Hard Algorithms

Heap sort is a comparison-based sorting algorithm that uses a binary heap data structure to build a max-heap and then repeatedly extracts the maximum element. With a guaranteed O(n log n) time complexity regardless of input data and O(1) auxiliary space, it combines many advantages of insertion sort and merge sort. While not stable and slightly slower than quick sort in practice, heap sort provides reliable performance without the risk of worst-case scenarios, making it valuable for systems requiring consistent performance.

Dijkstra's Algorithm is a graph search algorithm that solves the single-source shortest path problem for a graph with non-negative edge weights. It works by maintaining a set of vertices whose shortest distance from the source is already known and repeatedly selecting the vertex with the minimum distance value, updating the distance values of its adjacent vertices. Dijkstra's algorithm is widely used in network routing protocols and as a subroutine in other graph algorithms.