Sorting

Arvind Devaraj

Sorting

• Given an array, put the elements in order– Numerical or lexicographic

• Desirable characteristics– Fast– In place (don’t need a second array)– Stability

Insertion Sort

• Simple, able to handle any data• Grow a sorted array from the beginning

– Create an empty array of the proper size– Pick the elements one at a time in any order– Put them in the new array in sorted order

• If the element is not last, make room for it

– Repeat until done• Can be done in place if well designed

Insertion Sort

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Insertion Sort

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Insertion Sort

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Insertion Sort

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Insertion Sort

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Insertion Sort

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Insertion Sort

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Insertion Sort

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Insertion Sort

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114

Merge Sort

• Fast, able to handle any data– But can’t be done in place

• View the array as a set of small sorted arrays– Initially only the 1-element “arrays” are sorted

• Merge pairs of sorted arrays– Repeatedly choose the smallest element in each– This produces sorted arrays that are twice as long

• Repeat until only one array remains

Merge Sort

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Merge Sort

90 11 27

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Merge Sort

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Merge Sort

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Merge Sort

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Merge Sort

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Merge Sort

11 27 31

11 16 27 31 37 90

37161190 4

114

Divide and Conquer

• Split a problem into simpler subproblems– Keep doing that until trivial subproblems result

• Solve the trivial subproblems• Combine the results to solve a larger problem

– Keep doing that until the full problem is solved• Merge sort illustrates divide and conquer

– But it is a general strategy that is often helpful

Quick Sort

• For example, given80 38 95 84 99 10 79 44 26 87 96 12 43 81 3

we can select the middle entry, 44, and sort the remaining entries into two groups, those less than 44 and those greater than 44:

38 10 26 12 43 3 44 80 95 84 99 79 87 96 81

• If we sort each sub-list, we will have sorted the entire array

A sample heap

• Each node is larger than its children

19

1418

22

321

14

119

15

25

1722

Sorting using Heaps• What do heaps have to do with sorting an array?

Because the binary tree is balanced and left justified, it can be represented as an array– All our operations on binary trees can be represented as

operations on arrays– To sort:

heapify the array; while the array isn’t empty { remove and replace the root; reheap the new root node;

}

Summary of Sorting Algorithms

• in-place, randomized• fastest (good for large inputs)

O(n log n)expectedquick-sort

• sequential data access• fast (good for huge inputs)O(n log n)merge-sort

• in-place• fast (good for large inputs)O(n log n)heap-sort

O(n2)

O(n2)

Time

insertion-sort

selection-sort

Algorithm Notes

• in-place• slow (good for small inputs)

• in-place• slow (good for small inputs)