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...

NP-Completeness Explained — Decision Problems, Polynomial-Time Reductions, and Computational Hardness | Chapter 34 in Introduction to Algorithms Chapter 34 of Introduction to Algorithms by Cormen et al. introduces the critical concepts of NP-completeness and computational intractability, focusing on classes like P, NP, and co-NP, and how polynomial-time reductions help define problem difficulty. This post breaks down the central ideas behind decision problems, verification, and problem reductions that are essential for understanding theoretical computer science and real-world algorithmic challenges. 📺 Watch the full breakdown on YouTube: What Is NP-Completeness? NP-completeness is a concept in computational complexity theory that identifies the hardest problems in the class NP—problems for which solutions can be verified in polynomial time. If any NP-complete problem is solved in polynomial time, then all problems in NP can also be solved in polynomial time, implying P = NP...

Machine Learning Algorithms — K-Means, Weighted Experts & Gradient Descent | Chapter 33 in Introduction to Algorithms Chapter 33 of Introduction to Algorithms introduces three foundational machine learning techniques: k-means clustering for unsupervised learning, multiplicative-weights algorithms for online decision-making, and gradient descent for optimization in large-scale data settings. These algorithms are essential tools across supervised, unsupervised, and online learning paradigms, and they serve as building blocks in modern data science and artificial intelligence. Watch the video for a full breakdown of how these algorithms work and where they apply. Subscribe to Last Minute Lecture for more data-driven summaries from classic textbooks. Types of Machine Learning The chapter begins by categorizing learning approaches: Supervised Learning: Learn from labeled data to predict outcomes (e.g., spam classification). Unsupervised Learning: Discover hidd...

String Matching Algorithms — KMP, Rabin-Karp, & Suffix Arrays Explained | Chapter 32 in Introduction to Algorithms Chapter 32 of Introduction to Algorithms tackles the fundamental problem of string matching — finding all occurrences of a pattern string P within a larger text string T. This topic underlies critical applications in text search, data compression, bioinformatics, and pattern recognition. The chapter presents a spectrum of algorithms including the naive method, Rabin-Karp , finite automata , Knuth-Morris-Pratt (KMP) , and suffix arrays , each offering different trade-offs between preprocessing complexity and matching efficiency. Watch the video above for a guided breakdown of each algorithm and when to use them. Subscribe to Last Minute Lecture for more textbook-based algorithm tutorials. Naive String-Matching Algorithm The naive approach checks for a match of the pattern P at every possible shift within T. For each shift s, it compares P[1 to m] to T[s+1...

Number-Theoretic Algorithms for Cryptography & Modular Arithmetic | Chapter 31 in Introduction to Algorithms Chapter 31 of Introduction to Algorithms explores foundational number-theoretic algorithms that power modern cryptography, modular arithmetic, and primality testing. From computing greatest common divisors to understanding public-key encryption like RSA, this chapter equips readers with the mathematical tools that underpin secure communication and cryptographic protocols. Watch the video above for a guided podcast-style summary of the chapter, and subscribe to Last Minute Lecture for more algorithm-focused textbook breakdowns every week. Core Number-Theoretic Concepts The chapter begins by reviewing the basics of number theory: Divisibility: d divides a if a = kd for some integer k Primes: Only divisible by 1 and themselves Greatest Common Divisor (gcd): The largest integer dividing two numbers Modular Equivalence: a ≡ b (mod n) if a and b hav...

Polynomials & the Fast Fourier Transform (FFT) Explained | Chapter 30 in Introduction to Algorithms Chapter 30 of Introduction to Algorithms explores the Fast Fourier Transform (FFT) , a groundbreaking algorithm for multiplying polynomials efficiently in Θ(n log n) time. The chapter walks through the transition between coefficient and point-value forms of a polynomial, explains the mathematical role of complex roots of unity, and introduces the convolution theorem that underlies many applications in digital signal processing, image analysis, and algorithm design. Watch the video above for a guided walkthrough of FFT and its applications in algorithmic polynomial operations, and don’t forget to subscribe to Last Minute Lecture for more expert chapter summaries from core textbooks. Polynomial Representations: Coefficient vs Point-Value Form A polynomial can be represented in two forms: Coefficient form: A(x) = a₀ + a₁x + ... + aₙ₋₁xⁿ⁻¹ — efficient for evaluation ...

