算法課_歸併排序應用

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.

類似歸併排序，從1-2-4，

保證兩個歸併前是隨機排序的，即N1和N2裏的元素都有 1/N1 和 1/N2 的概率排到每一個位置，那麼N1和N2歸併的時候，N1/(N1+N2)的概率選擇N1，N2/(N1+N2)的概率選擇N2，那麼每個元素在每個位置的概率都是1/(N1+N2)。例如，兩個長度爲1的歸併，每個都有1/2的概率排在兩個位置上。