Chapter 5: Probabilistic Analysis
Average-Case Analysis
Finding the Maximum Example
Randomizing an Array Example
Average-Case Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Example 1: Finding the Maximum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Example 1: Probablistic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Example 1: Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Example 2: Random Permutation of an Array . . . . . . . . . . . . . . . . . . . . . . . 6
Example 2: Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
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Average-Case Analysis
In practice, many algorithms perform better than their worst-case.
The average case is analyzed by:
1. construct a probabilistic model of the input
2. determine the probabilities and running times (or costs) of alternate
executions
3. calculate expected running time (or cost)
Through randomization, one can often ensure that the probabilisitic model
is true.
CS 5633 Analysis of Algorithms Chapter 5: Slide – 2
Example 1: Finding the Maximum
Maximum(A)
max ←A[1]
for i←2to length(A)do
if max < A[i]then
max ←A[i]
return max
Problem: How many assignments to max?
Best-case: 1 (When does this happen?)
Worst-case: n(When does this happen?)
Average-case: (n+ 1)/2is incorrect
CS 5633 Analysis of Algorithms Chapter 5: Slide – 3
Example 1: Probablistic Model
Assume Ahas ndistinct numbers. (What is the effect of duplicates?)
Assume each permutation of the numb ers is equally likely. (How can
randomization guarantee this?)
How many permutations are there?
What is the probability of the best case?
What is the probability of the worst case?
CS 5633 Analysis of Algorithms Chapter 5: Slide – 4
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