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Sorting in Linear Time

Sorting in Linear Time. Lecture 5 Asst. Prof. Dr. İlker Kocabaş. How fast can we sort?. Decision-tree example. Decision-tree example. Decision - tree example. Decision-tree example. Decision-tree example. Decision-tree example. Lower Bound for decision - tree sorting.

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Sorting in Linear Time

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  1. Sorting in Linear Time Lecture 5 Asst. Prof. Dr. İlker Kocabaş

  2. How fast can we sort?

  3. Decision-tree example

  4. Decision-tree example

  5. Decision-treeexample

  6. Decision-tree example

  7. Decision-tree example

  8. Decision-tree example

  9. LowerBoundfordecision-treesorting

  10. Lowerboundforcomparisonsorting

  11. Sorting in linear time • Counting something, likehistograms, can help us to sort an array. • clr = getcolor(x,y) • count[ clr ]++; Limitation: count[0..255]

  12. Sorting in linear time

  13. Counting sort COUNTING-SORT(A,B,k) // Initialize aux. store // Count how many times occurred. // Cumulative sums of aux. store // ???

  14. Counting-sort example

  15. Counting-sortexample

  16. Counting-sort example

  17. Counting-sort example

  18. Counting-sort example

  19. Counting-sort example

  20. Counting-sort example

  21. Counting-sort example

  22. Counting-sort example

  23. Counting-sort example

  24. Counting-sort example

  25. Counting-sort example

  26. Counting-sort example

  27. Counting-sort example

  28. Counting-sort example

  29. Analysis

  30. Running Time

  31. Stable sorting

  32. Countingsortarray size limitation problem • If we need sort 32 bits integers, size of count array 232 = 4GB ! • Unsorted array size is usually smaller than this (1KB << 4GB) • Solution... Use smaller parts like digits and sort these parts particularly : Radix sort

  33. Radix sort

  34. Operation of radix sort

  35. Analysis of radix sort • Given n d-digit number in which each digit can take on up to k-possible values, the running time of RADIX-SORT sorts these numbers is (d*(n+k))

  36. Analysis of radix sort • Assume counting sort is the auxilary sort. • Sort n computer words of b-bits each. • Each word can be viewed as having d=b/r (base-2r) digits. • Example: Each key having d-digits of r bits. • Let d=4, r=8 (b=32) then each digit is an integer in the range 0 to 2r-1 = 28-1=255.

  37. Analysis of radix sort: Choosing r • How many passes should we make? • Each word of length b=rd bit is broken into r-bit pieces. • For a given word length of b, the running time of an array of length n that is T(r|n,b) can be minimized with respect to r.

  38. Analysis of radix sort: Choosing r

  39. Analysis of radix sort: Choosing r

  40. Conclusions

  41. Whataboutfloatingpoints? • Integer can countable or divisible to parts. • How can speed up sorting array of floating point numbers. • We can group into k-parts • For numbers beetween 0.0 ≤ f ≤ 1.0 • k*f have an integer part like 8.71 • int(k*f) is part number. • k can be length of array.

  42. Bucket Sort

  43. Bucket Sort A B 1 0 2 1 3 2 4 3 5 4 6 5 7 6 8 7 9 8 9 10

  44. Bucket Sort Running time:

  45. Bucket Sort

  46. Bucket Sort

  47. Bucket Sort

  48. Bucket Sort

  49. Bucket Sort

  50. Bucket Sort

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