This lecture introduces weighted shortest paths, generalizing previous unweighted algorithms (BFS, DFS) used for single-source shortest paths, connected components, and topological sort. The key challenge with weighted graphs is the possibility of negative-weight cycles, which can lead to infinitely decreasing path weights. The lecture covers representing weights, defining weighted path lengths, and handling negative weight cycles. A linear-time algorithm for single-source shortest paths in Directed Acyclic Graphs (DAGs) using topological sort and "dag relaxation" is presented, along with a method to reconstruct shortest path trees from distances. Future lectures will address general graph algorithms (Bellman-Ford, Dijkstra) for weighted shortest paths. This segment provides several compelling real-world examples illustrating the practical significance of weighted graphs. It covers diverse applications such as road networks (travel time), computer networks (latency), and social networks (relationship strength), showcasing the versatility and relevance of weighted graph analysis. This segment introduces the concept of weighted graphs, where edges have associated integer weights, generalizing the previous notion of distance as the number of edges. It highlights the shift from counting edges to summing edge weights to determine path length, setting the stage for exploring algorithms that handle weighted graphs. This segment defines the weight of a path as the sum of its constituent edge weights. It clarifies that the shortest path is the one with the minimum weight, introducing the concept of weighted shortest paths and setting the stage for discussing algorithms to find them. The discussion also addresses the possibility of using the same edge multiple times in a path. This segment delves into the complexities introduced by negative edge weights, particularly the potential for infinitely decreasing path weights due to negative weight cycles. It introduces the concept of infimum to handle cases where the shortest path weight might not be well-defined, setting the stage for discussing algorithms that handle negative weights. This segment visually and algorithmically explains how to determine if an edge is part of a shortest path. It uses a diagram to illustrate the condition where the shortest path distance from the source to a vertex equals the shortest path distance to its predecessor plus the edge weight, thus confirming the edge's inclusion in a shortest path and enabling parent pointer assignment. This segment uses a concrete graph example to illustrate the problem of negative weight cycles. It demonstrates how traversing a negative weight cycle repeatedly can lead to arbitrarily small path weights, highlighting the need for algorithms that can detect and handle such cycles. This segment details an algorithm to reconstruct parent pointers along shortest paths in linear time, given shortest path distances. It focuses on the algorithm's logic for assigning parent pointers based on shortest path distances, handling cases with finite shortest path distances, and initializing the parent pointer data structure. The explanation clarifies the conditions under which a vertex becomes a parent along the shortest path. This segment discusses the limitations of using Breadth-First Search (BFS) for weighted shortest paths, highlighting scenarios where BFS can be adapted (positive, uniform weights) and emphasizing the open problem of finding a linear-time algorithm for general weighted graphs. It sets the stage for introducing more sophisticated algorithms. This segment introduces the DAG relaxation algorithm for computing single-source shortest paths in a Directed Acyclic Graph (DAG) in linear time. It explains the concept of maintaining distance estimates, initializing them to infinity, and gradually lowering them until they equal the shortest path distances. The algorithm leverages the topological order of a DAG for efficient computation.This segment explains the triangle inequality principle in the context of shortest path algorithms. It describes how the algorithm lowers distance estimates when the triangle inequality is violated, a process called "relaxation." The segment emphasizes the safety of relaxation, ensuring that distance estimates always represent the weight of some path or remain infinite.This segment proves the safety of the relaxation step in the DAG relaxation algorithm. It demonstrates that the property of distance estimates being either infinite or the weight of some path is maintained after relaxation, ensuring the algorithm's correctness.This segment presents the DAG relaxation algorithm step-by-step. It explains the initialization of distance estimates, the processing of vertices in topological order, and the relaxation of edges when the triangle inequality is violated. The algorithm's simplicity and efficiency are highlighted.This segment demonstrates the correctness of the DAG relaxation algorithm using an example graph and a topological sort order. It proves by induction that the algorithm correctly computes shortest path distances for all vertices in linear time, emphasizing the algorithm's efficiency and its reliance on the topological ordering of the DAG.