A max-min heap is a specialized binary heap that alternates between levels of maximum and minimum heaps.

In a max-min heap, which is structured as a complete binary tree, the nodes at even levels maintain the max heap property, meaning that each node is greater than all of its descendants. Conversely, nodes at odd levels adhere to the min heap property, indicating that each node is less than all of its descendants. The root node, located at level $0$, represents the maximum element of the heap.

To implement a max-min heap, the first step is to construct a binary heap using an array, where each element corresponds to a node in the heap. The parent-child relationships in this structure can be defined by their indices. For a node located at index $i$, its left child can be found at index $2i + 1$, while its right child is positioned at index $2i + 2$. The parent of a node at index $i$ is located at index $\frac{i - 1}{2}$.

Once the binary heap is constructed, the next task is to enforce the max-min property. This is achieved by traversing the heap from the bottom to the top and comparing each node with its descendants. If any node violates the max-min property, it is swapped with the offending descendant. This procedure is repeated until the entire heap satisfies the max-min property.

Insertion into a max-min heap involves placing the new element at the end of the array (at the bottom of the heap) and then “bubbling up” to find its correct position. On the other hand, deletion entails removing the root element (the maximum element), replacing it with the last element in the array, and then “bubbling down” to restore the heap’s properties.

The max-min heap is a versatile data structure that enables efficient retrieval of both the maximum and minimum elements. It is particularly beneficial for applications that require frequent access to these extreme values. However, maintaining the max-min property requires careful attention following each insertion or deletion to ensure the integrity of the heap structure.