Grokking Graph Algorithms for Coding Interviews
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Introduction to Minimum Spanning Tree
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A Minimum Spanning Tree (MST) is a subset of the edges of a connected, weighted, and undirected graph. It connects all the vertices together without any cycles and with the minimum possible total edge weight. The MST includes all the vertices of the original graph but only a subset of the edges.

To understand this, imagine you have a network of cities connected by roads, each with a different length (or cost). The goal is to connect all cities with the shortest total road length without forming any loops. The MST provides this optimal solution.

Here, we have coverted the graph in MST. Now, we have the shortest path to reach at all vertices.

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Properties of a Spanning Tree

A Spanning Tree of a graph has the following properties:

  1. Connectivity: A Spanning Tree connects all the vertices in the graph.
  2. No Cycles: A Spanning Tree cannot contain cycles.
  3. Unique Path: There is exactly one path between any pair of vertices in a Spanning Tree.
  4. Vertices and Edges: If a graph has ( V ) vertices, each Spanning Tree will have exactly (V-1) edges.

Algorithms to Find Minimum Spanning Tree

There are several algorithms to find the Minimum Spanning Tree in a graph:

Kruskal's Algorithm

Kruskal’s algorithm builds the MST by adding edges in order of their weights, ensuring no cycles are formed.

Steps:

  1. Sort all edges in non-decreasing order of their weight.
  2. Pick the smallest edge. Check if it forms a cycle with the MST formed so far. If not, add it to the MST.
  3. Repeat step 2 until there are ( V-1 ) edges in the MST.

Prim's Algorithm

Prim’s algorithm starts with a single vertex and grows the MST one edge at a time, always choosing the smallest edge that expands the tree.

Steps:

  1. Start with any vertex as the initial MST.
  2. Select the smallest edge that connects a vertex in the MST to a vertex outside the MST.
  3. Add this edge to the MST.
  4. Repeat until all vertices are included in the MST.

Boruvka's Algorithm

Boruvka’s algorithm repeatedly adds the smallest edge from each component to another component until only one component remains.

Steps:

  1. Start with each vertex as a separate component.
  2. For each component, find the smallest edge connecting it to another component.
  3. Add these edges to the MST.
  4. Merge the components connected by these edges.
  5. Repeat until only one component remains.

Applications of Minimum Spanning Trees

Minimum Spanning Trees have various practical applications:

  1. Network Design: Used in designing efficient network systems like telecommunication, computer networks, and electrical grids.
  2. Approximation Algorithms: MSTs are used in algorithms to approximate solutions for problems like the Traveling Salesman Problem.
  3. Cluster Analysis: In data clustering, MSTs help in hierarchical clustering.
  4. Image Processing: MSTs assist in image segmentation and other image processing tasks.
  5. Civil Engineering: MSTs are useful in planning road networks, water supply systems, and other infrastructure projects.

This introduction sets the stage for deeper exploration into the algorithms and their implementation in the following lessons.

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