Computer Science 112

1. Computer Science 112 Fundamentals of Programming II Bucket Sort: An O(N) Sort Algorithm

2. N2 and Nlog2N Sort Algorithms • Selection sort and bubble sort are O(N2), because they run nested loops over the entire list • Quicksort and heap sort are O(Nlog2N), because one executes a linear process log2N times and the other executes a log2N process N times

3. An O(N) Sort Algorithm Consider a sorted list of unique integers, ranging from 0 to N - 1: 0 1 2 3 4 0 1 2 3 4

4. An O(N) Sort Algorithm Consider a sorted list of unique integers, ranging from 0 to N - 1: 0 1 2 3 4 0 1 2 3 4 Shuffle the list to randomize the numbers: 2 1 0 4 3 0 1 2 3 4

5. An O(N) Sort Algorithm Consider a sorted list of unique integers, ranging from 0 to N - 1: 0 1 2 3 4 0 1 2 3 4 Shuffle the list to randomize the numbers: 2 1 0 4 3 0 1 2 3 4 How can we sort this randomly ordered list in linear time?

6. An O(N) Sort Algorithm Create a temporary array of length N: 0 1 2 3 4 2 1 0 4 3 0 1 2 3 4 How can we sort this randomly ordered list in linear time?

7. An O(N) Sort Algorithm For each integer in the unsorted list: Copy the integer to the array at that position 0 1 2 3 4 2 2 1 0 4 3 0 1 2 3 4 How can we sort this randomly ordered list in linear time?

8. An O(N) Sort Algorithm For each integer in the unsorted list: Copy the integer to the array at that position 0 1 2 3 4 1 2 2 1 0 4 3 0 1 2 3 4 How can we sort this randomly ordered list in linear time?

9. An O(N) Sort Algorithm For each integer in the unsorted list: Copy the integer to the array at that position 0 1 2 3 4 0 1 2 2 1 0 4 3 0 1 2 3 4 How can we sort this randomly ordered list in linear time?

10. An O(N) Sort Algorithm For each integer in the unsorted list: Copy the integer to the array at that position 0 1 2 3 4 0 1 2 4 2 1 0 4 3 0 1 2 3 4 How can we sort this randomly ordered list in linear time?

11. An O(N) Sort Algorithm For each integer in the unsorted list: Copy the integer to the array at that position 0 1 2 3 4 0 1 2 3 4 2 1 0 4 3 0 1 2 3 4 How can we sort this randomly ordered list in linear time?

12. An O(N) Sort Algorithm For each integer in the unsorted list: Copy the integer to the array at that position Copy ‘em back to the list 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 How can we sort this randomly ordered list in linear time?

13. Complexity Analysis For each integer in the unsorted list: Copy the integer to the array at that position Copy ‘em back to the list 0 1 2 3 4 No comparisons! 2 * N assignments O(N) memory 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 How can we sort this randomly ordered list in linear time?

14. Lists with Duplicate Items Create a temporary array of linked lists of length K, for the integers in the list ranging from 0 to K - 1: 0 1 2 3 4 Each linked list will serve as a bucket to receive items from the original list 2 1 0 4 3 0 1 1 0 1 2 3 4 5 6 7

15. Lists with Duplicate Items Copy items from the original list to the corresponding buckets in the array 0 1 2 3 4 0 1 2 3 4 0 1 1 2 1 0 4 3 0 1 1 0 1 2 3 4 5 6 7

16. Lists with Duplicate Items Copy ‘em back to the original list 0 1 2 3 4 No comparisons! 2 * N assignments O(N + K) memory 0 1 2 3 4 0 1 1 0 0 1 1 1 2 3 4 0 1 2 3 4 5 6 7

17. Generalize to a Keyed List • Each item in the list must have an integer key • The keys can be repeated, but must be integers from 0 through a positive upper bound • The keys can be stored with the items, or computed as needed

18. Bucket Sort of a Keyed List from arrays import Array from node import Node defbucketSort(keyedList): # Create an array to accommodate the keys array = Array(keyedList.getMaxKey()) # Copy items from the list to the buckets for item inkeyedList: key = item.getKey() array[key] = Node(item, array[key]) # Copy items from buckets back to the list index = 0 for node in array: while node != None: keyedList[index] = node.data node = node.next index += 1 Computer Science 112

19. Some Buckets Can Be Empty 0 1 2 3 4 0 1 3 4 0 1 3 1 3 1 0 4 3 0 1 1 0 1 2 3 4 5 6 7

20. For Wednesday Hashing and O(k) Sets and Dictionaries