Deletion in min/max heaps

I think I'm confused about deletion in heaps, and since I have an exam today, I'm looking for your help to correct me.

I will post photos since it will makes it a bit more clear.

What I understand is that , heaps only deletes the root element or the top. So, I made 2 solutions and I'm kindly asking which solution is the correct one.

Q:the heapness will be violated, I will have to replace it by the rightmost element bottom down right? (32)

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It's hard to get what you are doing. But from my understanding the second pictures looks right. In the first picture: what whould be your final heap? – A.Schulz Nov 28 '12 at 10:08
I'm sorry for that, I tried my best, in the first pic I'll have to do precolate/bubble operations to achieve the heap property again. I think you confirm it. Me too thinking is that the second approach is the correct one. – Sobiaholic Nov 28 '12 at 11:09

A min-heap typically only supports a delete-min operation, not an arbitrary delete(x) operation. I would implement delete(x) as a composition of decrease-key(x, $-\infty$), and delete-min. Recall, that to implement decrease-key, you would bubble up the element to maintain the heap property (in this case all the way to the root). In a binary heap, to implement the delete-min operation, you replace the root by the last element on the last level, and then percolate that element down.

To summarize, to delete(x), bubble-up the element all the way to the root, then delete the element and put the last element in the heap at the root, then percolate down to restore the heap property.

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There's no need to bubble-up the element all the way to the root. You can replace the item to be removed with the last element in the heap. If that new element is less than its parent, then bubble it up. If it's greater than its children, bubble it down.

I implemented this in my DevSource article, A Priority Queue Implementation in C#. Full source is available at http://www.mischel.com/pubs/priqueue.zip.