Give an O(n log k)-time algorithm to merge k sorted lists into one sorted list,

Question

Give an O(n log k)-time algorithm to merge k sorted lists into one sorted list, where n is the total number of elements in all the input lists. Use a min-heap for k-way merging. Note that if you need to use any dened operation on a heap, just treat it as a function and don't need to worry about the details. For example, if you want to build a heap on an array A, just say buildHeap(A). However, you need to describe on what elements you build this heap. For your designed algorithm, analyze its time complexity.

Explanation / Answer

The total time complexity is thus O(k?logk+n?2logk)=O(nlogk)

The main thrust of the correctness proof is that the least element remaining is always the one to be extracted. The key invariant in support of this claim is that the priority queue contains, for each list, the least element remaining from that list. From this invariant, it follows that, since the least element remaining is also the least element remaining from its list, it is returned by extract-min.

We need to choose an element from the same list in part 3 to maintain the invariant that each list have its least element in the queue.

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