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Sorting Conclusions

Sorting Conclusions. David Kauchak cs302 Spring 2013. Administrative. Find a partner to bounce ideas off of Assignment 2 Proofs should be very concrete Proving big-O, must satisfy definition

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Sorting Conclusions

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  1. Sorting Conclusions David Kauchak cs302 Spring 2013

  2. Administrative • Find a partner to bounce ideas off of • Assignment 2 • Proofs should be very concrete • Proving big-O, must satisfy definition • When comparing (i.e. ordering) functions you’re not sure about, set them equal to eachother and do some manipulations • e.g. compare n and 3^{\sqrt{\log n}} or n^n and 3^{3^n} • Be careful of things that seem “too good to be true”, e.g. a new O(n log n) sorting algorithm that you’ve never heard of 

  3. Administrivia continued • Assignment 3 • Substitution method is a proof by induction. Follow the steps we talked about before. • Make sure to prove/justify any non-trivial steps (e.g. showing big-O, Ω and θ) • If you’re stuck on an algorithm, brainstorm different possibilities • Even start with the naïve algorithm and think about where it’s inefficient • Latex/writeups • Use vertical space to help make your argument/steps more clear • Look at what is being generated and make sure it looks like you expect (e.g. n^2.5 vs. n^{2.5} or n^1+\epsilon vs n^{1 + \epsilon}

  4. Sorting bounds Mergsort is O(n log n) Quicksort is O(n log n) on average Can we do better?

  5. Comparison-based sorting Sorted order is determined based only on a comparison between input elements • A[i] < A[j] • A[i] > A[j] • A[i] = A[j] • A[i] ≤ A[j] • A[i] ≥ A[j] Do any of the sorting algorithms we’ve looked at use additional information? • No • All the algorithms we’ve seen are comparison-based sorting algorithms

  6. Comparison-based sorting Sorted order is determined based only on a comparison between input elements • A[i] < A[j] • A[i] > A[j] • A[i] = A[j] • A[i] ≤ A[j] • A[i] ≥ A[j] In Java (and many languages) for a class of objects to be sorted we define a comparator What does it do?

  7. Comparison-based sorting Sorted order is determined based only on a comparison between input elements • A[i] < A[j] • A[i] > A[j] • A[i] = A[j] • A[i] ≤ A[j] • A[i] ≥ A[j] In Java (and many languages) for a class of objects to be sorted we define a comparator What does it do? • Just compares any two elements • Useful for comparison-based sorting algorithms

  8. Comparison-based sorting Sorted order is determined based only on a comparison between input elements • A[i] < A[j] • A[i] > A[j] • A[i] = A[j] • A[i] ≤ A[j] • A[i] ≥ A[j] Can we do better than O(n log n) for comparison based sorting approaches?

  9. Decision-tree model • Full binary tree representing the comparisons between elements by a sorting algorithm • Internal nodes contain indices to be compared • Leaves contain a complete permutation of the input • Tracing a path from root to leave gives the correct reordering/permutation of the input for an input 1:3 ≤ > [3, 12, 7] | 1,3,2 | [3, 7, 12] | 1,3,2 | [3, 7, 12] | 2,1,3 | [7, 3, 12]

  10. A decision tree model 1:2 ≤ > 1:3 2:3 ≤ > ≤ > |2,1,3| 1:3 2:3 |1,2,3| ≤ > ≤ > |1,3,2| |3,1,2| |2,3,1| |3,2,1|

  11. A decision tree model 1:2 ≤ > 1:3 2:3 ≤ > ≤ > |2,1,3| 1:3 2:3 |1,2,3| ≤ > ≤ > |1,3,2| |3,1,2| |2,3,1| |3,2,1| [12, 7, 3]

  12. A decision tree model 1:2 ≤ > 1:3 2:3 ≤ > ≤ > |2,1,3| 1:3 2:3 |1,2,3| ≤ > ≤ > |1,3,2| |3,1,2| |2,3,1| |3,2,1| [12, 7, 3] Is 12 ≤ 7 or is 12 > 7?

  13. A decision tree model 1:2 ≤ > 1:3 2:3 ≤ > ≤ > |2,1,3| 1:3 2:3 |1,2,3| ≤ > ≤ > |1,3,2| |3,1,2| |2,3,1| |3,2,1| [12, 7, 3] Is 12 ≤ 7 or is 12 > 7?

  14. A decision tree model 1:2 ≤ > 1:3 2:3 ≤ > ≤ > |2,1,3| 1:3 2:3 |1,2,3| ≤ > ≤ > |1,3,2| |3,1,2| |2,3,1| |3,2,1| [12, 7, 3] Is 12 ≤ 3 or is 12 > 3?

