Rekha Saripella - Radix and Bucket Sort

8/3/2019 Rekha Saripella - Radix and Bucket Sort

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Radix and Bucket Sort

Rekha Saripella

CS566


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History of Sorting

Herman Hollerith (February 29, 1860 November 17, 1929) is first known to havegenerated an algorithm similar to Radix sort.

He was the son of German immigrants, born in Buffalo, New York and was a CensusStatistician. He developed a Punch Card Tabulating Machine.

Holleriths machine included punch, tabulator and sorter, and was used to generatethe official 1890 population census. The census took six months, and in another twoyears, all the census data was completed and defined.

Hollerith formed the Tabulating Machine Company in 1896. The company mergedwith International Time Recording Company and Computing Scale Company to formComputer Tabulating Recording Company (CTR) in 1911. CTR was IBM'spredecessor. CTR was renamed International Business Machines Corporation in

1924.

Hollerith served as a consulting engineer with CTR until retiring in 1921.

There are references to Harold H.Seward, a computer scientist, as being thedeveloper of Radix sort in 1954 at MIT. He also developed the Counting sort.


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History of Sortingcontd

Quicksort algorithm was developed in 1960 by Sir Charles Antony Richard Hoare(Tony Hoare or C.A.R. Hoare, born January 11, 1934) while working at Elliot BrothersLtd. in the UK.

He also developed Hoare logic, and Communicating Sequential Processes (CSP), aformal language used to specify the interactions of concurrent processes.

Herman Hollerith Sir Charles Antony Richard Hoare


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Introduction to Sorting

Sorting is the fundamental algorithmic problem inmathematics and computer science.

It puts elements in a certain order. The mostcommonly used orders are numerical andlexicographical(alphabetical) order.

Efficient sorting is important to optimize the useof other algorithms, as it is the first step in mostof them.

There are many sorting algorithms, but knowingwhich one to use depends on the specific

problem at hand.

Some factors that help decide the sort to useare:


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Introduction to Sortingcontd How many elements need to be sorted?

Will there be duplicate elements in the data?

If there are duplicate items in the array, does their order need to be maintained aftersorting ?

What do we know about the distribution of elements? Are they partly ordered, ortotally random ? Based on the execution times of available sorting algorithms, we can

decide which sorts should or should not be used. In class, weve seen that quick sortcan be much worse than O(n^2) if used to sort elements that are partially or nearlyordered.

What resources are available for executing sorts ? Can we use more memory, morenumber of processors ?

Most of the time, we do not know enough information about the elements to besorted. In such cases, we need to look at the existing sorting algorithms, and figure

out which one would be a good match. An algorithm whose worst case execution time is acceptable may be chosen when

instance details are not known.


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Classification of Sorting algorithms

Sorting algorithms are often classified using different metrics: Computational complexity: classification is based on worst, average and best

behavior of sorting a list of size (n). For typical sorting algorithms acceptable/good behavior is O(n log n) and unacceptable/bad

behavior is (n^2). Ideal behavior for a sort is O(n).

Memory usage (and use of other computer resources): Some sorting algorithms are in place", such that only O(1) or O(log n) memory is needed

beyond the items being sorted. Others need to create auxiliary data structures for data to be temporarily stored. Weve seen

in class that mergesort needs more memory resources as it is not an in place algorithm,while quicksort and heapsort are in place. Radix and bucket sorts are not in place.

Recursion: some algorithms are either recursive or non-recursive.(e.g., mergesort isrecursive).

Stability: stable sorting algorithms maintain the relative order of elements/recordswith equal keys/values. Radix and bucket sorts are stable. General method: classification is based on how sort functions internally.

Methods used internally include insertion, exchange, selection, merging, distribution etc.Bubble sort and quicksort are exchange sorts. Heapsort is a selection sort.


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Classification of Sorting algorithmscontd

Comparison sorts: A comparison sort examines elements with a comparisonoperator, which usually is the less than or equal to operator(). Comparison sortsinclude: Bubble sort Insertion sort Selection sort Shell sort Heapsort Mergesort Quicksort.

Non-Comparison sorts: these use other techniques to sort data, rather than usingcomparison operations. These include:

Radix sort (examines individual bits of keys) Bucket sort (examines bits of keys) Counting sort (indexes using key values)


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Radix Sort

Radix is the base of a number system or logarithm. Radix sort is a multiple pass distribution sort.

It distributes each item to a bucket according to part of the item's key. After each pass, items are collected from the buckets, keeping the items in order, then

redistributed according to the next most significant part of the key.

This sorts keys digit-by-digit (hence referred to as digital sort), or, if the keys are

strings that we want to sort alphabetically, it sorts character-by-character. It was used in card-sorting machines. Radix sort uses bucket or count sort as the stable sorting algorithm, where the initial

relative order of equal keys is unchanged. Integer representations can be used to represent strings of characters as well as

integers. So, anything that can be represented by integers can be rearranged to be inorder by a radix sort.

