Discrete Math II - 11.4.1 Spanning Trees - Depth-First Search

Summary

In this engaging video by Kimberly Brehm, viewers are introduced to the concept of spanning trees and the depth-first search algorithm. A spanning tree is explained as a tree derived from a graph that includes all vertices without forming any cycles. The video illustrates how to form a spanning tree by removing circuits and suggests using a depth-first search method, which systematically explores all vertices starting from a single node until all are visited. The tutorial demonstrates this approach using examples and emphasizes that there can be multiple correct spanning trees for a given graph.

Highlights

• Start with understanding what a spanning tree is – it's a subgraph of 'G' that is a tree containing every vertex without forming cycles. 🌲
• The depth-first search algorithm builds spanning trees by connecting the current vertex to an adjacent one and backtracking if needed. ↔️
• Discover how a stack can be utilized to efficiently manage and conduct a depth-first search for building a spanning tree. 📋
• Practice manually forming spanning trees through examples and see how stack organization changes the search dynamics. 💡
• Multiple spanning trees can result from different vertex visiting orders, providing flexibility and creativity in solving graph problems. 🎨

Key Takeaways

• Understand the basics of spanning trees – subgraphs that include all vertices without cycles. 🌳
• Learn how to remove circuits to create a spanning tree quickly. ✂️
• Explore the depth-first search algorithm as a handy tool for forming spanning trees. 🕵️‍♀️
• Depth-first search involves backtracking when all possible paths are explored. 🔄
• Using a stack helps organize the search process for spanning trees. 📚

Overview

The video begins by exploring the concept of a spanning tree, defining it as a tree formed from any graph which includes all the vertices without forming cycles. Spanning trees are significant in graph theory as they simplify complex networks while maintaining their basic structure. In this context, Brehm clarifies that a spanning tree is essentially a subgraph that connects all vertices with a minimum number of edges, omitting any loops or circuits.

Brehm illustrates the depth-first search algorithm's role in constructing these spanning trees, which involves systematically visiting each vertex starting from a root node. The process involves selecting an edge, moving to the next vertex, and using backtracking when reaching a dead end. This algorithm can efficiently traverse a graph and can be implemented using a stack to track the vertices visited and those still pending exploration.

As the session progresses, viewers are encouraged to try forming spanning trees themselves to better grasp the concepts. Through interactive exercises, they can explore different strategies and understand that there may be numerous correct spanning trees for the same graph, depending on which vertices are chosen first. This hands-on practice solidifies the understanding that while the structure of spanning trees is systematic, it demands strategic decision-making and flexibility.

Chapters

• 00:00 - 00:30: Introduction to Spanning Trees In the initial chapter titled 'Introduction to Spanning Trees', the concept of spanning trees is unfolded. It starts with the basic definition, where a spanning tree is identified as a subgraph of a graph (denoted as 'g') that is itself a tree. Notably, as a tree, it does not contain any cycles and it must include every vertex from the original graph 'g'. The chapter sets the stage for further explorations into algorithms by establishing a thorough understanding of what constitutes a spanning tree.
• 00:30 - 01:00: Example of a Spanning Tree The chapter discusses an example of a spanning tree in graph theory. It describes the process of creating a spanning tree using a set of connections: A is connected to E, D, and B, while A is not directly connected to C. However, it is possible to connect B to C, forming the spanning tree. The chapter briefly mentions a method for creating a spanning tree by identifying circuits, typically where there are more direct paths than necessary.
• 01:00 - 02:00: Concept of a Spanning Tree A spanning tree is formed by starting with a node and adding edges from it while eliminating any circuits (loops). To do this, you identify circuits in a graph and remove extra edges, ensuring there's only one path between any two vertices. This process effectively removes cycles to create a tree structure over a graph, where all nodes are connected without forming a loop.
• 02:00 - 02:30: Depth-First Search Algorithm Steps The chapter explains the steps involved in implementing the Depth-First Search (DFS) algorithm. It highlights the significance of conducting this process algorithmically, allowing computers to execute it instead of manual execution. The process begins with selecting a root vertex to form a rooted graph, although the graph is fundamentally undirected. It sets the groundwork for creating subsequent edges as part of the DFS implementation.
• 02:30 - 03:30: Backtracking in Depth-First Search The chapter discusses the concept of backtracking in the context of depth-first search (DFS) algorithms in graph theory. It emphasizes that the order of examining the vertices doesn't necessarily matter, even if the edges are labeled in an ordered fashion like alphabetical order. For example, when faced with a choice of connecting vertex 'A' with connected vertices such as 'E,' 'D,' or 'B,' the decision process in DFS might opt for 'B' over 'E' based on the specific method or rules being applied.
• 04:00 - 04:30: Using a Stack for Depth-First Search The chapter introduces the concept of using a stack for implementing Depth-First Search (DFS) in graph traversal. It starts with the example of making a choice from a set of edges, selecting 'b' as the starting node. The key strategy is not to return to the starting node 'a' but to move forward from 'b', thus demonstrating the DFS principle. From 'b', the traversal proceeds to 'c'. The algorithm points out that when 'c' has no further connections, it signifies the end of this path exploration, adhering to the DFS technique.
• 04:30 - 07:00: Practical Example Using Depth-First Search The chapter provides a practical example of using the depth-first search (DFS) algorithm. It explains the process of backtracking when you have visited all possible vertices from a current node and need to revisit a previous one to explore unvisited vertices. In the example, starting from vertex C, moving to vertex B, then connecting B to D, and D to A, illustrates how DFS traverses and backtracks. When vertex D leads back to a previously visited vertex A, the algorithm backtracks to B to try a different connection, ultimately connecting B to E as it continues exploring.
• 08:00 - 09:00: Closing Remarks and Introduction to Breadth-First Search This chapter discusses the concept of a spanning tree, using an example that begins with a point 'a' extending to 'b', and branches further into 'c', 'd', and 'e'. It notes that this particular spanning tree was constructed through the depth-first search algorithm.

