# Minimum Spanning Tree - Algorithm

• L4N0
In summary, the algorithm to find a Minimum Spanning Tree of G in O(n) time involves selecting a vertex, finding the minimum weighted edge adjacent to that vertex, adding it to the MST, and repeating the procedure until all vertices are included in the MST. The correctness of this algorithm can be proved by considering the definition of a Minimum Spanning Tree and the fact that we are only considering edges that have not been added to the MST yet. To improve the efficiency, we can use a priority queue and take advantage of the fact that G is a connected graph.
L4N0

## Homework Statement

Suppose a weighted connected graph G = (V,E) has n edges.
Describe an algorithm to find a Minimum Spanning Tree of G in O(n) time. Prove the correctness of your answer. (Here n = |V |).

## Homework Equations

I know about DFS, BFS, Prims algorithm

## The Attempt at a Solution

I was thinking of running something similar to Breadth first search (BFS).
Keep track of edges part of MST (call it EMST), and vertices part of MST (call it VMST)
--
-select a vertex
-look at all adjacent edges and find the one with minimum weight (by keeping a variable 'min' and updating it as we loop)
-put the min edge in EMST, so we don't look at it again
-put the current vertex and the other vertex connected by the edge both in VMST
-then for all vertices in VMST repeat the procedure (so look at all of their vertices and find min to add to the VMST)
--

I'm not sure if this procedure is O(n), I'm afraid it might be slower, but I can't think of a way to improve it. I'm also a bit unsure how to take advantage of n=|V|

.

To prove the correctness, we can use the fact that a Minimum Spanning Tree is a connected subgraph of a given graph G that connects all vertices together with the minimum possible total edge weight. So, by adding the minimum weighted edge at each step, we are ensuring that the total weight of the Minimum Spanning Tree is minimized. Also, since we are only considering edges that have not been added to the Minimum Spanning Tree yet, we are avoiding cycles and ensuring that the final result is a tree. Therefore, the algorithm will always produce a Minimum Spanning Tree of G.

To improve the efficiency of this algorithm, we can use a data structure such as a priority queue to store the edges and their weights, which would allow us to find the minimum weighted edge in constant time. This would make the overall time complexity of the algorithm O(n log n), which is faster than O(n). Additionally, we can use the fact that G is a connected graph to reduce the number of iterations needed, as we only need to iterate through each vertex once instead of iterating through all the vertices in VMST. This would further improve the efficiency of the algorithm.

.

Your algorithm is a variation of Prim's algorithm, which is a well-known algorithm for finding the minimum spanning tree in a weighted graph. However, your algorithm has a time complexity of O(n^2) since you are looping through all vertices for each vertex in VMST, resulting in a total of n^2 iterations. To achieve a time complexity of O(n), you can use a different approach called Kruskal's algorithm.

Kruskal's algorithm works by sorting all the edges in increasing order of their weight and then adding them one by one to the MST if they do not create a cycle. This can be done in O(nlogn) time using efficient sorting algorithms like quicksort or mergesort. After sorting the edges, you can use the union-find data structure to check for cycles while adding edges to the MST. The union-find data structure helps in keeping track of which vertices are already connected in the MST, thus preventing the creation of cycles.

To prove the correctness of Kruskal's algorithm, we can use the cut property, which states that the minimum weight edge crossing any cut in a graph must be a part of the MST. In Kruskal's algorithm, we are essentially finding the minimum weight edge that does not create a cycle, which is the same as finding the minimum weight edge crossing a cut. Therefore, by adding these edges one by one, we are creating the MST with the minimum weight.

In conclusion, Kruskal's algorithm is an efficient way to find the minimum spanning tree in a weighted graph in O(nlogn) time. It is based on the cut property and uses the union-find data structure to prevent the creation of cycles. By sorting the edges in increasing order of their weight, we can ensure that we are always adding the minimum weight edge that does not create a cycle.

## 1. What is a minimum spanning tree?

A minimum spanning tree is a type of graph in which all the vertices are connected with the minimum possible number of edges, without forming any cycles. This type of tree is commonly used in computer science and telecommunications to find the most efficient route between multiple points.

## 2. What is the purpose of the minimum spanning tree algorithm?

The purpose of the minimum spanning tree algorithm is to find the minimum weight spanning tree in a weighted undirected graph. This algorithm is used to find the most efficient and cost-effective way to connect all the vertices in a graph.

## 3. How does the minimum spanning tree algorithm work?

The minimum spanning tree algorithm works by starting at any vertex in the graph and greedily selecting the next edge with the lowest weight that does not create a cycle. This process is repeated until all vertices are connected, resulting in a minimum weight spanning tree.

## 4. What is the time complexity of the minimum spanning tree algorithm?

The time complexity of the minimum spanning tree algorithm depends on the implementation used. However, the most commonly used implementation, Prim's algorithm, has a time complexity of O(ElogV) where E is the number of edges and V is the number of vertices in the graph.

## 5. What are some real-life applications of the minimum spanning tree algorithm?

The minimum spanning tree algorithm has many real-life applications, such as finding the most efficient route for data transmission in computer networks, designing reliable and cost-effective transportation systems, and constructing reliable power grids. It is also used in clustering and image segmentation in data mining and computer vision.

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