Execution of Radix sort is in (d(n + k)), where n is instance size or number ofelements that need to be sorted. k is the number of buckets that can be generatedand d is the number of digits in the element, or length of the keys.


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Classification of Radix Sort

Radix sort is classified based on how it works internally:

least significant digit (LSD) radix sort: processing starts from the least significantdigit and moves towards the most significant digit.

most significant digit (MSD) radix sort: processing starts from the most significantdigit and moves towards the least significant digit. This is recursive. It works in thefollowing way: If we are sorting strings, we would create a bucket for a,b,c upto z.

After the first pass, strings are roughly sorted in that any two strings that begin with differentletters are in the correct order.

If a bucket has more than one string, its elements are recursively sorted (sorting into bucketsby the next most significant character).

Contents of buckets are concatenated.

The differences between LSD and MSD radix sorts are In MSD, if we know the minimum number of characters needed to distinguish all the strings,we can only sort these number of characters. So, if the strings are long, but we candistinguish them all by just looking at the first three characters, then we can sort 3 instead ofthe length of the keys.


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Classification of Radix Sortcontd

LSD approach requires padding short keys if key length is variable, and guarantees that alldigits will be examined even if the first 3-4 digits contain all the information needed toachieve sorted order.

MSD is recursive. LSD is non-recursive. MSD radix sort requires much more memory to sort elements. LSD radix sort is the preferred

implementation between the two.

MSD recursive radix sorting has applications to parallel computing, as each of thesub-buckets can be sorted independently of the rest. Each recursion can be passedto the next available processor.

The Postman's sort is a variant of MSD radix sort where attributes of the key aredescribed so the algorithm can allocate buckets efficiently. This is the algorithm usedby letter-sorting machines in the post office: first states, then post offices, then routes,

etc. The smaller buckets are then recursively sorted.

Lets look at an example of LSD Radix sort.


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Example of LSD-Radix Sort

12 44 41 34 11 32 23

Input is an array of 15 integers. For integers, the number of buckets is 10, from 0to 9. The first pass distributes the keys into buckets by the least significant digit(LSD). When the first pass is done, we have the following.

23

44

34

12

42

32

41

11

0 1 2 3 4 5 6 7 8 9

5087 77

77

50 87 58

58

08

0842


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Example of LSD-Radix Sortcontd

50

We collect these, keeping their relative order:

Now we distribute by the next most significant digit, which is the highest digit in our

example, and we get the following.

11

12

23

32

34

41

42

44

When we collect them, they are in order.

12 42 444111 3223 34

12 42 4441 3411 32 23 77 58 08

0 1 2 3 4 5 6 7 8 9

50 877708

08 50 77 87

58

58

87


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Radix Sort

Running time for this example is:

T(n) = (d(n+k)) k = number of buckets = 10(0 to 9). n = number of elements to be sorted = 15 d = digits or maximum length of element = 2

Thus in our example, the algorithm will takeT(n) = (d(n+k))= (2(15+10))

= (50) execution time.

Pseudo code of Radix sort is:


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Bucket Sort

Bucket sort, or bin sort, is a distribution sorting algorithm. It is a generalization of Counting sort, and works on the assumption that keys to be

sorted are uniformly distributed over a known range (say 1 to m). It is a stable sort, where the relative order of any two items with the same key is

preserved. It works in the following way:

set up m buckets where each bucket is responsible for an equal portion of the range of keysin the array.

place items in appropriate buckets. sort items in each non-empty bucket using insertion sort. concatenate sorted lists of items from buckets to get final sorted order.

Analysis of running time of Bucket sort:

Buckets are created based on the range of elements in the array. This is a linear timeoperation. Each element is placed in its corresponding bucket, which takes linear time. Insertion sort takes a quadratic time to run. Concatenating sorted lists takes a linear time.


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Bucket Sortcontd

Execution time for Bucket sort is (n) for all the linear operations + O(n^2) time taken for insertion sort in each bucket.

n-1T(n) =(n) + O(n^2)

i=0

Using mathematical solutions, the above running time comes to be linear.

Running time of bucket sort is usually expressed asT(n) = O(m+n) where m is the range of input values n is the number of elements in the array.

If the range is in order of n, then bucket sort is linear. But if range is large, then sortmay be worse than quadratic.


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Example of Bucket Sort

The example uses an input array of 9 elements. Key values are in the range from 10 to19. It uses an auxiliary array of linked lists which is used as buckets.Items are placed in appropriate buckets and links are maintained to point to the nextelement. Order of the two keys with value 15 is maintained after sorting.


