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Questions and Answers
What is the primary purpose of Prim's algorithm?
What is the primary purpose of Prim's algorithm?
What is the time complexity of Prim's algorithm using a binary heap?
What is the time complexity of Prim's algorithm using a binary heap?
In Prim's algorithm, how do you select the next edge to add to the MST?
In Prim's algorithm, how do you select the next edge to add to the MST?
What is the space complexity of Prim's algorithm?
What is the space complexity of Prim's algorithm?
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What is the result of applying Prim's algorithm to the example graph?
What is the result of applying Prim's algorithm to the example graph?
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What is the first step in Prim's algorithm?
What is the first step in Prim's algorithm?
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Study Notes
Prim's Algorithm
Overview
- A greedy algorithm used to find the minimum spanning tree (MST) of a connected, undirected, and weighted graph.
- It is an efficient solution for finding the MST in a graph with non-negative edge weights.
How it Works
- Choose a starting node: Select an arbitrary node from the graph as the starting point.
- Initialize the MST: Create an empty minimum spanning tree and add the starting node to it.
- Grow the MST: In each iteration, select the edge with the minimum weight that connects a node in the MST to a node not in the MST.
- Add the edge and node: Add the selected edge and node to the MST.
- Repeat until complete: Continue growing the MST until all nodes in the graph are included.
Key Properties
- Time Complexity: O(|E|log|V|) using a binary heap, where |E| is the number of edges and |V| is the number of vertices.
- Space Complexity: O(|V| + |E|) to store the graph and the MST.
Example
Suppose we have a graph with nodes A, B, C, and D, and edges with weights as follows:
A --2--> B
| / | |
| 3 / | 4 |
|/ | |
C --5--> D
Applying Prim's algorithm, we can find the MST as follows:
- Start with node A.
- Add edge AB (weight 2) and node B to the MST.
- Add edge BC (weight 3) and node C to the MST.
- Add edge CD (weight 5) and node D to the MST.
The resulting MST is:
A --2--> B --3--> C --5--> D
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Description
Learn about Prim's algorithm, a greedy approach to find the minimum spanning tree in a connected, undirected, and weighted graph. Understand its time and space complexity and see an example of how to apply it.
