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Quicksort

Quicksort. David Kauchak cs062 Spring 2010. public static int partition( double [] nums , int start, int end){ int lessThanIndex = start-1; for ( int i = start; i < end; i ++ ){ if ( nums[i ] <= nums[end ] ){ lessThanIndex ++; swap(nums , lessThanIndex , i ); }

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Quicksort

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  1. Quicksort David Kauchak cs062 Spring 2010

  2. public static intpartition(double[] nums, int start, int end){ intlessThanIndex = start-1; for( inti = start; i < end; i++ ){ if( nums[i] <= nums[end] ){ lessThanIndex++; swap(nums, lessThanIndex, i); } } swap(nums, lessThanIndex+1, end); return lessThanIndex+1; } what does this method do?

  3. public static intpartition(double[] nums, int start, int end){ intlessThanIndex = start-1; for( inti = start; i < end; i++ ){ if( nums[i] <= nums[end] ){ lessThanIndex++; swap(nums, lessThanIndex, i); } } swap(nums, lessThanIndex+1, end); return lessThanIndex+1; } • nums[end] is called the pivot • Partitions the elements A[start…end-1] in to two sets, those ≤ pivot and those > pivot • Operates in place • Final result: end start pivot nums > pivot ≤ pivot

  4. start end … 5 7 1 2 8 4 3 6 … public static intpartition(double[] nums, int start, int end){ intlessThanIndex = start-1; for( inti = start; i < end; i++ ){ if( nums[i] <= nums[end] ){ lessThanIndex++; swap(nums, lessThanIndex, i); } } swap(nums, lessThanIndex+1, end); return lessThanIndex+1; }

  5. lessThanIndex … 5 7 1 2 8 4 3 6 … start end public static intpartition(double[] nums, int start, int end){ intlessThanIndex = start-1; for( inti = start; i < end; i++ ){ if( nums[i] <= nums[end] ){ lessThanIndex++; swap(nums, lessThanIndex, i); } } swap(nums, lessThanIndex+1, end); return lessThanIndex+1; }

  6. i lessThanIndex … 5 7 1 2 8 4 3 6 … start end public static intpartition(double[] nums, int start, int end){ intlessThanIndex = start-1; for( inti = start; i < end; i++ ){ if( nums[i] <= nums[end] ){ lessThanIndex++; swap(nums, lessThanIndex, i); } } swap(nums, lessThanIndex+1, end); return lessThanIndex+1; }

  7. i lessThanIndex … 5 7 1 2 8 4 3 6 … start end public static intpartition(double[] nums, int start, int end){ intlessThanIndex = start-1; for( inti = start; i < end; i++ ){ if( nums[i] <= nums[end] ){ lessThanIndex++; swap(nums, lessThanIndex, i); } } swap(nums, lessThanIndex+1, end); return lessThanIndex+1; }

  8. i lessThanIndex … 5 7 1 2 8 4 3 6 … start end public static intpartition(double[] nums, int start, int end){ intlessThanIndex = start-1; for( inti = start; i < end; i++ ){ if( nums[i] <= nums[end] ){ lessThanIndex++; swap(nums, lessThanIndex, i); } } swap(nums, lessThanIndex+1, end); return lessThanIndex+1; }

  9. i lessThanIndex … 5 7 1 2 8 4 3 6 … start end public static intpartition(double[] nums, int start, int end){ intlessThanIndex = start-1; for( inti = start; i < end; i++ ){ if( nums[i] <= nums[end] ){ lessThanIndex++; swap(nums, lessThanIndex, i); } } swap(nums, lessThanIndex+1, end); return lessThanIndex+1; }

  10. i lessThanIndex … 5 7 1 2 8 4 3 6 … start end public static intpartition(double[] nums, int start, int end){ intlessThanIndex = start-1; for( inti = start; i < end; i++ ){ if( nums[i] <= nums[end] ){ lessThanIndex++; swap(nums, lessThanIndex, i); } } swap(nums, lessThanIndex+1, end); return lessThanIndex+1; }

