Last Minute Lecture

Bipartite Matching Algorithms — Hopcroft-Karp, Gale-Shapley & Hungarian Method | Chapter 25 in Introduction to Algorithms Chapter 25 of Introduction to Algorithms (CLRS) presents a rich collection of algorithms for solving matching problems in bipartite graphs. These problems involve pairing elements from two disjoint sets—such as jobs and workers or students and schools—based on constraints like maximum size, stability, or optimal assignment weights. This chapter explores the powerful Hopcroft-Karp algorithm for maximum matching, the Gale-Shapley algorithm for stable marriage problems, and the Hungarian algorithm for solving the assignment problem with weights. 📺 Watch the full podcast-style summary above, or continue reading for a complete breakdown of the key matching strategies, theorems, and complexity results introduced in Chapter 25. Maximum Bipartite Matching A matching is a set of edges with no shared vertices. In bipartite graphs, the goal is often to fin...

Maximum Flow Algorithms — Ford-Fulkerson, Edmonds-Karp & Bipartite Matching | Chapter 24 in Introduction to Algorithms Chapter 24 of Introduction to Algorithms (CLRS) tackles the maximum-flow problem —how to move the greatest possible amount of material through a network from a source to a sink without exceeding edge capacities. This problem has critical applications in transportation, communication networks, and resource allocation. The chapter introduces essential flow concepts, residual networks, augmenting paths, and foundational algorithms like Ford-Fulkerson and Edmonds-Karp . It also explains how flow models apply to bipartite matching and special network structures. 📺 Watch the full chapter summary above, or continue reading for a complete breakdown of every core concept and algorithm covered in Chapter 24. What Is a Flow Network? A flow network is a directed graph where each edge has a nonnegative capacity, and the goal is to determine how much material can ...

All-Pairs Shortest Paths — Floyd-Warshall, Johnson’s Algorithm & Matrix Methods | Chapter 23 in Introduction to Algorithms Chapter 23 of Introduction to Algorithms (CLRS) dives into solving one of the most fundamental problems in graph theory: the all-pairs shortest paths (APSP) problem. This chapter introduces several algorithmic strategies—ranging from matrix multiplication using the tropical semiring to the efficient Floyd-Warshall and Johnson’s algorithms—that compute shortest path distances between every pair of vertices in a weighted, directed graph. These methods are widely applicable in routing, scheduling, reachability analysis, and computational optimization. 📺 Watch the full video above for a comprehensive summary, or continue reading for a detailed, searchable guide to every major algorithm and concept in Chapter 23. What Is the All-Pairs Shortest Paths Problem? The APSP problem aims to compute the shortest paths between all pairs of vertices in a graph. T...

Shortest Path Algorithms — Bellman-Ford, Dijkstra & DAG Methods Explained | Chapter 22 in Introduction to Algorithms Chapter 22 of Introduction to Algorithms (CLRS) provides a comprehensive guide to solving the single-source shortest-path problem in weighted, directed graphs. These algorithms are the foundation of many practical applications in computer science, such as GPS navigation, internet routing, and task scheduling. The chapter introduces essential algorithmic tools—including relaxation and optimal substructure—before explaining three powerful solutions: the Bellman-Ford algorithm, DAG shortest paths, and Dijkstra’s algorithm. 📺 Watch the full podcast-style summary above or continue reading for a detailed breakdown of key concepts, methods, and properties that make shortest-path algorithms so essential in graph theory. Understanding the Shortest Path Problem The single-source shortest path problem asks: given a weighted, directed graph and a source vertex s , w...

Minimum Spanning Trees — Kruskal’s & Prim’s Algorithms Explained | Chapter 21 in Introduction to Algorithms Chapter 21 of Introduction to Algorithms (CLRS) dives deep into one of the most elegant and widely used topics in graph theory: the minimum spanning tree (MST). MST algorithms are essential in network design, clustering, and optimization problems. This chapter introduces the MST problem, explains the greedy strategy behind its solutions, and breaks down two classic algorithms—Kruskal’s and Prim’s—used to efficiently construct MSTs. 📺 Watch the video above for a podcast-style walkthrough, or read below for a complete written summary and key definitions you need to understand this foundational algorithmic concept. What Is a Minimum Spanning Tree? A minimum spanning tree is a subset of a graph’s edges that connects all the vertices together without forming cycles and with the least possible total edge weight. The MST of a connected, undirected, weighted graph ensur...

Graph Algorithms Explained — BFS, DFS, Topological Sort & SCCs | Chapter 20 in Introduction to Algorithms Graph algorithms are at the heart of many critical operations in computer science, from web crawling and pathfinding to task scheduling and dependency resolution. Chapter 20 of Introduction to Algorithms (CLRS) provides a comprehensive overview of graph representations, traversal algorithms like BFS and DFS, and advanced applications such as topological sorting and strongly connected components (SCCs). Whether you're studying for AP Computer Science or brushing up on algorithm fundamentals, this chapter is essential reading. 📺 Watch the full video above for a visual breakdown of each algorithm and its application, or keep reading for an in-depth written explanation. Graph Representations Graphs can be represented in two main ways: adjacency lists and adjacency matrices. Each has trade-offs in space and time complexity depending on the graph’s density. Adja...