So now we have done the first attempt at computing pairwise distancing using brute force. We saw it from an efficiency perspective, that's not good. Then we came up with that algorithm that uses basically breadth-first search, and we showed that it is actually efficient, right? That the running time is linear in the number of edges. Quadratic in the number of nodes in the worst case. And now that we have, we have come up with, with an efficient algorithm, we now go back to our original question, which is again, we want to, if we are given a graph that corresponds to our network, Facebook network, computer network and so on. We want to check if it is, if it satisfies or exhibits the small world effect. And for that, remember that the computers or the computational solution was, given the graph, let's compute all pairwise distances and plot the distribution of these distances. Because once we plot the distribution of these distances, we can look at what type of distances we see, what is the frequency of every distance, and so on. For example, if, if all the distances between every pair of nodes, we see distance one, this, then we, we know that it is very, we have very, very short distances. And if we find lots of pairs whose distances are very large, then we know, we don't have a small world [INAUDIBLE] with that, with that graph. Okay? So now, we have an algorithm based on B F S. We are going to use it to compute what we're going to call to compute the distance distribution. Right? So now we can introduce, we can come up with this algorithm, compute. Okay? So we want to compute distance distribution. What is the input to the, to this algorithm? Now, we have been dealing with algorithm that took also as part of the input a node. And we computed distances from this node. But when we are looking at this question of the small world phenomenon, we want to compute pair-wise distances between every pair of nodes. So there is no specific node here. We want to compute for every pair of nodes. So the input is only a graph, okay? What is the output? The output is the frequencies of each possible distance. Now how many possible distances before we, we write this we have to figure out how many possible distances can we have in a graph. We have talked about this before that if we have n nodes, then the maximum possible distance that we can have, the maximum possible distance that we can have is n minus 1. So if you have only one node in the graph, the distances you're going to see are zero. If you have two nodes, either there is no path between them and the, all the distances are zero, or they're connected by an edge and the distance is 1, which is 2 minus 1. If you have three nodes, for any pair of nodes, the maximum distance that you can have, is two. If you have n, it is n minus 1. So basically we need to return for n minus 1 possible values. We need to return the frequency with which we see each one of these values, okay? So for distance zero, how many pairs of nodes we see at distance zero? For distance one, how many pairs of nodes we see distance one? For distance N minus 1, how many pairs of nodes have distance N minus 1. So the output is, I'm going to store all these in an array. I'm going to call it F for frequency, for example. It's going to be indexed from zero to size of V, minus 1. This notation is going to say if is an array is indexed from zero to, to big v minus 1, the size of v minus 1. Where I'm going to say that if of, of D for example, is the frequency of distance D, okay? With a frequency with which distance D appears over all possible pairs of nodes, okay? So, how would we compute this this output from a graph? So the way to do it, is that again usually algorithms have to initialize things. That before I start computing anything, what are the frequencies? Before I look at the graphs, I have to initialize the, the, this, this array for D is 0 to [SOUND]. For every possible distance, I'm going to say that ori, initially, for that distance the frequency is going to be zero. I haven't touched the graph yet, I haven't computed any pairwise distance. I know that every possible distance now appears with frequency, with frequency zero. After this, what do we want to do? We want to compute pairwise distance between every pair of nodes. So the way we are going to do it, since we have seen BFS. How does BFS work? BFS takes a node, and compute the distance from that node to every other node. So now, if I want to compute the pairwise distance of every pair of nodes that involve node i, I can run BFS when, for starting from node i, compute the distance to every other one. Now I know the pairwise distance between i and every other node. But I'm not done because now I have to compute the dis, pair-wise distance from every node to every other node. So the way to do this, is by basically going, looping for each node I in V, okay? We can say do here. For each one of them, we are going to do BFS on G and Node I. When we run BFS, so I'm calling a function here, BFS, that we had introduced before, we're going to call BFS on G and i. Remember, what does BFS return? It returns it the distances from i to every other node, and I'm going to tell it to return something like D. That I'm going to call now, it returns something that I'm going to store it. N in dist for example, okay? So I'm going to store in some sort of list or array that's called dist. Now what, what are the values in dist? Remember that dist is going to be indexed by every node in the graph. So dist sub j is going to be the distance from node i to node j. Right? So that value, if I look again at dist of sub G, this equals the distance between I and G, right? So this might be like five. It says that the distance between I and G is five. If the distance is five, then I need to have that f of five. I need to add one to it because now I know there is one node that has this times five. So I now need to go through every element here. And this one. For each j and V do. If I look at this g, I know this is the distance between IMJ and that distance whatever it is, I need to update the frequency of that distance. So now the frequency of this distance. So the frequency of five for example, is now incremented by 1 because we have seen one more pair that has that distance. Okay? So for each i, I do BFS from i to every other node computing the distance from i to each one of them. Then for each these, for these distances, I take them one at a time, and I update the frequency of the distances. So if I see dist of j is five, then I add one to f of five. If I see for another node, dist of j is three, I add one to f of three, and I keep updating them. Now once we are done with the first node i, then we go to another node, and do the same thing, another node, we do the same thing. Once I'm done here, I can return f, right? So this function here, this algorithm is going to run BFS from every node in the graph, thus computing the pairwise distances between i and every other node. And looping over all possible nodes is going to do it for every pair of nodes. And I will update the, the distances in F, their frequencies, I always add one every time I see it and return F. Notice that I am indexing this F by this dist of J because dist of J is the distance, it's the value of a distance to node J, so I can use it as the index. This dist of g, what are the possible values of dist of g? It can be zero, one, two, three, all the way to the size of v minus 1. So I'm not going to be exceeding or doing anything wrong with indexing this array. Okay? So this is an algorithm that now after, after this sequence of attempts at computing distances, analyzing efficiency and so on. Now we finally have an algorithm that is going to make use of an efficient algorithm for computing distances, that will basically tabulate, or the distance distribution for us, okay? Now that we have done this, what is the running time, how long does it take? And again, now it is very simple because we have done this exercise before. For D equals 0 to V minus 1, do this. If I assume the graph is given by its adjacentcy list, it has N nodes, M edges and connected and so on. This is going to do on the order of N operations, right? For every node is going to, is going to update this value, so it's on the order of N operations. This is going to be on the order of N operations. This here, is order of n. This, how long does this take? Some might think it takes one operation because I'm just calling a function and returning a value, and this is a common mistake. No it doesn't do one operation, it takes as long as BFS takes. Okay? How long does BFS take on a graph that has n nodes and m edges? As we said, if I assume the number of edges is greater than the number of nodes, this takes O of m operations. And this here takes O of n. Okay? So, and this is of course, 1 and 1. So what does this add up to? We have O of n plus, for this one, then we have O of n times m plus n. All right. Now this is the dominant term so we can remove this. And as I said if m is greater than n, if m is greater than n, I can remove this n and basically we get an O of N times m times, okay. So we will get here. Again I can drop this n because if m is greater than n, I end up with n times m, and this will be the running time. So when again, when the graph is very dense, m is a node of n squared, in the worst case it's going to be cubic in the number of nodes, okay? Which is still very good algorithm for computing all pairwise distances in an undirected graph with no weights on its edges. So we have this and again for, if the graph is not very dense, it's going to be actually much better than cubic. Okay? So now, again we started with the problem of is the, is the world small for a given network? We for, we, we understood the problem, we formulated it computationally and then we started attempting to come up with algorithm. We came up with brute force algorithm, showed it's not the efficient, came up with BFS, show that it is efficient. Once we have shown that it is efficient, now I can make use of this function and to, to solve our original problem. And we get a very efficient solution for it.