  11. i lessThanIndex … 5 7 1 2 8 4 3 6 … start end public static intpartition(double[] nums, int start, int end){ intlessThanIndex = start-1; for( inti = start; i < end; i++ ){ if( nums[i] <= nums[end] ){ lessThanIndex++; swap(nums, lessThanIndex, i); } } swap(nums, lessThanIndex+1, end); return lessThanIndex+1; }

  12. i lessThanIndex … 5 7 1 2 8 4 3 6 … start end public static intpartition(double[] nums, int start, int end){ intlessThanIndex = start-1; for( inti = start; i < end; i++ ){ if( nums[i] <= nums[end] ){ lessThanIndex++; swap(nums, lessThanIndex, i); } } swap(nums, lessThanIndex+1, end); return lessThanIndex+1; }

  13. i lessThanIndex … 5 1 72 8 4 3 6 … start end public static intpartition(double[] nums, int start, int end){ intlessThanIndex = start-1; for( inti = start; i < end; i++ ){ if( nums[i] <= nums[end] ){ lessThanIndex++; swap(nums, lessThanIndex, i); } } swap(nums, lessThanIndex+1, end); return lessThanIndex+1; }

  14. i lessThanIndex … 5 1 7 2 8 4 3 6 … start end public static intpartition(double[] nums, int start, int end){ intlessThanIndex = start-1; for( inti = start; i < end; i++ ){ if( nums[i] <= nums[end] ){ lessThanIndex++; swap(nums, lessThanIndex, i); } } swap(nums, lessThanIndex+1, end); return lessThanIndex+1; }

  15. i lessThanIndex … 5 1 7 2 8 4 3 6 … start end public static intpartition(double[] nums, int start, int end){ intlessThanIndex = start-1; for( inti = start; i < end; i++ ){ if( nums[i] <= nums[end] ){ lessThanIndex++; swap(nums, lessThanIndex, i); } } swap(nums, lessThanIndex+1, end); return lessThanIndex+1; }

  16. i lessThanIndex … 5 1 2 78 4 3 6 … start end public static intpartition(double[] nums, int start, int end){ intlessThanIndex = start-1; for( inti = start; i < end; i++ ){ if( nums[i] <= nums[end] ){ lessThanIndex++; swap(nums, lessThanIndex, i); } } swap(nums, lessThanIndex+1, end); return lessThanIndex+1; }

  17. i lessThanIndex … 5 1 2 78 4 3 6 … start end public static intpartition(double[] nums, int start, int end){ intlessThanIndex = start-1; for( inti = start; i < end; i++ ){ if( nums[i] <= nums[end] ){ lessThanIndex++; swap(nums, lessThanIndex, i); } } swap(nums, lessThanIndex+1, end); return lessThanIndex+1; }

  18. i lessThanIndex … 5 1 2 78 4 3 6 … start end public static intpartition(double[] nums, int start, int end){ intlessThanIndex = start-1; for( inti = start; i < end; i++ ){ if( nums[i] <= nums[end] ){ lessThanIndex++; swap(nums, lessThanIndex, i); } } swap(nums, lessThanIndex+1, end); return lessThanIndex+1; }

  19. i lessThanIndex … 5 1 2 78 4 3 6 … start end What’s happening?

