Okay. Recall now that our original problem that we are interested in is testing whether the, the small world effect or phenomenon holds for a given system of interactions. For example give a Facebook network we want to see. If we see, if we observe the small world phenomenon. Or if we have even a computer network, a network of computers, we want to see if the small world phenomenon also holds there. Because, again, you can think, for example, for a network of computers, if the, small world phenomenon holds for that system, it means that information can flow between these computers very quickly. Or, you can think about, about bad situations where an attack on the network can flow also through the system very quickly. So if one of the computers in the system is infected with a virus, that virus might spread throughout the network very quickly. So that was our original problem, is given a network. Which in it, by now, we know that we can represent its connectivity in terms of a graph, of nodes and, and edges, we want to check if that network, satisfies the war, the small world phenomenon and we, as we said, we took that problem, we formulated it as a graph theoretic problem of computing pair wise distances between every pair of nodes in the graph. And then plotting a distribution. And our first attempt was, okay let's come up with an algorithm that follows the definition of what is a distance in a graph. And that led us to what we call a brute force algorithm. And, that algorithm, we showed that it is easy to implement, it is easy to reason about its correctness. However, we also showed, that from the perspective of time efficiency, how long. The algorithm will take on any reasonable input of any reasonable size. We showed that that algorithm is not feasible in practice. We couldn't even run it on, on a graph, that had 100 nodes. Okay? So, the question is what do we do then? I mean, is that, is that the end of story? We cannot test if a network, if a large network has a small world phenomenon or satisfies a small world phenomenon. The answer is fortunately no. We don't, this is not the end of the story. In fact, we can think about computing distances in a undirected, unweighted or even directed and unweighted graph. Very quickly, okay? So we can come up with with an algorithm that's much more intelligent than brute force way of looking at every possible path of every possible length that led to, to to exponential, at least exponential amount of time. So, let's reason about how would we go about coming up with an algorithm that computes distances in a graph, in an unweighted graph, graph that has no. No weights on the edges, much more intelligently. And I will illustrate with a, with a, with a specific graph, just for the sake of illustration. [SOUND] So I will illustrate with this graph here. This is a, an undirected graph that has six nodes. And, we will, we will try to for example here I'm going to focus on one case of let's compute distances for example from zero to every other node. Remember that in the original problem we want to compute the, distance between every pair of node. Okay? So let me now focus on the problem that I want to compute the distance between zero and every other node in this graph. Now, [SOUND] the way we can do it more intelligently is to start thinking about looking at this graph and thinking how can I find the distance from node zero to every other node. I will denote by dj for example the distance, [SOUND] between node 0 and node j okay. So here the source node. Is, is fixed. I'm, I'm always focusing on node 0 and I want to compute the distance from 0 to every other node. The distance from node 0 to node 5 is going to be represented by d sub 5 for example. So, when I look at this graph, the easiest thing to say is what is the distance from node 0 to itself. So the route from d, from node 0 to itself, which we call d0, is 0, okay. So now I know that at least, I can compute the distance from the node to itself. Then, what are the next nodes that are easy for me to compute distances? If I look at this graph and I know that the distance to node 0 is 0, now I can look at the neighbors of that node, remember the neighbors are nodes that are connected by an edge, okay. So if, if the, if my distance, if the distance to me from node 0 is 0, now I look at my neighbor. Going from node 0 to my neighbor is going to go through me, okay? If I am a node here the, the distance is going to go through me to that node. Now, a distance that goes through from 0 to one is going to be the distance to node 0 plus one edge, right? So if I look at all the neighbors of node 0, their