quick sort, lower bound on sorting, bucket sort,

radix sort, comparison of algorithms, code, …

Sorting: part 2

sortingquick sort

Demo with cards?

1. Divide S into 3 parts as follows• Select an element e• Create L with all element is S less than e• Create E with all element in S equal to e• Create G with all element in S greater than e

2. Recur by quick sorting L and quick sorting G3. Conquer by appending L to E to G

Divide & Conquer (with quick sort)

sortingquick sort

Demo with cards?

sortingquick sort

sortingquick sort

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sortingquick sort

sortingquick sort

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sortingquick sort

sortingquick sort

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sortingquick sort

Stopping condition

sortingquick sort

Less than, Equal to, Greater than

sortingquick sort

pivot

sortingquick sort

Get the data sets L, E, and G

sortingquick sort

Recurse & conquer

sortingquick sort

sortingquick sort

.. and this happens if S is sorted!

sortingquick sort

sortingquick sort

Two enhancements• improve selection of pivot

• randomise• “in place” sorting

• uses no more space and is fast

sorting

A lower bound on comparison based sorting

i.e. how good can it get?

sortingA lower bound on comparison based sorting

• Assume we are given a sequence S, of length n, to be sorted (no duplicates)• an algorithm compares pairs of elements with two outcomes

• x < y TRUE• x < y FALSE

• we can represent a comparison based sorting algorithm as a decision tree• i.e. represent its behaviour• internal node represents a decision• left is TRUE, right is FALSE

• there are n! possible orderings of S• there must be n! leaves in our decision tree

• one for each possible ordering of S• running time of algorithm must be at least the height of the decision tree• the height of the decision tree is at least log(n!) base 2• log(n!) is O(n.log(n))

sorting

stable sorting

sortingstable sorting

A sorting algorithm is stable if for any two entries x and yof S such that

• x.getKey() == y.getKey() and • x precedes y in S before sorting• x precedes y after S is sorted

sorting

bucket sortaka

pigeonhole sort

sorting

bucket sortaka

pigeonhole sort

PIGEONHOLE SORT

sortingbucket sort (aka pigeonhole sort)

Use the sey of an object as an index into a bucket array

Assume keys are in range [0,N-1]

Assume we have n object to sort

insert object e into bucket[e.getKey()]

Complexity is O(n+N)

sortingbucket sort (aka pigeonhole sort)

Use the sey of an object as an index into a bucket array

Assume keys are in range [0,N-1]

Assume we have n object to sort

insert object e into bucket[e.getKey()]

Complexity is O(n+N)

How might we do this in java?What data structures would we use?

Is it stable?What kind of data sets would be suitable?

sorting

radix sort

We will describe by example

sortingradix sort

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89

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81

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81

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89, 69, 29

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89, 69, 29

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89, 69, 29

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89, 69, 29

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89, 69, 29,39

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89, 69, 29,39

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89, 69, 29,39

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89, 69, 29,39

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Sorted on least significant digit!

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81, 89

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81, 89

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81, 89

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Sorted on next (second) significant digit!

81, 89

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69

sortingradix sort

• We can extend this to numbers of any length• We can do this on strings

Considering numbers to base b• require b buckets• require d = log(m) scans (d is number of digits)

• log is to base b• m is largest number to be sorted

• each scan is of order n• therefore O(d.n)

sorting

comparison of algorithms

sortingcomparison of algorithms

Bubble sort

Insertion sort

Selection sort

Merge sort

Quick sort

Heap sort

Bucket sort

Radix sort

sortingcomparison of algorithms

Bubble sort

Insertion sort

Selection sort

Merge sort

Quick sort

Heap sort

Bucket sort

Radix sort

sortingcomparison of algorithms

Bubble sort

Insertion sort

Selection sort

Merge sort

Quick sort

Heap sort

Bucket sort

Radix sort

sortingcomparison of algorithms

Bubble sort

Insertion sort

Selection sort

Merge sort

Quick sort

Heap sort

Bucket sort

Radix sort

sortingcomparison of algorithms

Bubble sort

Insertion sort

Selection sort

Merge sort

Quick sort

Heap sort

Bucket sort

Radix sort

sortingcomparison of algorithms

Bubble sort

Insertion sort

Selection sort

Merge sort

Quick sort

Heap sort

Bucket sort

Radix sort

sortingcomparison of algorithms

Bubble sort

Insertion sort

Selection sort

Merge sort

Quick sort

Heap sort

Bucket sort

Radix sort

sortingcomparison of algorithms

Bubble sort

Insertion sort

Selection sort

Merge sort

Quick sort

Heap sort

Bucket sort

Radix sort

sortingcomparison of algorithms

Bubble sort

Insertion sort

Selection sort

Merge sort

Quick sort

Heap sort

Bucket sort

Radix sort

sortingcomparison of algorithms

Bubble sort

Insertion sort

Selection sort

Merge sort

Quick sort

Heap sort

Bucket sort

Radix sort

sorting

Yes, but how did you sort out your CD’s?

not covered sorting

Sift sortShell sortBead sort

not covered sorting

Sift sortShell sortBead sort

not covered sorting

Sift sortShell sortBead sort

not covered sorting