Algorithms: Union Find and Minimum Spanning Trees

This lecture covers the concepts of Union-Find data structures and Minimum Spanning Trees (MST). It begins with an overview of the Ford-Fulkerson method for finding maximum flow and minimum cut in networks. The instructor explains the importance of augmenting paths and the relationship between max flow and min cut. The lecture then transitions to the Union-Find data structure, detailing its operations such as make-set, union, and find-set, along with their complexities. The instructor emphasizes the efficiency of the weighted-union heuristic and path compression techniques to optimize these operations. Following this, the lecture introduces Minimum Spanning Trees, discussing algorithms like Kruskal's and Prim's, which are used to find the least costly way to connect all vertices in a graph without cycles. The instructor illustrates these concepts with examples, including applications in network design and clustering. The lecture concludes with a discussion on the cut property and its significance in proving the correctness of MST algorithms.