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Randomized Algorithms Randomized Algorithms CS648 1

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Overview 2 To design efficient algorithm A new tool to design an efficient algorithm

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RECAP OF THE PREVIOUS LECTURE 3

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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 4

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Min-Cut 5

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SOME BASIC FACTS 6

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Min-Cut 7 How about cuts in multi-graph ?

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Algorithm for min-cut 13

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Algorithm for min-cut 14

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Algorithm for min-cut 15

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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 Masters 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. 16

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Common recurrences 17

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Common recurrences 18

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FASTER MIN-CUT ALGORITHM 19

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Revisiting algorithm for min-cut 20 We shall modify this algorithm to improve its success probability. But we shall not allow any significant blow up in the running time.

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Algorithm for min-cut 21

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Key observations about Min-Cut algorithm 22 recursively

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Revised algorithm for min-cut 23

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Faster algorithm for min-cut 24

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Solving the recurrence 26

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Solving the recurrence 27

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Solving the recurrence 28

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Solving the recurrence 29 proof

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This lecture introduced many concepts which can be used for design and analysis of efficient randomized algorithms. Try to reflect upon these concept… 30

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