Radix sorting
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This document discusses radix sorting and provides details on its implementation. Radix sorting is a non-comparative sorting algorithm that uses bucket sorting to sort integers by their individual digits in multiple passes. Pseudocode and C code examples are provided to demonstrate how radix sorting works by distributing numbers into buckets based on their digits and recombining the buckets in sorted order. The time complexity of radix sorting is linear for a constant number of digits.
![Bucket Sort
ALGORITHM
1.Keep an array count of size m(Maximum range of output)
2.Repeat following steps till input is exhausted
a) Read input number as x.
b) Increment count [x] by 1
3.Repeat the following for i=0 to m
a) Assign j to zero
b) 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.
9](https://image.slidesharecdn.com/radixsorting-130818053243-phpapp01/85/Radix-sorting-9-320.jpg)
![Pseudo code for radix sort
ALGORITHM
1.Create an array a[ 0…..n-1] elements.
2.Call bucket sort repeatedly on least to most significant digit of each
element as the key.
3.Return the sorted array.
10](https://image.slidesharecdn.com/radixsorting-130818053243-phpapp01/85/Radix-sorting-10-320.jpg)
![C code for radix sort
void radixsort(int *a, int n)
{
int i, b[MAX], m = a[0], exp = 1;
for (i = 1; i < n; i++)
{
if (a[i] > m)
m = a[i];
}
while (m / exp > 0)
{
int bucket[10] =
{ 0 };
for (i = 0; i < n; i++)
bucket[(a[i] / exp) % 10]++;
for (i = 1; i < 10; i++)
bucket[i] += bucket[i - 1];
for (i = n - 1; i >= 0; i--)
b[--bucket[(a[i] / exp) % 10]] =
a[i];
for (i = 0; i < n; i++)
a[i] = b[i];
exp *= 10;
#if def SHOWPASS
printf("nPASS : ");
print(a, n);
#endif
}
}
11](https://image.slidesharecdn.com/radixsorting-130818053243-phpapp01/85/Radix-sorting-11-320.jpg)
Radix sorting
- 1. Radix Sorting IESL College of Engineering 1
- 2. Group Members P.D. Bulathsinhala W.N.S. Fernando K.K.H. Rangana Rajeswaran Indravarman M.K.H. Gunasekara 2
- 3. Introduction • Radix sort is non comparative sorting method • Two classifications of radix sorts are least significant digit (LSD) radix sorts and most significant digit (MSD) radix sorts. • LSD radix sorts process the integer representations starting from the least digit and move towards the most significant digit. MSD radix sorts work the other way around. 3
- 4. Introduction • Radix sort is generalization of bucket sort. • It uses several passes of bucket sort. • Radix sort is stable and fast. 4
- 5. Simulation 173 256 548 326 753 478 222 144 721 875 0 1 2 3 4 5 6 7 8 9 173256548326753478222144721875 173 753 256 326 548 478 222 144721 875 5
- 6. Simulation 0 1 2 3 4 5 6 7 8 9 173256548326753478222144721875 173 753 256 326 548 478 222 144721 875 0 1 2 3 4 5 6 7 8 9 173753 256 326 548 478 222 144721 875 6
- 7. Simulation 0 1 2 3 4 5 6 7 8 9 173256548326753478222144721875 173 753256 326 548478222144 721 875 0 1 2 3 4 5 6 7 8 9 173753 256 326 548 478 222 144721 875 7
- 8. Simulation 0 1 2 3 4 5 6 7 8 9 173 753256 326 548478222144 721 875 144 173 222 256 326 478 548 721 753 875 8
- 9. Bucket Sort ALGORITHM 1.Keep an array count of size m(Maximum range of output) 2.Repeat following steps till input is exhausted a) Read input number as x. b) Increment count [x] by 1 3.Repeat the following for i=0 to m a) Assign j to zero b) 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. 9
- 10. Pseudo code for radix sort ALGORITHM 1.Create an array a[ 0…..n-1] elements. 2.Call bucket sort repeatedly on least to most significant digit of each element as the key. 3.Return the sorted array. 10
- 11. C code for radix sort void radixsort(int *a, int n) { int i, b[MAX], m = a[0], exp = 1; for (i = 1; i < n; i++) { if (a[i] > m) m = a[i]; } while (m / exp > 0) { int bucket[10] = { 0 }; for (i = 0; i < n; i++) bucket[(a[i] / exp) % 10]++; for (i = 1; i < 10; i++) bucket[i] += bucket[i - 1]; for (i = n - 1; i >= 0; i--) b[--bucket[(a[i] / exp) % 10]] = a[i]; for (i = 0; i < n; i++) a[i] = b[i]; exp *= 10; #if def SHOWPASS printf("nPASS : "); print(a, n); #endif } } 11
- 12. Iterative version of using queues • We can use queues as buckets ALGORITHM 1. The integers are enqueued into an array of ten separate queues based on their digits from right to left. 2. The queues are dequeued back into an array of integers, in increasing order. 3. Iterate this method till the last left digit. 12
- 13. Example 23 69 72 85 90 35 58 95 48 0 => 1 => 2 => 3 => 4 => 5 => 6 => 7 => 8 => 9 => 23 35 90 85 58 72 69 95 48 13
- 14. Example 0 => 1 => 2 => 3 => 4 => 5 => 6 => 7 => 8 => 9 => 23 35 90 85 58 72 69 95 48 9090 7290 72 2390 72 23 8590 72 23 85 3590 72 23 85 35 9590 72 23 85 35 95 5890 72 23 85 35 95 58 4890 72 23 85 35 95 58 48 68 14
- 15. Example 90 72 23 85 35 95 58 48 69 0 => 1 => 2 => 3 => 4 => 5 => 6 => 7 => 8 => 9 => 23 35 90 85 58 72 69 95 48 15
- 16. Recursive Version - Inefficient • Each bucket is dynamically allocated and resized as needed • This runs the risk of serious memory fragmentation • Degrade performances. 16
- 17. String variation • The keys are strings of d characters each • We represent each key by a d- tuple of integers, where is the ASCII (8-bit integer) or Unicode (16-bit integer) representation of the i-th character and apply radix sort. 17
- 18. Analysis • Each pass over n d-digit numbers and k base keys then takes time O(n+k). (Assuming counting sort is used for each pass.) • There are d passes, so the total time for radix sort is O(d (n+k)). • When d is a constant and total run time = O(n) • radix sort runs in linear time. 18
- 19. Comparison 19 Merge sort Radix sort sorted the overall relation between input size and execution time (in seconds).
- 20. Applications • Mostly used in parallel computing 20
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