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‎Graphs Algorithims/Minimum Spanning Tree Algorthims/Prim's Algorithm/.ipynb_checkpoints/Prim's Algorithm-checkpoint.ipynb

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## Prim's Algorithm
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### Introduction
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Prim's Algorithm is a popular greedy algorithm used for finding the minimum spanning tree (MST) of a connected, undirected graph. The goal of the algorithm is to connect all the vertices of the graph with the minimum possible total edge weight. This makes it particularly useful in network design, such as laying out an efficient and cost-effective network of roads, pipelines, or communication channels.
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## Prim's Algorithm Pseudocode
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Prim's Algorithm(graph):
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Initialize an empty set MST to store the minimum spanning tree
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Choose a starting vertex, add it to MST
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Initialize a priority queue (min heap) with edges and their weights from the starting vertex
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while MST does not include all vertices:
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Select the edge with the minimum weight from the priority queue
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If adding the selected edge does not create a cycle in MST:
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Add the edge to MST
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Add the destination vertex of the edge to MST
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Enqueue the edges connected to the destination vertex with their weights
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return MST
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### Purpose
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The main purpose of Prim's Algorithm is to find a subset of edges that forms a tree including every vertex, where the total weight of all the edges in the tree is minimized. This helps in optimizing various network-related problems, ensuring that the connected infrastructure is both cost-effective and efficient.
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### Applications
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Prim's Algorithm finds applications in various domains:
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- **Network Design**: Designing cost-effective and efficient networks for communication or transportation.
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- **Cluster Analysis**: Identifying clusters or groups of related entities in data.
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- **Maze Generation**: Creating mazes with corridors that are as short as possible.
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- **Circuit Design**: Optimizing the layout of electronic circuits.
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### Time Complexity
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The time complexity of Prim's Algorithm is O(E log V), where E is the number of edges and V is the number of vertices in the graph. This is mainly due to the priority queue operations.
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### Space Complexity
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The space complexity of Prim's Algorithm is O(V + E), where V is the number of vertices and E is the number of edges in the graph. This accounts for the space used to store the minimum spanning tree (MST) and the priority queue.
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‎Graphs Algorithims/Minimum Spanning Tree Algorthims/Prim's Algorithm/Prim's Algorithm.ipynb

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## Prim's Algorithm
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### Introduction
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Prim's Algorithm is a popular greedy algorithm used for finding the minimum spanning tree (MST) of a connected, undirected graph. The goal of the algorithm is to connect all the vertices of the graph with the minimum possible total edge weight. This makes it particularly useful in network design, such as laying out an efficient and cost-effective network of roads, pipelines, or communication channels.
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## Prim's Algorithm Pseudocode
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Prim's Algorithm(graph):
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Initialize an empty set MST to store the minimum spanning tree
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Choose a starting vertex, add it to MST
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Initialize a priority queue (min heap) with edges and their weights from the starting vertex
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while MST does not include all vertices:
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Select the edge with the minimum weight from the priority queue
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If adding the selected edge does not create a cycle in MST:
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Add the edge to MST
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Add the destination vertex of the edge to MST
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Enqueue the edges connected to the destination vertex with their weights
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return MST
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### Purpose
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The main purpose of Prim's Algorithm is to find a subset of edges that forms a tree including every vertex, where the total weight of all the edges in the tree is minimized. This helps in optimizing various network-related problems, ensuring that the connected infrastructure is both cost-effective and efficient.
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### Applications
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Prim's Algorithm finds applications in various domains:
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- **Network Design**: Designing cost-effective and efficient networks for communication or transportation.
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- **Cluster Analysis**: Identifying clusters or groups of related entities in data.
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- **Maze Generation**: Creating mazes with corridors that are as short as possible.
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- **Circuit Design**: Optimizing the layout of electronic circuits.
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### Time Complexity
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The time complexity of Prim's Algorithm is O(E log V), where E is the number of edges and V is the number of vertices in the graph. This is mainly due to the priority queue operations.
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### Space Complexity
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The space complexity of Prim's Algorithm is O(V + E), where V is the number of vertices and E is the number of edges in the graph. This accounts for the space used to store the minimum spanning tree (MST) and the priority queue.
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