无法理解K路合并算法（给出了计数器示例）

Question

In K way merge sort, the solution that uses a heap: essentially maintains a heap and constantly extracts max from that heap. I have a counterexample for why this won't work well.

5 -> 1 -> 0
4 -> 2 -> 1
3 -> 2 -> 0

Suppose we initialize our heap. It contains {5, 4, 3}.

We run extract max, we obtain 5 and add that into our new list (that represents the final solution). Our heap now looks like {4,3}. We then refill our heap with the head of list that we extracted the max element from.

This implies that we get something like this: {4, 3, 1}.

This doesn't make sense to me. This heap doesn't represent the top K elements anymore. 1 shouldn't be used to refill the heap, it should have been 2. So, this O(nlgk) method doesn't make much sense to me.

I hope someone can shed light on how this algorithm works because I'm stuck here.

Answer 1

The max heap always contains the max elements of k lists (or arrays). For your 'counter' example:

5 -> 1 -> 0
4 -> 2 -> 1
3 -> 2 -> 0

The heap is {5, 4, 3} contains max elements of these three lists.

Now you extract 5 from the heap, means you also remove 5 from the first list:

5-->1-->0: after extract 5, the list now is 1-->0: so 1 now is the top of the list.

Then the new heap is {4, 3, 1}, still contains max elements of lists.

Lets continue your example: the current heap after extracting 5 and heapifying is:

{4, 3, 1}

Extract 4 from the heap, means you also remove 4 from:

4-->2-->1: remove 4 you have 2-->1. 2 now is the top element of the list.

Then a new heap now is

{3, 2, 1}

Keep doing this, you get what you want (descending list).