1. Great Ideas in Computing Average Case Analysis
Richard Anderson
University of Washington
July 2, 2008
IUCEE: Data Structures Activities

2. What happens “on the average”
July 2, 2008
IUCEE: Data Structures Activities

3. Worst case versus average case
T(n) = max {T(I) | I is a problem instance of size n}
T(n) = ave {T(I) | I is a problem instance of size n} = sum {p(I) T(I) | I is a problem instance of size n}
July 2, 2008
IUCEE: Data Structures Activities

4. Is real world data random?
Is real world data worst case?
July 2, 2008
IUCEE: Data Structures Activities

5. IUCEE: Data Structures Activities
Average case analysis
max := A[0]
for i := 1 to n-1
if A[ i ] > max
max := A[ i ] *
How many times is line * executed
July 2, 2008
IUCEE: Data Structures Activities

6. IUCEE: Data Structures Activities
Quicksort
Worst case runtime is n2
Average case is n log n
July 2, 2008
IUCEE: Data Structures Activities

7. IUCEE: Data Structures Activities
Random Trees
July 2, 2008
IUCEE: Data Structures Activities

8. IUCEE: Data Structures Activities
Euclidean TSP
July 2, 2008
IUCEE: Data Structures Activities

9. IUCEE: Data Structures Activities
Asymmetric TSP
Random Asymmetric TSP
n  n matrix with random entries in [0, 1]
Theorem
There is a polynomial time algorithm that gives a very good approximation for a random ATSP
July 2, 2008
IUCEE: Data Structures Activities

10. IUCEE: Data Structures Activities
Assignment Problem
Minimum weight perfect matching in a complete, bipartite graph
Solvable in polynomial time
How is the assignment problem related to the ATSP?
July 2, 2008
IUCEE: Data Structures Activities

11. IUCEE: Data Structures Activities
ATSP Algorithm
Solve relaxed version with Assignment Problem
Splice together cycles
Solution to the Assignment problem is a random permutation
Random permutations have a small number of cycles
July 2, 2008
IUCEE: Data Structures Activities

12. IUCEE: Data Structures Activities
Random Graphs
What is a random graph?
July 2, 2008
IUCEE: Data Structures Activities

13. IUCEE: Data Structures Activities
Standard model
Each edge is present with probability p
Expected degree of a vertex is pn
July 2, 2008
IUCEE: Data Structures Activities

14. Does a random graph have a Hamiltonian circuit
July 2, 2008
IUCEE: Data Structures Activities

15. Hamiltonian Circuit Algorithm
July 2, 2008
IUCEE: Data Structures Activities

16. What does the Web Graph look like?
July 2, 2008
IUCEE: Data Structures Activities

17. Degree distribution in the web graph
July 2, 2008
IUCEE: Data Structures Activities