1.6.3. Week 3: Graph Introduction

Moving on from searching and sorting, we come to graphs and graph algorithms. So, we will introduce graphs. We will see how we can use them to model certain types of problems.

In this chunk, we focus on introducing graphs as a fundamental data structure in computer science. Graphs consist of nodes (also known as vertices) and edges (the connections between the nodes). We explore the idea that graphs can be used to represent a variety of problems in different domains, such as social networks, transportation networks, and even internet connections. This representation allows us to visualize relationships and interactions in a way that algorithms can manipulate effectively.

Think of graphs as a map of a city. Each intersection is a node (vertex), and the roads connecting those intersections are the edges. Just like you can use the map to find the best route from your house to a friend's house, we use graphs in algorithms to find optimal solutions to various problems.

We need to know how to represent a graph. How do you? Graph is essentially a picture, so how do you translate this picture into something that an algorithm can manipulate? So, that is representation.

This chunk discusses the various ways to represent graphs in computer algorithms. Common representations include adjacency matrices and adjacency lists. An adjacency matrix is a 2D array where each cell indicates if there is an edge between two vertices. An adjacency list consists of an array or list where each element contains a list of the nodes that are adjacent to it. Choosing the right representation depends on the specific problem and the properties of the graph, such as its density and the operations we want to perform.

Imagine you have a list of friends and how they are connected. An adjacency list would be like writing down each friend and listing who they know (for example, 'Alice knows Bob and Charlie'). An adjacency matrix, on the other hand, would be like creating a chart that marks with a 'yes' or 'no' if two friends know each other.

We will look at standard problems in graphs; reachability and connectedness. We will look at a special class of graphs called directed acyclic graphs, which are very useful for modelling certain kinds of problems.

In this chunk, we delve into classic problems associated with graphs, such as reachability (determining if there is a path between two nodes) and connectedness (checking if all nodes in a graph are part of the same component). We also introduce directed acyclic graphs (DAGs), which are graphs that have directed edges and do not contain cycles, meaning you can't return to the same node if you follow the direction of the edges. DAGs are particularly useful in scenarios like scheduling tasks where certain tasks must precede others.

Consider a family tree as an example of a directed acyclic graph. Each person can be a node, and the lines (or edges) representing parent-child relationships only go one way – from parents to children. You cannot go back to a parent if you follow the direction of the parent-child lines, hence there are no cycles.

And then we will look at other canonical problems on graphs including shortest paths and spanning trees.

This chunk introduces some of the most well-known problems involving graphs, such as finding the shortest path between two nodes (which may represent the least distance or least cost) and determining the minimum spanning tree of a graph (a subset of edges that connects all vertices with the minimum total edge weight). These problems are foundational in network design and optimization.

Think of the shortest path problem as planning a road trip from one city to another while wanting to spend the least amount of time on the road. The spanning tree problem can be likened to creating the most cost-effective network of roads that still connects every city without loops, ensuring you reach all destinations without redundancy.

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