Radix Sort-Algorithm Design and Analysis for Sorting-Lecture Slides, Slides of Design and Analysis of Algorithms

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Radix Sort

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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( behavior is Ω( n^ 2). n log n ) and unacceptable/bad  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(logbeyond the items being sorted. n ) memory is needed  Others need to create auxiliary data structures for data to be temporarily stored. We‟ve seenin 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 is

recursive).

 Stability : stable sorting algorithms maintain the relative order of elements/records

with 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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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, thenredistributed 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 in

order by a radix sort.

 Execution of Radix sort is in Ө(d(n + k)), where n is instance size or number of

elements that need to be sorted. k is the number of buckets that can be generated

and 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 significant

digit and moves towards the most significant digit.

 most significant digit (MSD) radix sort: processing starts from the most significant

digit and moves towards the least significant digit. This is recursive. It works in the

following 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 different letters 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 can distinguish 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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Example of LSD-Radix Sort

Input is an array of 15 integers. For integers, the number of buckets is 10, from 0

to 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.

0 1 2 3 4 5 6 7 8 9

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Example of LSD-Radix Sort…contd

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.

When we collect them, they are in order.

0 1 2 3 4 5 6 7 8 9 08 50 77 87