiment.

To calculate E[X], the following approach is sometimes useful:

• Partition the experiment into stages carefully.

• Calculate expected duration of each stage.

• Using linearity of expectation, calculate E[X].

In the next class, we shall discuss more non-trivial randomized algorithms which are analyzed using this method.

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