Discrete Math II - 11.4.1 Spanning Trees - Depth-First Search Transcription

• 00:00 - 00:30 for this first video of section 11.4 we're going to talk about spanning trees exactly what is a spanning tree and then talk about a depth first search algorithm of course first it's important to understand what is a spanning tree so a spanning tree is a tree created from a graph so it's actually a subgraph of g that is a tree which means we don't have any cycles and it contains every vertex of g so for instance i could choose
• 00:30 - 01:00 to connect a to e and connect a to d and connect a to b and a doesn't connect directly to [Music] c but i could connect b to c so this would be an example of a spanning tree one method for creating a spanning tree is to simply identify anywhere that you have a circuit where there's more than
• 01:00 - 01:30 one way to get to one vertex and simply remove those edges and so essentially that's what i've done here is i've created a spanning tree by getting rid of any circuits so instead of simply identifying where the circuits are and removing any extra edges we can actually do this by starting with a node and then creating edges from there that are obviously already in the
• 01:30 - 02:00 original graph and the important thing here is that we can do this in an algorithmic fashion which means instead of having to do this by hand we can let a computer do it for us but let's look at how the algorithm would work essentially we're going to start with a root vertex so we're going to create a rooted graph but really we're looking at this as to be undirected so i'm just going to start with a the next step says create an edge by
• 02:00 - 02:30 connecting the current vertex to an incident vertex so it really doesn't matter if you have edges that are labeled in an ordered fashion so for instance we have the alphabet and so if we wanted to connect these in alphabetical order obviously i would choose b over e so i would examine from a that i could connect either e or d or b because those are all the vertices that are connected to a
• 02:30 - 03:00 and i can choose any one that i want so for instance let's say i choose b so i'm going to choose b and connect that edge now the trick here for the depth i'm sorry the depth first search is that i don't go back to a i start at b and say where can i go from b well b would take me to c so then i look at c and say well c is not connected to anything else so then the algorithm says
• 03:00 - 03:30 if you've gone as far as you can and you still haven't visited all the vertices go back to the previous vertex so i'm going to go back from c to b so that's why it's backtracking so i'm back to b and now i'm going to connect anything from b so i'm going to connect b to d and then d connects to a but i've already visited a so i can't connect anything to d so i would go back to b again and say well that connects to e
• 03:30 - 04:00 so one example of a spanning tree is starting at a going to b and if you'll notice b branches off into c and d and e so this is one example of a spanning tree that was created using our depth first search algorithm
• 04:00 - 04:30 what you'll see quite often and this is not the way that it's shown in your textbook but quite often you'll see using a stack to keep yourself organized and so here's how that would work we would start with our root node so we're going to start with a and i'm going to go ahead and put visited over here so if i start with a i have visited a and in our stack we're going to add any
• 04:30 - 05:00 vertices that are next to or adjacent to a so in this case that would be b and c and we can choose either b or c so let's say i choose b so now b is going to leave the stack and i will say that i have visited b now c is still in the stack meaning it's still something i can get to but remember our algorithm says now you're going to go from b
• 05:00 - 05:30 so instead of leaving c at the top of the stack c is going to move down oops and we're going to put any vertices adjacent to b which would be d and so then i'm going to visit d and then i'm going to write anything adjacent to d so i'm going to replace d with e so now e is visited
• 05:30 - 06:00 and then from e i can get to either f or g well c is still in our stack but it keeps getting moved to the bottom again because because it is adjacent to a so we're going to keep c in the stack but again we're going to keep going from e we're going to go as far as we can with e so i'm going to visit f and then i'm going to write anything
• 06:00 - 06:30 adjacent to f so i'm going to have g or c so now notice g and c were already in our stack so now i'm going to visit g again from f and then from g over to c so now i have visited all of the vertices what does my spanning tree look like well it's just a path a b d e
• 06:30 - 07:00 f g c so that is what my spanning tree would look like in this case starting from a and making the decisions that i did here's a practice for you to try on your own so go ahead and press pause try this question and then press play to see how you did again it's important to remember that there's no right answer there's no one right answer as long as you're able to visit each of the vertices you have a spanning tree
• 07:00 - 07:30 so i'm going to start at a and instead of creating a stack this time i'm just going to circle potential vertices as we go so from a the only potential vertices c and so we're going to visit c from there i can visit either b or e and just for fun this time i'm going to visit e [Music] so now e has been visited b is still in the stack from e i can visit d or f i'm going to
• 07:30 - 08:00 choose d and from d i can only choose f so f again leaves the stack and is now visited from f i can visit g or h i'm going to choose g oops and then from g i can only choose h so now i have visited h and then from h i can only visit oops i should have been creating the
• 08:00 - 08:30 actual tree as i go from h i can only visit i and now i have to backtrack all the way back to c to get this very last vertex of b oops b so this is one possible spanning tree that you could have come up with creating our depth first search up next we're going to take a look at another method for creating a spanning
• 08:30 - 09:00 tree using a breadth first search