  20. i lessThanIndex … 5 1 2 78 4 3 6 … start end ≤ pivot > pivot unprocessed

  21. i lessThanIndex … 5 1 2 78 4 3 6 … start end public static intpartition(double[] nums, int start, int end){ intlessThanIndex = start-1; for( inti = start; i < end; i++ ){ if( nums[i] <= nums[end] ){ lessThanIndex++; swap(nums, lessThanIndex, i); } } swap(nums, lessThanIndex+1, end); return lessThanIndex+1; }

  22. i lessThanIndex … 5 1 2 48 73 6 … start end public static intpartition(double[] nums, int start, int end){ intlessThanIndex = start-1; for( inti = start; i < end; i++ ){ if( nums[i] <= nums[end] ){ lessThanIndex++; swap(nums, lessThanIndex, i); } } swap(nums, lessThanIndex+1, end); return lessThanIndex+1; }

  23. i lessThanIndex … 5 1 2 4 3 7 86 … start end public static intpartition(double[] nums, int start, int end){ intlessThanIndex = start-1; for( inti = start; i < end; i++ ){ if( nums[i] <= nums[end] ){ lessThanIndex++; swap(nums, lessThanIndex, i); } } swap(nums, lessThanIndex+1, end); return lessThanIndex+1; }

  24. i lessThanIndex … 5 1 2 4 3 6 8 7… start end public static intpartition(double[] nums, int start, int end){ intlessThanIndex = start-1; for( inti = start; i < end; i++ ){ if( nums[i] <= nums[end] ){ lessThanIndex++; swap(nums, lessThanIndex, i); } } swap(nums, lessThanIndex+1, end); return lessThanIndex+1; }

  25. Partition running time? • O(n) public static intpartition(double[] nums, int start, int end){ intlessThanIndex = start-1; for( inti = start; i < end; i++ ){ if( nums[i] <= nums[end] ){ lessThanIndex++; swap(nums, lessThanIndex, i); } } swap(nums, lessThanIndex+1, end); return lessThanIndex+1; }

  26. Quicksort How can we use this method to sort nums? public static intpartition(double[] nums, int start, int end){ intlessThanIndex = start-1; for( inti = start; i < end; i++ ){ if( nums[i] <= nums[end] ){ lessThanIndex++; swap(nums, lessThanIndex, i); } } swap(nums, lessThanIndex+1, end); return lessThanIndex+1; }

  27. Quicksort private static voidquicksortHelper(double[] nums, int start, int end){ if( start < end ){ int partition = partition(nums, start, end); quicksortHelper(nums, start, partition-1); quicksortHelper(nums, partition+1, end); } } public static intpartition(double[] nums, int start, int end){ intlessThanIndex = start-1; for( inti = start; i < end; i++ ){ if( nums[i] <= nums[end] ){ lessThanIndex++; swap(nums, lessThanIndex, i); } } swap(nums, lessThanIndex+1, end); return lessThanIndex+1; }

  28. 8 5 1 3 6 2 7 4 private static voidquicksortHelper(double[] nums, int start, int end){ if( start < end ){ int partition = partition(nums, start, end); quicksortHelper(nums, start, partition-1); quicksortHelper(nums, partition+1, end); } }

  29. 8 5 1 3 6 2 7 4 private static voidquicksortHelper(double[] nums, int start, int end){ if( start < end ){ int partition = partition(nums, start, end); quicksortHelper(nums, start, partition-1); quicksortHelper(nums, partition+1, end); } }

  30. 1 3 2 4 6 8 7 5 private static voidquicksortHelper(double[] nums, int start, int end){ if( start < end ){ int partition = partition(nums, start, end); quicksortHelper(nums, start, partition-1); quicksortHelper(nums, partition+1, end); } }

  31. 1 3 2 4 6 8 7 5 private static voidquicksortHelper(double[] nums, int start, int end){ if( start < end ){ int partition = partition(nums, start, end); quicksortHelper(nums, start, partition-1); quicksortHelper(nums, partition+1, end); } }

  32. 1 3 2 4 6 8 7 5 private static voidquicksortHelper(double[] nums, int start, int end){ if( start < end ){ int partition = partition(nums, start, end); quicksortHelper(nums, start, partition-1); quicksortHelper(nums, partition+1, end); } }

  33. 1 2 3 4 6 8 7 5 private static voidquicksortHelper(double[] nums, int start, int end){ if( start < end ){ int partition = partition(nums, start, end); quicksortHelper(nums, start, partition-1); quicksortHelper(nums, partition+1, end); } }

