Last Minute Lecture

Approximation Algorithms for NP-Complete Problems — Vertex Cover, TSP & FPTAS | Chapter 35 in Introduction to Algorithms In Chapter 35 of Introduction to Algorithms by Cormen et al., we explore the critical topic of approximation algorithms—an essential strategy for dealing with NP-complete problems where exact solutions are computationally impractical. These methods provide provable performance guarantees and serve as the backbone of efficient algorithmic design for intractable problems. 📺 Watch the full chapter breakdown here: What Are Approximation Algorithms? Approximation algorithms are designed to deliver near-optimal solutions for hard problems where exact algorithms would be too slow. Instead of perfect accuracy, they offer trade-offs between speed and solution quality, measured by an approximation ratio . This chapter dives into several classic NP-complete problems and the strategies used to approximate them. Vertex Cover: A 2-Approximation Strategy The vert...

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...

Greedy Algorithms — Activity Selection, Huffman Codes, Knapsack, and Caching Explained | Chapter 15 of Intro to Algorithms Greedy algorithms provide fast and elegant solutions to many optimization problems by making the best local decision at each step, hoping this strategy leads to an optimal global solution. Chapter 15 of introduction to algorithms dives into the greedy approach, contrasting it with dynamic programming, and walks through key applications: activity selection, fractional and 0-1 knapsack, Huffman coding for data compression, and caching strategies. Understanding when the greedy-choice property holds is essential to recognizing which problems are truly “greedy-amenable.” For a quick video breakdown and worked examples, watch the full chapter summary below. Don’t forget to subscribe to Last Minute Lecture for new algorithm tutorials. The Greedy Strategy A greedy algorithm chooses the best available option at each stage, never reconsidering earlier decis...