Randomized AlgorithmsCS648

Lecture 19• algorithm for Min-cut in a graph

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Overview

1. Recap of the previous lecture time Monte Carlo algorithm for min-cut

2. Knowledge of recurrence

3. time Monte Carlo algorithm for min-cut

4. An important bi-product:

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To design efficient algorithm

A new tool to design an efficient algorithm

RECAP OF THE PREVIOUS LECTURE

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Graph and Multi-graph

A multi-graph may have:• More than one edge between a pair of vertices• No self loop

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𝒖

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𝒂

𝒃 𝒉

𝒍

Min-Cut

: undirected connected graph Definition (cut):A subset whose removal disconnects the graph.

Definition (min-cut): A cut of smallest size.Problem Definition: Design algorithm to compute min-cut of a given graph.

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SOME BASIC FACTS

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Min-Cut

Question: How many cuts ?Answer: Question : what is relation between degree() and size of min-cut ?Answer: size of min-cut degree() Question : If size of min-cut is , what can be minimum value of ?Answer:

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𝒖

How about cuts in multi-graph ?

Contract()

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Contract(

Contract()

Contract(){ Let ; Merge the two vertices and into one vertex; Preserve multi-edges; Remove the edge ; Let be the modified graph; return ;}

Time complexity of Contract():

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Relation between and ?

Relation between and ?

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Contract(

Observations about Contract()

Let be the size of min-cut of . Let be any min-cut of . Let be the graph after Contract().Observation 1: may be a multi-graph.Observation 2: Every cut of is also a cut of .Observation 3: remains a cut of if .

Question: If is selected randomly uniformly, what is the probability that is preserved in ?Answer:

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≤𝒌

𝒏𝒌 /𝟐≤𝟐𝒏

Recall that

Contract()

Let be the size of min-cut of . Let be any min-cut of . Lemma: If edge to be contracted is selected randomly uniformly, is preserved with probability at least .

Let ; Contract().Let ; Contract().

Question: What is probability that is preserved in ?Answer:

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(1− 𝟐𝒏−𝟏 )

Algorithm for min-cut

Min-cut():{ Repeat ?? times {

Let ; Contract().

} return the edges of multi-graph ;}

Running time:

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𝒏−𝟐

Algorithm for min-cut

Question: What is probability that is preserved during the algorithm ?Answer:

= = >

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Algorithm for min-cut

Min-cut-high-probability():{ Repeat Min-cut() algorithm log times and report the smallest cut computed;} Running time: Error Probability : <

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REVISITING RECURRENCES

In the following slides, we shall revisit common recurrences. Try to solve these recurrences in a simple manner instead of using Master’s theorem. This will help you develop a useful insight into recurrences. This insight will help you fine-tune

the previous inefficient algorithm and eventually lead to design (and analysis) of a more efficient algorithm for min-cut.

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Common recurrences•

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•

•

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𝑶 (𝒏𝟐)

𝑶 (𝒏)

𝑶 (𝒏𝐥𝐨𝐠𝒏)

𝑶 (𝒏)

Common recurrences•

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•

Question: What is the largest value of for which the solution of the following recurrence ?•

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𝑶 (𝒏𝟐)

𝑶 (𝒏𝟐)

𝑶 (𝒏𝟑)

𝒂=√𝟐

FASTER MIN-CUT ALGORITHM

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Revisiting algorithm for min-cut

Min-cut():{ Repeat ?? times {

Let ; Contract().

} return the edges of multi-graph ;}

Running time:

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𝒏−𝟐

We shall modify this algorithm to improve its success probability. But we shall not allow any significant blow up in the running time.

Algorithm for min-cut

Probability that min-cut is preserved during the algorithm : =

Question: What is probability that min-cut is preserved in first iteration ? Answer: Try to make careful observations based on the above answer to improve the

success probability of the min-cut algorithm.

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≈ (𝒏−𝒊𝒏 )𝟐

Key observations about Min-Cut algorithm

Algorithm performs contractions :

After contractions:• Min-cut is preserved with probability ¼ .• There are only vertices left. (the graph size is significantly reduced).

IDEA: Invoke two calls of Min-cut algorithm after contractions and return the smaller of the two cuts computed. Since the graph size is reduced significantly, we can afford to do it.

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𝒏 /𝟐

recursively

Revised algorithm for min-cutFast-Min-cut(): If has at most 10 vertices compute exact min-cut else { ; Repeat times { Let ; Contract(). } Fast-Min-cut(); Fast-Min-cut(); Let be smaller of the cuts , ; } return ;

• for

Question: Prob. min-cut is preserved at the end of the Repeat loop = ?

Question: How to increase this prob. ?

Answer: reduce .

Question: What is the smallest value of to ensure ?

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1/4

Faster algorithm for min-cutFast-Min-cut(): If has at most 10 vertices compute exact min-cut else { ; Repeat times { Let ; Contract(). } Fast-Min-cut(); Fast-Min-cut(); Let be smaller of the cuts , ; } return ;

Running time of the algorithm :

Question: Prob. min-cut is preserved at the end of the Repeat loop = ?

: the probability that algorithm returns the min-cut.

For , For , ??

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1/2

Success probability of Fast-Min-cut()

For , For , = We shall now show that recurrence has a solution: If we repeat Fast-Min-cut total times and report the smallest of all the output, probability of error= ??

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(1− 𝟏log𝒏 )

𝟒 log𝟐𝒏<

Solving the recurrence

For , For , = Substituting , we get the recurrenceFor , For ,

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Solving the recurrence

For , For , = It suffices to consider the above recurrence only for those which are powers of .

For the sake of neatness, let us use to denote . Thus the above recurrence can be expressed as

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Solving the recurrence

It suffices to solve the following recurrence., For ,

The recurrence still looks difficult since it does not belong to the pool of recurrences you are familiar with.

However, it has a very short and elementary solution. In fact the solution is very inspiring.

First try on your own before proceeding to the following slides.

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Solving the recurrence

,For ,

Observation 1: Observation 2:

Hence,

// by unfolding and using the fact that // and we are done.

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//since

proof

This lecture introduced many concepts which can be used for design and analysis of efficient randomized algorithms. Try to

reflect upon these concept…

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