算法课_归并排序应用

Question 1

Merging with smaller auxiliary array. Suppose that the subarray a[0] to a[N-1] is sorted and the subarray a[N] to a[2*N-1] is sorted. How can you merge the two subarrays so that a[0] to a[2*N-1] is sorted using an auxiliary array of size N (instead of 2N)?

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Question Explanation

Hint: copy only the left half into the auxiliary array.

Question 2

Counting inversions. An inversion in an array a[] is a pair of entries a[i] and a[j] such that i<j but a[i]>a[j]. Given an array, design a linearithmic algorithm to count the number of inversions.

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Question Explanation

Hint: count while mergesorting.

Question 3

Shuffling a linked list. Given a singly-linked list containing N items, rearrange the items uniformly at random. Your algorithm should consume a logarithmic (or constant) amount of extra memory and run in time proportional to NlogN in the worst case.

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Question Explanation

Hint: design a linear-time subroutine that can take two uniformly shuffled linked lists of sizes N1 and N2 and combined them into a uniformly shuffled linked lists of size N1+N2.

类似归并排序，从1-2-4，

保证两个归并前是随机排序的，即N1和N2里的元素都有 1/N1 和 1/N2 的概率排到每一个位置，那么N1和N2归并的时候，N1/(N1+N2)的概率选择N1，N2/(N1+N2)的概率选择N2，那么每个元素在每个位置的概率都是1/(N1+N2)。例如，两个长度为1的归并，每个都有1/2的概率排在两个位置上。