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Lower Bound on Sorting

Lower Bound on Sorting. David Kauchak cs062 Spring 2010. Decision-tree model. B inary 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

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Lower Bound on Sorting

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  1. Lower Bound on Sorting David Kauchak cs062 Spring 2010

  2. Decision-tree model • 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]

  3. 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] • This is why most built-in sorting approaches only require you to define the comparison operator (i.e. compareTo in Java) • Can we do better than O(n log n)?

  4. 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|

  5. 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]

  6. 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?

  7. 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?

  8. 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?

  9. 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?

  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| [12, 7, 3] Is 7 ≤ 3 or is 7 > 3?

  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| 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| 3, 2, 1 [3, 7, 12] [12, 7, 3]

  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| [7, 12, 3]

  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| [7, 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| [7, 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| [7, 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| [7, 12, 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| [3, 7, 12] [7, 12, 3]

  19. 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 • For an input of size n, how many leaves? • n! leaves

  20. 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|

  21. 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|

  22. 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 log is monotonically increasing 

  23. Other heaps

  24. Other heaps

  25. Other heaps

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