An interval heap is a data structure that combines the properties of a min-heap and a max-heap. It is used to efficiently maintain a collection of intervals, where each interval has a start and end value.

## Structure of an Interval Heap

An interval heap can be represented as a binary tree, where each node contains an interval and two additional values: low and high. The low value represents the minimum value within the subtree rooted at that node, while the high value represents the maximum value. Additionally, each node may have one or two children, depending on whether it is a leaf or an internal node.

### Operations on Interval Heap

The main operations supported by an interval heap are:

**Insertion:**To insert an interval into the heap, a new node is created and inserted at the bottom level of the tree. The values of low and high are updated for all nodes on the path from the newly inserted node to the root.**Deletion:**To delete an interval from the heap, the node containing that interval is located and removed from the tree. The remaining nodes are then restructured to maintain the heap property.**Find Min/Max:**The minimum and maximum intervals in the heap can be found by accessing the root node.

### Maintaining Heap Property

The key property of an interval heap is that for any internal node, both its low value and high value must be greater than or equal to those of its children. This property ensures that finding minimum or maximum intervals takes constant time.

To maintain this property during insertion or deletion operations, we perform cascading updates starting from nodes affected by changes in the tree structure. This ensures that the low and high values are always correct.

## Applications of Interval Heap

Interval heaps have several applications in computer science, including:

__Interval Scheduling:__Interval heaps can be used to efficiently schedule tasks or events based on their start and end times.__Event-driven Simulation:__Interval heaps can be used to manage events in a simulation system, where each event is associated with a specific interval of time.__Interval Tree Construction:__Interval heaps can be used as a building block for constructing interval trees, which are data structures used for efficient range queries.

In conclusion, an interval heap is a versatile data structure that combines the properties of min-heaps and max-heaps. It allows efficient insertion, deletion, and retrieval of minimum and maximum intervals. Understanding interval heaps can greatly benefit programmers working on applications involving intervals or scheduling.