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Great Ideas in Computing Average Case Analysis

Great Ideas in Computing Average Case Analysis. Richard Anderson University of Washington. What happens “on the average”. Worst case versus average case. T(n) = max {T(I) | I is a problem instance of size n}

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Great Ideas in Computing Average Case Analysis

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  1. Great Ideas in ComputingAverage Case Analysis Richard Anderson University of Washington IUCEE: Data Structures Activities

  2. What happens “on the average” 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} IUCEE: Data Structures Activities

  4. Is real world data random? • Is real world data worst case? IUCEE: Data Structures Activities

  5. 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 IUCEE: Data Structures Activities

  6. Quicksort • Worst case runtime is n2 • Average case is n log n IUCEE: Data Structures Activities

  7. Random Trees IUCEE: Data Structures Activities

  8. Euclidean TSP IUCEE: Data Structures Activities

  9. 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 IUCEE: Data Structures Activities

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

  11. 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 IUCEE: Data Structures Activities

  12. Random Graphs • What is a random graph? IUCEE: Data Structures Activities

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

  14. Does a random graph have a Hamiltonian circuit IUCEE: Data Structures Activities

  15. Hamiltonian Circuit Algorithm IUCEE: Data Structures Activities

  16. What does the Web Graph look like? IUCEE: Data Structures Activities

  17. Degree distribution in the web graph IUCEE: Data Structures Activities

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