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Bucket Sort

Pseudo code of Bucket sort is:


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Advantages and Disadvantages

Advantages Radix and bucket sorts are stable, preserving existing order of equal keys. They work in linear time, unlike most other sorts. In other words, they do not bog

down when large numbers of items need to be sorted. Most sorts run in O(n logn) or O(n^2) time.

The time to sort per item is constant, as no comparisons among items are made.With other sorts, the time to sort per time increases with the number of items.

Radix sort is particularly efficient when you have large numbers of records to sortwith short keys.

Drawbacks Radix and bucket sorts do not work well when keys are very long, as the total

sorting time is proportional to key length and to the number of items to sort. They are not in-place, using more working memory than a traditional sort.


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Addendum Count sort

Count sortis a sorting algorithm that takes linear time (n), which is the bestpossible performance for a sorting algorithm.

It assumes that each of the n input elements is an integer in the range 0 to k, where kis an integer. When k = O(n), the sort runs in (n) time.

This is a stable, non-comparison sort.

It works as follows: Set up an array of initially empty values, its length being the range of keys in input array.

This is the count array. Suppose input array = {0,5,2,8,3,1,0,4} Count array size = 9, and has placeholders for occurrences of keys from 0 (minimum

element value) to 8 (maximum element value).

Each element in count array will store the number of times elements occur in input array,starting from least key value to the maximum key value. Go over the input array, counting occurrences of elements. Populate count array with counts

of the elements. After population, count array = {2,1,1,1,1,1,0,0,1}

Iterate over input array in order, and put elements from input array into the result array, usingcount array for the number of occurrences.


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AddendumCount sortcontd

Count sort uses auxiliary data structures internally (for count array and result array),and is a resource-intensive algorithm.

Pseudo code of Count sort is:


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Addendum - some uses of Sorting

Indexes in relational databases.

Since index entries are stored in sorted order, indexes help in processingdatabase operations and queries. Without an index the database has to loadrecords and sort them during execution. An index on keys will allow the databaseto simply scan the index and fetch rows as they are referenced. To order records

in descending order, the database can simply scan the index in reverse.

File comparisons.

Data in files is first sorted, and then occurrences in both files are compared andmatched.

Grouping items.

Items with the same identification are grouped together using sorting. Thisrearrangement of data allows for better identification of the data, and aids instatistical studies.


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Bibliography and References:

http://www.cs.umass.edu/~immerman/cs311/applets/vishal/RadixSort.html - demonstration ofRadix Sort.

http://users.cs.cf.ac.uk/C.L.Mumford/tristan/CountingSort.html - demonstration of Count sort. Art of Programming Volume 3 by Donald Knuth. Introduction to Algorithms by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford

Stein. http://www.cs.ubc.ca/~harrison/Java/sorting-demo.html - demonstration of different sorting

algorithms by James Gosling, Jason Harrison, Jack Snoeyink. Jim Boritz, Denis Ahrens, Alvin Raj http://en.wikipedia.org/wiki/Sorting_algorithm http://www.cs.cmu.edu/~adityaa/211/Lecture12AG.pdf - Introduction to Sorting http://www-03.ibm.com/ibm/history/history/year_1911.html - History of IBM http://www.w3c.rl.ac.uk/pasttalks/A_Timeline_of_Computing.html - timeline of computing history. http://www.nist.gov/dads/HTML/radixsort.html - radix sort http://www.cs.purdue.edu/homes/ayg/CS251/slides/chap8c.pdf - Radix and Bucket sorts.

http://www.cse.iitk.ac.in/users/dsrkg/cs210/applets/sortingII/radixSort/radix.html - Radix sort. http://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15451-

s07/www/lecture_notes/lect0213.pdf- Radix and Bucket sorts. http://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15451-f03/www/lectures/lect0923.txt -

Radix and Bucket sorts. http://www.cs.berkeley.edu/~kamil/sp03/042803.pdf - Sorting.
http://users.cs.cf.ac.uk/C.L.Mumford/tristan/CountingSort.htmlhttp://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15451-s07/www/lecture_notes/lect0213.pdfhttp://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15451-s07/www/lecture_notes/lect0213.pdfhttp://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15451-f03/www/lectures/lect0923.txthttp://www.cs.berkeley.edu/~kamil/sp03/042803.pdfhttp://www.cs.berkeley.edu/~kamil/sp03/042803.pdfhttp://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15451-f03/www/lectures/lect0923.txthttp://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15451-f03/www/lectures/lect0923.txthttp://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15451-f03/www/lectures/lect0923.txthttp://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15451-s07/www/lecture_notes/lect0213.pdfhttp://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15451-s07/www/lecture_notes/lect0213.pdfhttp://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15451-s07/www/lecture_notes/lect0213.pdfhttp://users.cs.cf.ac.uk/C.L.Mumford/tristan/CountingSort.html

Rekha Saripella - Radix and Bucket Sort

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