distance from node 0. Is 1 plus the distance from 0 to itself. Okay. So I know that if I look at node 0 the distance itself is 0. Then I know that the distance from 0 to all its neighbors is 1. So I can immediately say, okay, for node 1, d1 [SOUND] is 1, d3 is one, d4 is 1. Now, notice I didn't go to node 1 and then node 2. I'm looking first at all the neighbors of 0. Okay? So, think about it as, I'm going to proceed in layers. At layer 0, [SOUND] I'm looking at the initial node. Okay? The node, node 0. Then, at, at layer 1. I'm going to look, [SOUND] I'm going to look at all nodes that are at distance 1, one edge away from node 0. And this is easy, again, because we have, if, if we for example, have, have an adjacent illustrative presentation of the graph. Then I look at node 0, I look at all the adjacent nodes of it. For each one of them I say the distance to these nodes are, is 1. Okay. After I'm done, I'm done with layer 1 now I can go to layer 2. What is layer 2? If I start from node 0 and I have computed the distance to its neighbors. I next move, I move in the next step, to the neighbors of the neighbors of node 0. So these are the neighbors at layer 1 of node 0. Now for each one of them, I look at its neighbors. Okay? So for node 1. It has two neighbors. Th, it has three neighbors. Node 0. Node 3. And node 2. Okay? So I look at node 1. It has three neighbors. But these two neighbors their distances has been computed. Okay? They have been computed already so I am done. Now I look at, at nodes for the which then the, the distance has not been computed yet. Now. What is the distance to node 2 from node 0? I know that I can get to node 1, from node 0 in one step, right? So the distance from 0 to 1 is 1. Node 2 is away from 0 by 1 edge from node 1, so I can take the length of the path to 1, add to it 1 and I will get the distance to node 2. [SOUND] Okay. So I can think of the distance to node 2 is the distance to node 1 plus one. Because 2 is a neighbor of 1. But what is the distance to node 1 if there's 1. So it is 2. Okay. Now, as I said, 3 is also a neighbor of 1, and I can say that d3 is d1 plus one, which will give it 2, but 2 is larger than the distance that they already have computed and I don't need to update it, okay? So we do, we do this. Node 3 has, now, we, we are done with node 1, we are done with all the neighbors of node 1. We go to node 3 that we have already processed, and I look at its neighbors. They have already been processed, their distances have been computed. I look at neigh, at node 4 this the distance to its neighbor 0 is already computed, we don't already have to, to update it. And it has this neighbor 2, the distance have been computed from node 0. We have another node, node 5, which is, [SOUND] we need to compute it's distance. The, the distance to 5 is that I can go through 4, through distance, [SOUND] from 0 to 4, plus 1 more edge, right? Which is 1 plus 1, which is 2. Okay. And, this way, I have dealt with these at layer 2, okay. And, given this spec, specific graph, I am done actually. The algorithm is done. So we, we, we traversed the graph in, in, in the in layers. At layer 0, we deal with the source node that we specified. To, to compute distances from. At layer 1 all the notes that are, that are neighbors of the original node. At layer 2, all the neighbors of the neighbors and so on. Okay? This kind of algorithm, that looks at, at the node, then at all it's neighbors. Then for each neighbor at all its neighbors. This is known as, breadth [SOUND] first search or BFS. Okay? This is a very, famous algorithm for graph exploration or traversing graphs. That works in the breadth-first manner. The, the the reason we call it breadth-first is that when we look at the node we al, always look at the neighbors of that node before we move from this node to its neighbors. Okay? So, a breadth-first search algorithm is going to explore the graph. And while exploring, the graphic it can update these values of these djs and give us the distances, okay? Now, [COUGH] this, this is how the algorithm works and it's easier to describe, but will before we go to the, go the point where we write the algorithm, we have to pay attention to a few details here. So, the first detail that we have to pay attention to is, when I, when we compute the distance d0, then we started looking at, at the neighbors. I said okay, we have d1 is 1, d3 is 1, d4 is 1. When we were done we said let's go to layer 2. How did we go to layer 2? Now we went to the neighbors of 1, then the neighbors of 3, then the neighbors of 4, in this order, because I compute the distance to 1 first, then