java - Binary Heap Implemented via a Binary Tree Structure

Question

For an assignment, we were instructed to create a priority queue implemented via a binary heap, without using any built-in classes, and I have done so successfully by using an array to store the queued objects. However, I'm interested in learning how to implement another queue by using an actual tree structure, but in doing so I've run across a bit of a problem.

How would I keep track of the nodes on which I would perform insertion and deletion? I have tried using a linked list, which appends each node as it is inserted - new children are added starting from the first list node, and deleted from the opposite end. However, this falls apart when elements are rearranged in the tree, as children are added at the wrong position.

Edit: Perhaps I should clarify - I'm not sure how I would be able to find the last occupied and first unoccupied leaves. For example, I would always be able to tell the last inserted leaf, but if I were to delete it, how would I know which leaf to delete when I next remove the item? The same goes for inserting - how would I know which leaf to jump to next after the current leaf has both children accounted for?

Answer 1

A tree implementation of a binary heap uses a complete tree [or almost full tree: every level is full, except the deepest one].
You always 'know' which is the last occupied leaf - where you delete from [and modifying it is O(logn) after it changed so it is not a problem], and you always 'know' which is the first non-occupied leaf, in which you add elements to [and again, modifying it is also O(logn) after it changed].

The algorithm idea is simple:
insert: insert element to the first non-occupied leaf, and use heapify [sift up] to get this element to its correct place in the heap.

delete_min: replace the first element with the last occupied leaf, and remove the last occupied leaf. then, heapify [sift down] the heap.

EDIT: note that delete() can be done to any element, and not only the head, however - finding the element you want to replace with the last leaf will be O(n), which will make this op expensive. for this reason, the delete() method [besides the head], is usually not a part of the heap data structure.