Post on 17-Dec-2015
transcript
CSE 2331/5331
CSE 2331/5331
Topic 5:
Prob. Analysis
Randomized Alg.
Expected Complexity
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Probabilistic method: Given a distribution for all possible inputs Derive expected time based on distribution
Randomized algorithm: Add randomness in the algorithm Analyze the expected behavior of the algorithm
First Example
What is worst case time complexity? What is expected / average time complexity?
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Expected Running Time
Expected / average running time
= probability of event I = running time given event I
To analyze, need to assume a probabilistic distribution for all inputs
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SeqSearch Alg
Expected running time =
If we assume = 0 All permutations are equally likely
implies Then expected running time =
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Remarks
For probabilistic analysis An input probabilistic distribution input model has be assumed! For a fixed input, the running time is fixed. The average / expected time complexity is for if we consider
running it for a range of inputs, what the average behavior is.
Randomized algorithm No assumption in input distribution! Randomness is added in the algorithm
For a fixed input, the running time is NOT fixed. The expected time is what we can expect when we run the
algorithm on ANY SINGLE input.
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Randomized Algorithms
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Expectation
X is a random variable The expectation of X is
E.g, coin flip
Linearity of expectation:
Conditional expectation:
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Linearity of Expectation
= expected running time for func2 What is ?
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+ cn
Conditional Expectation
= expected running time of func2 = expected running time of func3 What is the expected running time of func1?
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A Randomized Example
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Worst case complexity?Expected case?
Running Time Analysis
Worst Case:
Expected running time:
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Time Analysis
Worst case:
Expected running time:
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A more complicatedvariation.
Analysis
Worst case:
Expected case:
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An Example with Recursion
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Worst case:
Expected running time
Solving this we have
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Another Example with Recursion
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Analysis
Worst case:
!
Expected running time:
!
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Another Example
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Analysis
Worst case:
Expected running time:
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