to 3, then to 4. When I want it to go to layer 2, I want it to go to 1, and it's neighbors, to 3, look at it's neighbors, to 4, look at it's neighbors. So how do I keep track of this information? How do I keep track. Of this, order, that I was the 1 then 3 then 4, so that next time when I go to layer 2, I go to neighbors of 1, then neighbors of 3, then neighbors of 4. So this is where, the, the concept of data structure has come into the picture, that we have to use some sort of data structure that is going. To, to allow me to save this information and keep, keep track about how the nodes were visited, so that the next time I look at their neighbors for the next layer, I also visit the neighbors of nodes in the right order. So, let's look again at this case. When I visit node 0, after that I look at it's neighbors. I looked at node 1, then 3, then 4, so I want once I, once I computed the d1, I want to put it in a data structure such that, this is node 1 here, then I, put 3, then I put 4 here. And next time when I look at for layer 2, I need to look at this node first, then this, then this, okay? So, remember what happened here. I insert, I insert node 1, then I insert node 3, then I insert node 4, okay? So when I, when I am done with this, I have this list of nodes that has these three nodes here. That next time, when I go to layer 2, I can come back here and say okay, now, let's look at node 1, then node 3, then ne, node 4. Okay? So, the way we, we deal with this kind of, of algorithm, or with this type of data structure is that, we have this type of list that we are, we are going to, to have. I'm going to have two endpoints to it. I'm going to label them as the head, [SOUND] and I'm going to label this as the tail, okay? And every time I see, when, when I'm processing the neighbors of a given node, I'm going to add, once I compute the distance to a neighbor, I'm going to add that neighbor to this list. In such a way that I maintain the order in which they were visited. So the way to do this is always to insert a node and a tail. Okay? So I insert one. So, if I want to work through this. [SOUND] In the beginning, the list was, empty. It had nothing. So the head. [SOUND] N tail are pointing to the same place. So I, I, I encounter node 1 first. I said insert it at the whatever the tail is pointing to. It's here so it becomes, [SOUND] node node 1 here. Sorry, node 1. Now this is the head. And this is the tail now, the tail moves to the right. When I see node 3, I add now node 3 here at the tail and I move the tail to the right. [SOUND] When I see node 4, I, put node 4 here and I move the tail here. [SOUND] Okay. So this is how we grow this list. Now this list, is maintained in a very specific way, that, I always add things at the tail. When I want, when next, time when I come to process la, layer 2, I want to start with node 1, then 3, then 4, so I look at the neighbors of 1, then the neighbors of 3, then the neighbors of 4. So then I come and access things from the head side, okay. So the next time, when I want to, when I want to process at layer 2, I want to look at all the neighbors of, of of node 1, which is exists at the head. So I can look at this, deal with all its neighbors, I get done with it, and I move the head. [SOUND] It's here. I deal with this. I get done with it. I move ahead and so on. But notice also that when I da, when I deal with, when I process node 1, I have to add all the neighbors that have not been processed to the queue. So once I'm done with 1, also node 2 was added at the tail. [SOUND] And the tail moved, [SOUND] okay? Then when we process 3, 3 has no neighbors that have not been processed, so we process it and we move the head without adding anything. [SOUND] Then, we process node 4. It has only one neighbor that has not been processed yet. We added it to the, to the tail, and we move the head here, and we move the tail to the right. [SOUND]. Okay. And we keep, proceeding like this. This type of list, where we add things to the tail. And we remove things from the head is known as a queue. Okay. So this data structure is called a queue. [SOUND] And every probably you have, you have seen a queue you have been in a queue when you go to a cafeteria and, and stand in line, the first person that shows up. Is added to the beginning of the queue. When you show up you come to the end of the queue. The person who gets service first is the one at the head of the queue and, you when you're added to the end of the queue your turn comes after everyone has been processed. This data structure is called a queue, and it is very, very essential for the, details of BFS.
