SORTTING IN LINEAR TIME - Radix Sort
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Radix sort is a non-comparative sorting algorithm that sorts data based on the individual digit positions of integer keys. It works by performing multiple passes over the data, where each pass groups keys by a specific digit (from least to most significant). A stable sorting method is used to group the keys in each pass. This results in a running time of O(dn) for d digit positions and n keys.
![EXAMPLE
• Let’s sort the following array using radix sort:
A[10] =
64 8 216 512 27 729 0 1 343 125](https://image.slidesharecdn.com/radixsort-150201095343-conversion-gate01/85/SORTTING-IN-LINEAR-TIME-Radix-Sort-4-320.jpg)
![PASS 1
A[10] =
8 216 512 27 729 0 1 343 12564](https://image.slidesharecdn.com/radixsort-150201095343-conversion-gate01/85/SORTTING-IN-LINEAR-TIME-Radix-Sort-5-320.jpg)
![PASS 1
Sorted Array after pass 1 (LSD)
A[10] =
1 512 343 64 125 216 27 8 7290](https://image.slidesharecdn.com/radixsort-150201095343-conversion-gate01/85/SORTTING-IN-LINEAR-TIME-Radix-Sort-6-320.jpg)
![PASS 2
A[10] =
01 512 343 64 125 216 27 08 72900](https://image.slidesharecdn.com/radixsort-150201095343-conversion-gate01/85/SORTTING-IN-LINEAR-TIME-Radix-Sort-7-320.jpg)
![PASS 2
Sorted Array after pass 2 :
A[10] =
01 08 512 216 125 27 729 343 6400](https://image.slidesharecdn.com/radixsort-150201095343-conversion-gate01/85/SORTTING-IN-LINEAR-TIME-Radix-Sort-8-320.jpg)
![PASS 3
A[10] =
001 008 512 216 125 027 729 343 064000](https://image.slidesharecdn.com/radixsort-150201095343-conversion-gate01/85/SORTTING-IN-LINEAR-TIME-Radix-Sort-9-320.jpg)
![PASS 3
Sorted Array after pass 3 :
A[10] =
001 008 027 064 125 216 343 512 729000](https://image.slidesharecdn.com/radixsort-150201095343-conversion-gate01/85/SORTTING-IN-LINEAR-TIME-Radix-Sort-10-320.jpg)
SORTTING IN LINEAR TIME - Radix Sort
- 1. RADIX SORT
- 2. RADIX SORT • Radix sort is a non-comparative integer sorting algorithm that sorts data with integer keys by grouping keys by the individual digits which share the same significant position and value. • A positional notation is required, but because integers can represent strings of characters (e.g., names or dates), radix sort is not limited to integers
- 3. PSEUDOCODE • RADIX_SORT (A, d) 1. for i = 1 to d 2. Stable_Sort (A) on digit i
- 4. EXAMPLE • Let’s sort the following array using radix sort: A[10] = 64 8 216 512 27 729 0 1 343 125
- 5. PASS 1 A[10] = 8 216 512 27 729 0 1 343 12564
- 6. PASS 1 Sorted Array after pass 1 (LSD) A[10] = 1 512 343 64 125 216 27 8 7290
- 7. PASS 2 A[10] = 01 512 343 64 125 216 27 08 72900
- 8. PASS 2 Sorted Array after pass 2 : A[10] = 01 08 512 216 125 27 729 343 6400
- 9. PASS 3 A[10] = 001 008 512 216 125 027 729 343 064000
- 10. PASS 3 Sorted Array after pass 3 : A[10] = 001 008 027 064 125 216 343 512 729000
- 11. ANALYSIS • Assume that we use counting sort as the intermediate sort. • Ө(n+k) per pass (digits in the range 0,….,k) • d passes • Total = Ө(d(n+k)) • If k=O(n), Time = Ө(dn)
- 12. Can we prove it will work? • Basis: If d = 1, there’s only one digit, so sorting on that digit sorts the array. • Inductive Step: • Assume lower-order digits {j: j<i} are sorted • Show that sorting next digit i leaves array correctly sorted • If two digits at position i are different, ordering numbers by that digit is correct. • If they are the same, numbers are already sorted on the lower-order digits. Since we use a stable sort, the numbers stay in the right order
- 13. How to break each key into digits? (Lemma 8.4) • n keys • b bits/key • Break into r-bit digits (r ≤ b). Have d = 𝒃 𝒓 • Use counting sort, k = 2 𝑟 -1
- 14. EXAMPLE • 32- bit keys, 8- bit digits. • b = 32, r = 8, d = 32/8 = 4 • k = 28 – 1 = 255 • Time = Ө(b/r(n + 2 𝑟 ))
- 15. • If b < lgn (n + 2 𝑟) = Ө(n). Thus, choosing r = b yields an optimal running time of (b/b(n + 2 𝑏)) = Ө(n) • If b ≥ lgn Choosing r = lgn yields a running time of Ө(bn/lgn) If lgn > r > lgn, Time = Ω(bn/lgn)