Linear Programming – Optimization, Duality & Applications | Chapter 29 in Introduction to Algorithms Chapter 29 of Introduction to Algorithms presents a comprehensive foundation in linear programming (LP) , a powerful optimization tool widely used in computer science, operations research, and economics. The chapter walks through how to formulate LP problems, introduces geometric intuition, and compares algorithmic approaches like the simplex method , ellipsoid algorithm , and interior-point methods . It also explores advanced topics such as duality theory and integer linear programming (ILP) . Watch the full video above for a visual summary of Chapter 29, and be sure to subscribe to Last Minute Lecture for more high-quality textbook breakdowns every week. What Is Linear Programming? Linear programming is a method for optimizing a linear objective function subject to linear equality and inequality constraints . A typical LP seeks to maximize or minimize an expression...

LUP Decomposition, Matrix Inversion & Least Squares in Algorithms | Chapter 28 in Introduction to Algorithms Chapter 28 of Introduction to Algorithms offers a deep dive into efficient techniques for solving systems of linear equations, computing matrix inverses, and applying least-squares methods using LU and LUP decomposition. These methods are central to numerical algorithms, especially in scientific computing, computer graphics, and machine learning. Watch the video above for a clear walkthrough, or continue reading for a detailed breakdown of the chapter's most important concepts and algorithms. Don’t forget to subscribe to Last Minute Lecture for more chapter-based study guides! Solving Linear Systems with LU and LUP Decomposition At the heart of this chapter is the equation Ax = b , a foundational form in linear algebra where the goal is to solve for x . While the theoretical solution x = A⁻¹b may seem straightforward, computing the inverse directly is ofte...

Online Algorithms, Competitive Analysis & Caching Strategies | Chapter 27 in Introduction to Algorithms Chapter 27 of Introduction to Algorithms explores the fascinating world of online algorithms—those that operate without access to future input. Unlike offline algorithms, which can optimize decisions with full knowledge of the input sequence, online algorithms must make real-time choices. This chapter introduces competitive analysis as a framework for evaluating their effectiveness and walks through key applications like list maintenance, caching, and scheduling. From the Move-to-Front (MTF) heuristic to randomized caching strategies like RANDOMIZED-MARKING, this chapter offers foundational concepts critical to understanding real-world systems such as memory management and web caching. Watch the video above for a complete walkthrough of Chapter 27, and read on for a full breakdown of the core models, algorithms, and theorems that define online decision-making in algorithm...

Fork-Join Parallelism and Task Scheduling Explained — Work-Span Model, DAGs, and Greedy Scheduling | Chapter 26 in Introduction to Algorithms In the era of multicore processors, understanding how to efficiently structure algorithms for parallel execution is more crucial than ever. Chapter 26 of Introduction to Algorithms (CLRS) presents a powerful model for parallel computing using fork-join parallelism. This blog post explains the core concepts of the work-span model, task scheduling, and computation DAGs, complete with real-world examples like parallel merge sort and matrix multiplication. These insights will help students and developers alike grasp the essentials of scalable parallel algorithm design. Watch the video above for a concise breakdown of the chapter, and keep reading for a deeper dive into the key concepts, models, and examples that define parallel algorithm design in modern computing. What Is Fork-Join Parallelism? Fork-join parallelism is a task-parallel mo...

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...

Disjoint-Set Data Structures and Union-Find Explained | Chapter 19 in introduction to algorithms Disjoint-set data structures are essential tools for efficiently managing collections of non-overlapping sets, enabling fast set partitioning, connectivity checks, and core graph algorithms. Chapter 19 of introduction to algorithms explores the inner workings of union-find structures, their practical applications, and the performance-boosting heuristics that make them indispensable in computer science. Watch the chapter summary below for an accessible breakdown of disjoint-set structures and their algorithmic uses. Subscribe to Last Minute Lecture for more introduction to algorithms tutorials! Core Disjoint-Set Operations MAKE-SET(x): Creates a new set containing element x . FIND-SET(x): Returns a pointer to the representative of the set containing x . UNION(x, y): Merges the sets containing x and y . A representative is any consistent member that uniquely i...