算法导论8-5思考题-平均排序-average sorting

Suppose that, instead of sorting an array, we just require that the elements increase on average. More precisely, we call an n-element array A k-sortedif, for all i = 1, 2, . . ., n - k, the following holds: What does it mean for an array to be 1-sorted? Give a permutation of the numbers 1, 2, . . ., 10 that is 2-sorted, but not sorted. Prove that an n-element array is k-sorted if and only if A[i] ≤ A[i + k] for all i = 1, 2, . . ., n - k. Give an algorithm that k-sorts an n-element array in O(n lg(n/k)) time. We can also show a lower bound on the time to produce a k-sorted array, when k is a constant. Show that a k-sorted array of length n can be sorted in O(n lg k) time. (Hint: Use the solution to Exercise 6.5-8.) Show that when k is a constant, it requires Θ(n lg n) time to k-sort an n-element array. (Hint: Use the solution to the previous part along with the lower bound on comparison sorts.) 解答： a. 如果是1-排序的，其实就是正常的排序的意思 b.6,2,7,3,8,4,9,5,10 c. 直接从定义推就可以了 d. 要求对一个给定的数组进行k-排序。根据c中的结论，我们只需要正常排序几个整数集合，他们分别是     1. a[0],a[k],a[2*k],....     2. a[1],a[1+k],a[1+2*k],...     3. a[2],a[2+k],a[2+2*k]...     .....     k. a[k-1],a[k-1+k],a[k-1+2*k]... 排序每个集合需要O(n/k *log(n/k)) ,所有的共O(n*log(n/k)) e. 这个题说，假设数组已经是k-排序了，如何在O(n log(k))内把该数组正常排序     这个就是k个有序表merge的过程了，可以使用一个堆来搞，这样就可以了 f. 略