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8.Sorting in linear time

Explore the lower bound for sorting using decision trees, Counting sort, Radix sort, and Bucket sort. Learn how these methods achieve linear time complexity and apply them to sort arrays effectively. Understand the running time analysis and expectations involved in implementing these sorting algorithms.

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8.Sorting in linear time

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  1. 8.Sorting in linear time Hsu, Lih-Hsing

  2. 8.1 Lower bound for sorting The decision tree model

  3. 8.2 Counting sort

  4. The operation of Counting-sort on an input array A[1..8]

  5. 8.3 Radix sort Used by the card-sorting machines you can now find only in computer museum. RADIX_SORT(A,d) 1 fori 1 tod 2 do use a stable sort to sort array A on digit i

  6. 8.4 Bucket sort

  7. Analysis The running time of bucket sort is taking expectations of both sides and using linearity of expectation, we have

  8. We claim that We define indicator random variables Xij = I {A[j] falls in bucket i} for i = 0, 1, …, n-1 and j = 1, 2,…,n. thus,

  9. Indicator random variable Xij is 1 with probability 1/n and 0 otherwise, and therefore When k j, the variables Xij and Xik are independent, and hence

  10. We can conclude that the expected time for bucket sort is (n)+n·O(2-1/n)= (n).

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