Many computational tasks, particularly in optimization and algorithm design, involve managing elements based on priority. Whether finding the k most relevant items, scheduling tasks, or implementing certain graph algorithms, we often need efficient access to the item with the highest or lowest priority. This chapter focuses on Heaps, a specialized tree-based data structure designed for exactly this purpose. We will examine how Heaps maintain their structural property to enable fast insertions and extraction of the minimum or maximum element, typically achieving $O(\log n)$ time complexity for these operations.

You will learn about:

• The defining properties of min-heaps and max-heaps.
• Implementing essential heap operations: insertion, extraction, and building a heap (heapify).
• Using heaps as the underlying structure for Priority Queues.
• Applying heaps to efficiently solve selection problems (e.g., finding the k-th smallest/largest value).
• Recognizing how priority queues contribute to the efficiency of algorithms like Dijkstra's shortest path.
• Utilizing Python's heapq module for practical heap management.

Completing this chapter will equip you with the knowledge to apply heaps and priority queues effectively when tackling relevant optimization problems in machine learning contexts.