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Randomized AlgorithmsCS648
Lecture 17
Miscellaneous applications of Backward analysis
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MINIMUM SPANNING TREE
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Minimum spanning tree
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Minimum spanning tree
Algorithms:
• Prim’s algorithm
• Kruskal’s algorithm
• Boruvka’s algorithm
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Less known but it is the first algorithm for MST
Minimum spanning tree
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Minimum spanning tree
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Minimum spanning tree
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Minimum spanning tree
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Light Edge
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USING BACKWARD ANALYSIS FORMISCELLANEOUS APPLICATIONS
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PROBLEM 1SMALLEST ENCLOSING CIRCLE
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Smallest Enclosing Circle
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Smallest Enclosing Circle
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PROBLEM 2SMALLEST LENGTH INTERVAL
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0 1
Sampling points from a unit interval
PROBLEM 3MINIMUM SPANNING TREE
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Light Edge
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USING BACKWARD ANALYSIS FORTHE 3 PROBLEMS :
A GENERAL FRAMEWORK
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A General framework
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PROBLEM 3MINIMUM SPANNING TREE
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A BETTER UNDERSTANDING OF LIGHT EDGES
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Minimum spanning tree
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Random sampling
Minimum spanning tree
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Minimum spanning tree
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Minimum spanning tree
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Light
First useful insight
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Minimum spanning tree
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Light heavy
Minimum spanning tree
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Light heavy
Second useful insight
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Light Edge
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We shall answer the above question using the Generic framework. But before that, we need to get a better understanding of the
corresponding random variable.
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32Light
33Light heavy
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Step 1
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Step 2
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Step 2
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Step 2
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Step 3
Expressing the entire experiment as Randomized Incremental Construction
A slight difficulty in this process is the following:
• The underlying experiment talks about random sample from a set.
• But RIC involves analyzing a random permutation of a set of elements.
Question: What is relation between random sample from a set and a random permutation of the set ?
Spend some time on this question before proceeding further.
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random sample and random permutation
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Step 3
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Step 3
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…
Step 3
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…
Step 3
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…
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Forward analysis
…
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Backward analysis
…
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Backward analysis
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Use Lemma 2.
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Backward analysis
…
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