How do you implement a max heap?

A max heap is implemented by using a binary tree structure and maintaining the heap property during insertions and deletions.

A max heap is a specialised tree-based data structure that satisfies the heap property. In a max heap, for any given node I, the value of I is greater than or equal to the values of its children. This property must be true across the entire tree. The heap is usually implemented as an array, where each element represents a node of the tree and the parent-child relationship is defined by their indices.

To implement a max heap, you start by defining the binary tree structure. Each node in the tree has a value and two children. The root node, which has no parent, contains the maximum value. Each child node has a value less than or equal to its parent node's value. This structure can be stored in an array, where for any index i, the left child is at index 2i+1 and the right child is at index 2i+2.

When inserting a new element into the max heap, you add it at the end of the array (the bottom of the tree) and then "bubble up" if necessary to maintain the heap property. This involves comparing the inserted element with its parent and swapping them if the element is larger. This process is repeated until the element is in a position where it is less than or equal to its parent, or it becomes the root of the tree.

Deletion in a max heap usually refers to removing the root node, as it contains the maximum value. After removing the root, the heap property may be violated. To restore the property, you can move the last element in the array to the root and then "bubble down", swapping it with its largest child until it is larger than both its children.

In summary, implementing a max heap involves creating a binary tree structure, usually represented as an array, and maintaining the heap property during insertions and deletions. This is achieved through the "bubble up" and "bubble down" processes, which ensure that each parent node is always greater than or equal to its child nodes.