Max heap is a complete binary tree. A complete binary tree is a binary tree in which all levels are completely filled and all the nodes in the last level are as left as possible. Max heap should also meets this criteria: the parent’s key is larger than both children’s keys. The largest value is at the root. Heap is mostly used to implement priority queue and heapsort.

Since it is complete (no holes in the tree),  it can be represented as an array. Each node’s position in the tree is corresponding to the index in the array.

The relationship between the nodes has the following formulas:
parentPos = (pos-1)/2
left = 2*pos+1
right=2*pos+2

So we can use array to implement max heap. The min heap is opposite order.

Part 1 – Max heap implementation
Part 2 – Min heap implementation
Part 3 – Priority queue implementation with heap

Insert

We declare three attributes: heap, length and maxSize. heap is an array. maxSize is the max length of the array. It is used to initialize the array. length is actual number of elements in the array. It starts from 0.

To insert a new element, we put the new value at the end of the array. Then we move it up based on its value compared to its parent. If it’s value larger than its parent, it should be switched the position with its parent, until it is below a larger node and above a smaller one. We call this process heap up (or trickle up).

Doodle

Remove

The remove operation is to remove the element with the highest value, which is the first one in the array. When we remove the first element in the array, we move the last element to fill out the empty spot. But this element’s value might be smaller than its children’s. To correct this, we compare its value with its children’s values, and switch with the child with the larger value, until it is below a larger node and above a smaller one. This process is called heap down (or trickle down). Heap down is more complicated than heap up because it has two children to compare.

Doodle

Search

Since the underlying data structure is an array, we iterate through all elements to find the one matched with the key.

Python

Traversal

Traversal is to visit all nodes in the tree in some specified order. Please note the heap is a tree in conceptual context, the underlying implementation is array. Therefore there are no actual nodes as we see in the binary tree implementation (there is no root node). But we still can traverse all elements in the same way as any tree traversal – by level order, preorder, inorder and postorder.

Level order is to visit node level by level from top to bottom. The order is the same as visiting array from index 0 to its length.

Python

Preorder is to visit the root, traverse the left subtree, traverse the right subtree. The implementation is using dfs recursion. To get the children “node”, we use the formula 2*pos+1 for left child, and 2*pos+2 for right child.

Python

Inorder is to traverse the left subtree, visit the root, traverse the right subtree. The implementation is using dfs recursion. To get the children “node”, we use the formula 2*pos+1 for left child, and 2*pos+2 for right child.

Python

Preorder is to visit the root, traverse the left subtree, traverse the right subtree. The implementation is using dfs recursion. To get the children “node”, we use the formula 2*pos+1 for left child, and 2*pos+2 for right child.