The document describes bucket sort and radix sort algorithms. It provides examples to illustrate how each algorithm works. Specifically: 1) Bucket sort works by counting the number of objects that fall into multiple "buckets" based on a certain value like size or weight. It then places the objects in the correct order. 2) Radix sort sorts data based on individual digits of a number, starting from the least significant digit to the most significant. It repeatedly performs bucket sorts on each digit to fully sort the data. 3) Examples show the step-by-step process of bucket sort and LSD radix sort on sample input data to produce a sorted output.

1. WINTER Template BUCKET AND RADIX SORT 01 PRESENTED BY AJAY CHAURASIA NIKIT CHINCHGHARE SANDEEP DAS

2. 02 ALGORITHM 1.Keep an array count of size m(Maximum range of output) 2.Repeat following steps till input is exhausted Read input number as x. Increment count [x] by 1 3.Repeat the following for i=0 to m Assign j to zero While count [ i ] is not equal to zero do i) a[ j ] = I and increment j ii) Decrement count[ i ] 4.Return the sorted elements stored in array a.

3. 03 EXAMPLE OF BUCKET SORT Initial Array Sorted Array Bucket Count 0 1 1 0 2 1 3 1 4 1 5 1 6 2 6 5 6 0 3 4 2 0 2 3 4 5 6 6

4. 04 Radix sort can process digits of each number either starting from the least significant digit (LSD) or starting from the most significant digit (MSD). ALGORITHM Create an array a[ 0…..n-1] elements. Call bucket sort repeatedly on least to most significant digit of each element as the key. Return the sorted array.

5. EXAMPLE OF LSD-RADIX SORT 05 Input is an array of 10 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. Initial Array We collect these, keeping their relative order: 13 34 42 32 44 11 23 87 08 58 0 1 2 3 4 5 6 7 8 9 11 42 13 34 87 58 32 23 44 08 11 42 32 13 23 34 44 87 58 08

6. WINTER Template 06 Now we distribute by the next most significant digit, which is the highest digit in our example, and we get the following. EXAMPLE CONTD………….. Sorted Array 0 1 2 3 4 5 6 7 8 9 08 11 23 32 42 58 87 13 34 44 08 11 13 23 32 34 42 44 58 87

7. 07 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 log n) 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 sort with 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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