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How do you use Prim's algorithm for finding minimum spanning trees?

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Finding the minimum spanning tree (MST) of a weighted graph is a common problem in algorithms and data structures. A MST is a subset of edges that connects all the vertices of the graph without forming any cycles and with the minimum possible total weight. One way to find a MST is to use Prim's algorithm, which is a greedy method that builds the MST edge by edge. In this article, you will learn how to use Prim's algorithm for finding MSTs and what are its advantages and disadvantages.

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Harshendra Subba Reddy Nallamilli

Knight?? @LeetCode (Top 2.17% & Max. Rating 2044) & Pupil?? @CodeForces (Max. Rating 1345) & 3? @CodeChef (Max. Rating…
Federico Felizzi

Global HEOR Franchise Lead - Oncology
Neha Pathak

Software Intern | Ex Senior Software Analyst at Accenture | SQL, Power BI Certified| Automation | Machine Learning|…

1 How Prim's algorithm works

Prim's algorithm starts with an arbitrary vertex and adds the edge with the smallest weight that connects it to another vertex that is not already in the MST. Then, it repeats this process until all the vertices are included in the MST. At each step, the algorithm maintains a set of vertices that are part of the MST and a priority queue of edges that are candidates to be added to the MST. The priority queue is sorted by the edge weight, so the algorithm always picks the edge with the lowest weight from the queue.

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Harshendra Subba Reddy Nallamilli

Knight?? @LeetCode (Top 2.17% & Max. Rating 2044) & Pupil?? @CodeForces (Max. Rating 1345) & 3? @CodeChef (Max. Rating 1719) & 4? @GeeksforGeeks (Max. Rating 1990)
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Prim's algorithm is used to find the Minimum Spanning Tree (MST) of a given weighted graph. MST means that connecting all the nodes of the given graph with the minimum weights of each connection. Simply, it works like this. Take any node as a root node. From the root node, check what are the adjacent nodes of that root node. Look for the one with minimum weight. Connect that node to the root node. Now, you have 2 nodes in your MST. Later, look for the minimum weighted adjacent node from both the nodes in your MST. Connect the minimum weighted node to your MST and so on until you have all the nodes covered in your MST. Finally the tree obtained is the Minimum Spanning Tree.

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Neha Pathak

Software Intern | Ex Senior Software Analyst at Accenture | SQL, Power BI Certified| Automation | Machine Learning| Software Testing | NLP |Purdue University
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In simple words to understand, this algorithm aims in getting an entire path covering all vertices of graph from one selected vertex to GET MINIMUM WEIGHT SUM ON ADDING WEIGHTS OF ALL SUB PATHS TRAVERSED. Everytime select next vertex by checking which sub path has smallest weight value ( cost of traveling that path from current vertex to destination vertex). And again recheck if total sum of weights is least.

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2 How to implement Prim's algorithm

Implementing Prim's algorithm requires a data structure to store the graph and its weights, such as an adjacency matrix or an adjacency list. Additionally, a priority queue data structure is needed, such as a binary heap or a Fibonacci heap. The algorithm can be summarized as follows: initially, initialize an empty minimum spanning tree (MST) and priority queue. Then, select an arbitrary vertex and mark it as visited. Add all the edges from the visited vertex to the priority queue. While the priority queue is not empty and the MST has less than n-1 edges, where n is the number of vertices in the graph, extract the edge with the lowest weight from the priority queue. If it connects a visited vertex and an unvisited one, add it to the MST and mark the unvisited vertex as visited. Add all edges from the newly visited vertex to the priority queue. Finally, return the MST. For example, in Python using an adjacency matrix and a binary heap, this algorithm can be implemented as shown in code.

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Bhavana P

Detail-Oriented Data Analyst | 5 ? SQL Hackerrank| 5 ? Python Hackerrank | Statistical Insights
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Prim's Algorithm(G): 1. Choose a starting vertex s. 2. Mark s as visited. 3. Initialize an empty priority queue (min-heap) Q. 4. Add all edges incident to s to Q. 5. While Q is not empty: a. Extract the minimum-weight edge (u, v) from Q. b. If v is not visited: i. Mark v as visited. ii. Add (u, v) to the MST. iii. Add all edges incident to v that connect to unvisited vertices to Q. 6. Return the MST.

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3 What are the advantages of Prim's algorithm

Prim's algorithm has several advantages when compared to other methods for finding MSTs, such as Kruskal's algorithm or Boruvka's algorithm. Prim's algorithm is straightforward to implement and understand, particularly if you use a simple data structure like an adjacency matrix and a binary heap. Additionally, it works optimally for dense graphs, where the number of edges is close to the maximum possible, since it only considers the edges that are adjacent to the vertices in the MST. Furthermore, Prim's algorithm can be easily altered to find the second-best MST, the MST of a subgraph, or the MST of a graph with negative weights.

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4 What are the disadvantages of Prim's algorithm

When using Prim's algorithm, it is important to be aware of its drawbacks. Prim's algorithm is not very efficient for sparse graphs, as it still needs to check all the edges in the priority queue. Additionally, it requires extra space to store the visited vertices and the priority queue, which can be significant for large graphs. Furthermore, Prim's algorithm is not suitable for parallel or distributed computing, since it depends on a single source vertex and a global priority queue.

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5 How to prepare for Prim's algorithm questions

If you're preparing for a data structures and algorithm interview, it's important to become familiar with Prim's algorithm. To do this, review the basic concepts of graphs, such as vertices, edges, weights, cycles, and connectivity. Additionally, practice coding Prim's algorithm in your preferred programming language with different data structures and test cases. It's also important to compare and contrast Prim's algorithm with other MST algorithms, such as Kruskal's or Boruvka's algorithms, and understand their trade-offs and use cases. Moreover, learn how to modify Prim's algorithm to solve variations and extensions of the MST problem. Lastly, think of some real-world scenarios where finding a MST can be useful.

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6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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Federico Felizzi

Global HEOR Franchise Lead - Oncology
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It is worth noting that Minimum Spanning Trees (MSTs) find applications in a wide range of areas. For instance, in the context of dependency parsing in NLP, there is a keen interest in identifying the Spanning Tree with the highest weight. MSTs have versatile applications that extend to logistics, circuit design, and have even proven valuable in predicting outbreaks during the COVID-19 pandemic. Additionally, they can be effectively employed in the allocation of resources, such as ICU units and medical equipment, to maximize public health outcomes.

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Jocelyn THIEBAUT

PhD in Theoretical Computer Science | Web Developer Passionate
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Prim algorithm (as well as any algorithm computing MST of a graph) can be used to generate mazes. The idea is the following: consider a grid G of size n_1 x n_2, and put random weight on the edges. If you compute a MST T of G, you can represent your maze as follows: the cells are the vertices of G, and you have a wall between the cells u and v if the edge (u,v) is not an edge of T, otherwise, there is a corridor between these two cells. Note that depending of the algorithm you use, the "shape" of the maze might be different.

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How do you use Prim's algorithm for finding minimum spanning trees?

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1 How Prim's algorithm works

2 How to implement Prim's algorithm

3 What are the advantages of Prim's algorithm

4 What are the disadvantages of Prim's algorithm

5 How to prepare for Prim's algorithm questions

6 Here’s what else to consider

Algorithms

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How do you use Prim's algorithm for finding minimum spanning trees?

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1 How Prim's algorithm works

2 How to implement Prim's algorithm

3 What are the advantages of Prim's algorithm

4 What are the disadvantages of Prim's algorithm

5 How to prepare for Prim's algorithm questions

6 Here’s what else to consider

Algorithms

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