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The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at cow.MIT.edu. PROFESSOR: Today we're going to introduce graph search in general and talk about one algorithm, which is breadth-first search, and understand how in principle you can solve a puzzle like the Rubik's Cube. So before I get to Rubik's Cubes let me remind you of some basic stuff about graphs. Or I can tell you to start out with, graph search is about exploring a graph. And there are many notions of exploring a graph. Maybe I give you some node in a graph, s, and some other node in a graph, t, and I'd like to find a path that's going to represent a problem like I give you a particular state of a Rubik's Cube and I want to know is there some path that gets me into a solved state? Do I really want to solve this on stage? What the hell? We started. So this is a particularly easy state to solve, which is why I set up this way. All right, so there you go. Seven by seven Rubik's Cube solved in 10 seconds. Amazing. New world record. So you're given some initial state of the Rubik's Cube. You're given the targets that you know what solved looks like. You want to find this path. Maybe you want to find all paths from s. Maybe you just want to explore all the nodes in a graph you can reach from s. Maybe you want to explore all the nodes in a graph or maybe all the edges in a graph. These are all exploration problems. They're all going to be solved by algorithms from this class and next class. So before we go further though, I should remind you what a graph is and sort of basic features of graphs that we're going to be using. This is also 6042 material, so you should know it very well. If you don't, there's an appendix in the textbook about it. We have a set of vertices. We have a set of edges. Edges are either unordered pairs-- some sets of two items-- or ordered pairs. In this case, we call the graph undirected. In this case, we call the graph directed. Usually, there's only one type. Either all the edges are directed or all the edges are undirected. There is a study of graphs that have both, but we are not doing that here. Some simple examples. Here is a graph. This is an undirected graph. This is a directed graph. The set of vertices here is a, b, c, d. The set of vertices here is a, b, c. The set of edges here is-- E is going to be things like a, b; b, c; c, d-- I think you get the idea. Just for completeness, V is a, b, c, d. Just so you remember notations and so on. One of the issues we're going to talk about in this class is how do you represent a graph like this for an algorithm? So it's all fine to say, oh, this is a set of things. This is a set of things. An obvious representation is, you have a list or an array of vertices. You have an array of edges. Each edge...
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