#### Overview of this book

Building your own applications is exciting but challenging, especially when tackling complex problems tied to advanced data structures and algorithms. This endeavor demands profound knowledge of the programming language as well as data structures and algorithms – precisely what this book offers to C# developers. Starting with an introduction to algorithms, this book gradually immerses you in the world of arrays, lists, stacks, queues, dictionaries, and sets. Real-world examples, enriched with code snippets and illustrations, provide a practical understanding of these concepts. You’ll also learn how to sort arrays using various algorithms, setting a solid foundation for your programming expertise. As you progress through the book, you’ll venture into more complex data structures – trees and graphs – and discover algorithms for tasks such as determining the shortest path in a graph before advancing to see various algorithms in action, such as solving Sudoku. By the end of the book, you’ll have learned how to use the C# language to build algorithmic components that are not only easy to understand and debug but also seamlessly applicable in various applications, spanning web and mobile platforms.
Chapter 1: Data Types
Free Chapter
Chapter 2: Introduction to Algorithms
Chapter 3: Arrays and Sorting
Chapter 4: Variants of Lists
Chapter 5: Stacks and Queues
Chapter 6: Dictionaries and Sets
Chapter 7: Variants of Trees
Chapter 8: Exploring Graphs
Chapter 9: See in Action
Chapter 10: Conclusion
Index
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# Minimum spanning tree

While talking about graphs, it is beneficial to introduce the subject of a spanning tree. What is it? A spanning tree is a subset of edges that connects all nodes in a graph without cycles. Of course, it is possible to have many spanning trees within the same graph. For example, let’s take a look at the following diagram:

Figure 8.16 – Illustration of spanning trees within a graph

On the left-hand side is a spanning tree that consists of the following edges: (1, 2), (1, 3), (3, 4), (4, 5), (5, 6), (6, 7), and (5, 8). The total weight is equal to 40. On the right-hand side, another spanning tree is shown. Here, the following edges are chosen: (1, 2), (1, 3), (2, 4), (4, 8), (5, 8), (5, 6), and (6, 7). The total weight is equal to 31.

However, neither of the preceding spanning trees is the minimum spanning tree (MST) of this graph. What does it mean that a spanning tree is minimum? The answer is really simple: it is...