This document discusses algorithms for finding minimum cuts in graphs. It begins with an overview and definitions of graphs and multi-graphs. It then discusses the min-cut problem and presents an initial algorithm for finding min-cuts that recursively partitions the graph. The document goes on to revisit common recurrence relations and use insights from analyzing recurrences to modify the initial algorithm, making it faster by improving its success probability at each step while maintaining the same running time. It concludes by analyzing the new faster algorithm's recurrence relation.

1. Randomized Algorithms
CS648
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2. Overview
To design efficient algorithm
A new tool to design an efficient algorithm
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3. RECAP OF THE PREVIOUS LECTURE
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4. 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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5. Min-Cut
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6. SOME BASIC FACTS
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7. Min-Cut
How about cuts in multi-graph ?
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13. Algorithm for min-cut
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14. Algorithm for min-cut
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15. Algorithm for min-cut
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16. 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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17. Common recurrences
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18. Common recurrences
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19. FASTER MIN-CUT ALGORITHM
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20. Revisiting algorithm for min-cut
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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21. Algorithm for min-cut
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22. Key observations about
Min-Cut algorithm
recursively
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23. Revised algorithm for min-cut
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24. Faster algorithm for min-cut
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26. Solving the recurrence
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27. Solving the recurrence
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28. Solving the recurrence
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29. Solving the recurrence
proof
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30. 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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