Can we use binary search tree to simulate heap operation?

Question

I was wondering if we can use a binary search tree to simulate heap operations (insert, find minimum, delete minimum), i.e., use a BST for doing the same job?

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Answer 1

Basically, I agree with @amit answer. I will elaborate more on the implementation of this modified BST.

Heap can do findMin or findMax in O(1) but not both in the same data structure. With a slight modification, the BST can do both findMin and findMax in O(1).

In this modified BST, you keep track of the the min node and max node every time you do an operation that can potentially modify the data structure. For example in insert operation you can check if the min value is larger than the newly inserted value, then assign the min value to the newly added node. The same technique can be applied on the max value. Hence, this BST contain these information which you can retrieve them in O(1). (same as binary heap)

In this BST (specifically Balanced BST), when you pop min or pop max, the next min value to be assigned is the successor of the min node, whereas the next max value to be assigned is the predecessor of the max node. Thus it perform in O(1). Thanks to @JimMischel comment below however we need to re-balance the tree, thus it will still run O(log n). (same as binary heap)

In my opinion, generally Heap can be replaced by Balanced BST because BST perform better in almost all of the heap data structure can do. However, I am not sure if Heap should be considered as an obsolete data structure. (What do you think?)

PS: Have to cross reference to different questions: https://stackoverflow.com/a/27074221/764592