If the elements of a set can be compared with total order semantics then these elements can be stored in a heap. A heap is a binary tree in which these two rules are followed:

• The element contained by each node is greater than or equal to the elements of that node’s children.
• The tee is a complete binary tree so that every level except the deepest must contain as many nodes as possible; at the deepest level all the nodes are as far left as possible.

To implement a priority queue is easy using heaps since:

The element contained by each node has a priority that is greater than or equal to the priorities of the elements of that node’s children.

The tree is a complete binary tree

Add always at the bottom in the last position open.

• Place the new element in the heap in the first available location. This keeps the structure as a complete binary tree, but it might no longer be a heap since the new element might have a higher priority than its parent.
• While (the new element has a priority that is higher than its parent) swap the element with its parent.

1. The element that we will be removing will always be the root. To do so, save the value of the root into a variable.
2. Copy the last element in the deepest level to the root This will be called the out-of-place element.
3. Take this last node out of the tree.
4. While the out-of-place element has a priority that is lower than the one of its children) Swap the out of place element with its highest priority child.
5. Return the value that you saved in step 1.