Understanding Two Heaps

Two Heaps is a pattern used to solve problems that involve maintaining a dynamic set of elements with the ability to find the median, the smallest or largest elements efficiently, or to balance two sets of elements. This pattern typically involves using two heaps: a min-heap and a max-heap.

Process

1. Initialization: Start by defining the problem space and initializing two heaps: a min-heap and a max-heap. The min-heap will store the larger half of the elements, and the max-heap will store the smaller half.

2. Balancing Heaps: Insert elements into the appropriate heap based on their values. For example, if you are maintaining a median, elements smaller than the median should go into the max-heap, and elements larger than the median should go into the min-heap.

3. Heap Maintenance: After inserting an element, ensure the heaps remain balanced. This typically involves adjusting the sizes of the heaps so that the difference in their sizes is at most 1. If the heaps become unbalanced, transfer the root element from one heap to another.

4. Retrieving Results: Depending on the problem, retrieve the desired result from the heaps. For example, to find the median, you can take the root of the max-heap (if the heaps are balanced) or the average of the roots of both heaps (if the heaps are of equal size).

When to Use

1. Median Maintenance: To efficiently find the median of a dynamically changing dataset.

2. Priority Queues: To maintain two sets of elements with different priorities.

How Does It Reduce Time Complexity?

1. Efficient Insertion and Extraction: Using heaps allows for efficient insertion and extraction of elements, with a time complexity of O(log N) for each operation. This results in O(log N) overall time complexity for maintaining the heaps.

2. Balancing: By maintaining two heaps, you can efficiently balance the elements and retrieve the desired result (e.g., median) in constant time O(1).

Example problem for better understanding

Find Median from Data Stream

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