19.1.5. Algorithm Strategies

To make this algorithm more precise, we need to make this algorithm more systematic, we have to keep track of the vertices which have been visited, so that we do not keep exploring the same vertex again. So, we do not want to keep going around in circle and doing the same problem again and again.

When analyzing a graph, it's crucial to keep track of which vertices (or nodes) have already been visited. This prevents the algorithm from going in circles and exploring the same vertices multiple times. Have you ever played a maze game where you mark paths you've taken? This is similar. By marking visited vertices, we ensure that the algorithm only moves forward towards unvisited vertices.

Imagine navigating a city by foot. You have a map, and every time you pass a street, you note it down. This way, you don’t walk down the same street repeatedly, ensuring you explore new paths instead. This is how graph traversal works!

The strategy that we executed is what is called breadth first, that is we first explore all the neighbors of the starting vertex from all the neighbors of these things which are one step away, all the neighbors that thing that two step away and so on.

The Breadth-First Search (BFS) algorithm explores all immediate neighbors of a vertex before moving to their neighbors. For example, if you start at a point in a network, BFS lets you examine all directly connected points first, ensuring you understand the next immediate connections before delving deeper. This method is useful in various applications such as finding the shortest path in an unweighted graph.

Consider a scenario where you're hosting a party and want to know your guests' friends. You would first ask your friends (the neighbors) about who they invited. Once you've listed all their friends, you'll then ask those new friends about their connections next. This method of expanding your guest list systematically mirrors a breadth-first approach.

The other strategy to go as far as possible in one direction, so you pick one neighbor at starting vertex, then you pick one neighbor of new vertex, then you pick one neighbor of that new vertex and so on.

Depth-First Search (DFS) focuses on exploring as far down a branch as possible before backtracking. It's like climbing down a tree; you climb down one branch until there's no more room, and then you come back up to try another branch. DFS is effective for scenarios where you want to explore all possible paths until you reach a goal or exhaust options.

Think of exploring a cave. You enter the cave system, and as you walk down a tunnel, you continue to follow it as far as it goes. Only upon hitting a dead end do you backtrack to explore another tunnel. This push-deep-then-go-back process characterizes depth-first exploration.

So, one of the things that you can observe is that most of the entries in this matrix are actually 0. So, remember that if you have n vertices, this matrixes size n square, because I have n rows and n columns.

When dealing with graph representations, especially adjacency matrices, it's noteworthy that most of the entries might be zero, indicating no direct edge between those vertices. For n vertices, the matrix occupies n squared space. This could space be inefficient if the graph is sparse (having less connectivity). Therefore, alternatives like adjacency lists become more efficient in such cases.

Think about filling out a seating chart for a large gathering. If most guests don’t sit next to each other, your chart would have many empty seats (zero entries). Instead, you could simply list who sits next to whom, rather than drawing out an entire chart with empty spots, thus saving space and focusing on relevant information.

So, these two representation have their advantages and disadvantages, in the adjacency matrix, we need much more space and then, adjacency list, but some questions are easier to answer than in the adjacency list.

Adjacency matrices and lists serve distinct purposes in graph representation. While matrices use more space and can quickly check if there's an edge between two nodes, lists are more efficient for sparse graphs and allow direct access to a vertex's neighbors. The choice between them depends on the specific needs of the problem being solved.

Think of it like organizing a library: an adjacency matrix would be like having a huge catalog page with every book listed even if some shelves are empty, while an adjacency list would only list the books actually on the shelves, saving space and making it easier to find what you need.

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