8.Sorting in linear time Hsu, Lih-Hsing. Computer Theory Lab. Chapter 8P.2 8.1 Lower bound for sorting The decision tree model.

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Presentation transcript:

8.Sorting in linear time Hsu, Lih-Hsing

Computer Theory Lab. Chapter 8P Lower bound for sorting The decision tree model

Computer Theory Lab. Chapter 8P.3

Computer Theory Lab. Chapter 8P Counting sort

Computer Theory Lab. Chapter 8P.5

Computer Theory Lab. Chapter 8P.6 The operation of Counting-sort on an input array A[1..8]

Computer Theory Lab. Chapter 8P Radix sort Used by the card-sorting machines you can now find only in computer museum. RADIX_SORT(A,d) 1 for i  1 to d 2 do use a stable sort to sort array A on digit i

Computer Theory Lab. Chapter 8P.8

Computer Theory Lab. Chapter 8P.9

Computer Theory Lab. Chapter 8P Bucket sort

Computer Theory Lab. Chapter 8P.11 Analysis The running time of bucket sort is taking expectations of both sides and using linearity of expectation, we have

Computer Theory Lab. Chapter 8P.12 We claim that We define indicator random variables X ij = I {A[j] falls in bucket i} for i = 0, 1, …, n-1 and j = 1, 2,…,n. thus,

Computer Theory Lab. Chapter 8P.13

Computer Theory Lab. Chapter 8P.14 Indicator random variable X ij is 1 with probability 1/n and 0 otherwise, and therefore When k  j, the variables X ij and X ik are independent, and hence

Computer Theory Lab. Chapter 8P.15 We can conclude that the expected time for bucket sort is  (n)+n·O(2-1/n)=  (n).

Computer Theory Lab. Chapter 8P.16