  15. A decision tree model 1:2 ≤ > 1:3 2:3 ≤ > ≤ > |2,1,3| 1:3 2:3 |1,2,3| ≤ > ≤ > |1,3,2| |3,1,2| |2,3,1| |3,2,1| [12, 7, 3] Is 12 ≤ 3 or is 12 > 3?

  16. A decision tree model 1:2 ≤ > 1:3 2:3 ≤ > ≤ > |2,1,3| 1:3 2:3 |1,2,3| ≤ > ≤ > |1,3,2| |3,1,2| |2,3,1| |3,2,1| [12, 7, 3] Is 12 ≤ 3 or is 12 > 3?

  17. A decision tree model 1:2 ≤ > 1:3 2:3 ≤ > ≤ > |2,1,3| 1:3 2:3 |1,2,3| ≤ > ≤ > |1,3,2| |3,1,2| |2,3,1| |3,2,1| [12, 7, 3] Is 7 ≤ 3 or is 7 > 3?

  18. A decision tree model 1:2 ≤ > 1:3 2:3 ≤ > ≤ > |2,1,3| 1:3 2:3 |1,2,3| ≤ > ≤ > |1,3,2| |3,1,2| |2,3,1| |3,2,1| [12, 7, 3] Is 7 ≤ 3 or is 7 > 3?

  19. A decision tree model 1:2 ≤ > 1:3 2:3 ≤ > ≤ > |2,1,3| 1:3 2:3 |1,2,3| ≤ > ≤ > |1,3,2| |3,1,2| |2,3,1| |3,2,1| [12, 7, 3] 3, 2, 1

  20. A decision tree model 1:2 ≤ > 1:3 2:3 ≤ > ≤ > |2,1,3| 1:3 2:3 |1,2,3| ≤ > ≤ > |1,3,2| |3,1,2| |2,3,1| |3,2,1| [3, 7, 12] [12, 7, 3] 3, 2, 1

  21. A decision tree model 1:2 ≤ > 1:3 2:3 ≤ > ≤ > |2,1,3| 1:3 2:3 |1,2,3| ≤ > ≤ > |1,3,2| |3,1,2| |2,3,1| |3,2,1| [7, 12, 3]

  22. A decision tree model 1:2 ≤ > 1:3 2:3 ≤ > ≤ > |2,1,3| 1:3 2:3 |1,2,3| ≤ > ≤ > |1,3,2| |3,1,2| |2,3,1| |3,2,1| [7, 12, 3]

  23. A decision tree model 1:2 ≤ > 1:3 2:3 ≤ > ≤ > |2,1,3| 1:3 2:3 |1,2,3| ≤ > ≤ > |1,3,2| |3,1,2| |2,3,1| |3,2,1| [7, 12, 3]

  24. A decision tree model 1:2 ≤ > 1:3 2:3 ≤ > ≤ > |2,1,3| 1:3 2:3 |1,2,3| ≤ > ≤ > |1,3,2| |3,1,2| |2,3,1| |3,2,1| [7, 12, 3]

  25. A decision tree model 1:2 ≤ > 1:3 2:3 ≤ > ≤ > |2,1,3| 1:3 2:3 |1,2,3| ≤ > ≤ > |1,3,2| |3,1,2| |2,3,1| |3,2,1| [7, 12, 3]

  26. A decision tree model 1:2 ≤ > 1:3 2:3 ≤ > ≤ > |2,1,3| 1:3 2:3 |1,2,3| ≤ > ≤ > |1,3,2| |3,1,2| |2,3,1| |3,2,1| [3, 7, 12] [7, 12, 3] 3, 1, 2

  27. How many leaves are in a decision tree? Leaves must have all possible permutations of the input What if decision tree model didn’t? Some input would exist that didn’t have a correct reordering Input of size n, n! leaves

  28. A lower bound What is the worst-case number of comparisons for a tree? 1:2 ≤ > 1:3 2:3 ≤ > ≤ > |2,1,3| 1:3 2:3 |1,2,3| ≤ > ≤ > |1,3,2| |3,1,2| |2,3,1| |3,2,1|

  29. A lower bound The longest path in the tree, i.e. the height 1:2 ≤ > 1:3 2:3 ≤ > ≤ > |2,1,3| 1:3 2:3 |1,2,3| ≤ > ≤ > |1,3,2| |3,1,2| |2,3,1| |3,2,1|

  30. A lower bound What is the maximum number of leaves a binary tree of height h can have? A complete binary tree has 2h leaves from hw

  31. Can we do better?

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