  34. 1 2 3 4 6 8 7 5 private static voidquicksortHelper(double[] nums, int start, int end){ if( start < end ){ int partition = partition(nums, start, end); quicksortHelper(nums, start, partition-1); quicksortHelper(nums, partition+1, end); } }

  35. 1 2 3 4 6 8 7 5 private static voidquicksortHelper(double[] nums, int start, int end){ if( start < end ){ int partition = partition(nums, start, end); quicksortHelper(nums, start, partition-1); quicksortHelper(nums, partition+1, end); } }

  36. 1 2 3 4 6 8 7 5 private static voidquicksortHelper(double[] nums, int start, int end){ if( start < end ){ int partition = partition(nums, start, end); quicksortHelper(nums, start, partition-1); quicksortHelper(nums, partition+1, end); } }

  37. 1 2 3 4 6 8 7 5 private static voidquicksortHelper(double[] nums, int start, int end){ if( start < end ){ int partition = partition(nums, start, end); quicksortHelper(nums, start, partition-1); quicksortHelper(nums, partition+1, end); } }

  38. 1 2 3 4 5 8 7 6 What happens here? private static voidquicksortHelper(double[] nums, int start, int end){ if( start < end ){ int partition = partition(nums, start, end); quicksortHelper(nums, start, partition-1); quicksortHelper(nums, partition+1, end); } }

  39. 1 2 3 4 5 8 7 6 private static voidquicksortHelper(double[] nums, int start, int end){ if( start < end ){ int partition = partition(nums, start, end); quicksortHelper(nums, start, partition-1); quicksortHelper(nums, partition+1, end); } }

  40. 1 2 3 4 5 8 7 6 private static voidquicksortHelper(double[] nums, int start, int end){ if( start < end ){ int partition = partition(nums, start, end); quicksortHelper(nums, start, partition-1); quicksortHelper(nums, partition+1, end); } }

  41. 1 2 3 4 5 6 7 8 private static voidquicksortHelper(double[] nums, int start, int end){ if( start < end ){ int partition = partition(nums, start, end); quicksortHelper(nums, start, partition-1); quicksortHelper(nums, partition+1, end); } }

  42. 1 2 3 4 5 6 7 8 private static voidquicksortHelper(double[] nums, int start, int end){ if( start < end ){ int partition = partition(nums, start, end); quicksortHelper(nums, start, partition-1); quicksortHelper(nums, partition+1, end); } }

  43. Running time of Quicksort? • Worst case? • Each call to Partition splits the array into an empty array and n-1 array

  44. Quicksort: Worst caserunning time • When does this happen? • sorted • reverse sorted • near sorted/reverse sorted n-1 + n-2 + n-3 + … + 1 = O(n2)

  45. Quicksort best case? • Each call to Partition splits the array into two equal parts … How much work is done at each “level”, i.e. running time of a level? O(n)

  46. Quicksort best case? • Each call to Partition splits the array into two equal parts … How many levels are there? Similar to binary search, each call to Parition will throw away half the data until we’re down to one element: log2n levels

  47. Quicksort best case? • Each call to Partition splits the array into two equal parts … Overall runtime? O(n log n)

  48. Quicksort Average case? • Two intuitions • As long as the Partition procedure always splits the array into some constant ratio between the left and the right, say L-to-R, e.g. 9-to-1, then we maintain O(n log n) • As long as we only have a constant number of “bad” partitions intermixed with a “good partition” then we maintain O(n log n)

  49. How can we avoid the worst case? • Inject randomness into the data private static voidrandomizedPartition(double[] nums, int start, int end){ inti = random(start, end); swap(nums, i, end); return partition = partition(nums, start, end); } Randomized quicksort is average case O(n log n)

  50. What is the wost case running time of randomized Quicksort? O(n2) We could still get very unlucky and pick “bad” partitions at every step

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