program kruskal_example implicit none integer, parameter:: pr = selected_real_kind(15,3) integer, parameter:: n = 7! Number of Vertice. Prim’s and Kruskal’s algorithms are two notable algorithms which can be used to find the minimum subset of edges in a weighted undirected graph connecting. Kruskal’s Algorithm (Python). GitHub Gist: instantly share code, notes, and snippets.

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A telecommunication company wants to connect all the blocks in a new neighborhood.

However, the easiest jruskal to install new cables is to bury them along roads. So the company decides to use hubs which are placed at algoritbme junctions. How can the cost be minimized if the price for connecting two hubs corresponds to the length of the cable ed them?

Considering the roads as a graph, the above example is an instance of the Minimum Spanning Tree problem. Prim’s and Kruskal’s algorithms are two notable algorithms which can be used to find the minimum subset of edges in a weighted undirected graph connecting all nodes.

You can open another browser window to read the description in parallel. Given a connected, undirected graph, a spanning tree of that graph is an acyclic subgraph that connects all vertices.

Obviously, a graph may have many spanning trees. Weights may be assigned to each edge of the graph, then the total weight of a subgraph is the sum of its edge weights.

The spanning tree with the weight less than or equal to all other spanning trees is called the minimum spanning tree MST. More generally, any undirected graph has a minimum spanning forest MSFwhich is a union of minimum trees for its connected components.

Kruskal’s algorithm is a greedy algorithm a problem solving heuristic of making the locally optimal choice at each stage with the hope of finding a global optimum that efficiently finds the minimum spanning tree for any connected weighted undirected graph.

In the case that the graph is not connected this algorithm will calculate minimum spanning forest without need for any further modifications. The algorithm initializes a forest consisting of trees with single nodes.

The edges are sorted kguskal a queue based on their weights. In each round an edge is removed from the queue.

If the edge’s endpoints belong to different trees then the trees are merged using the edge. The process continues until all the nodes are in the same tree or the edge-queue is empty. The most efficient way to implement the algorithm is using a Union-Find data structure which provides efficient set operations such as the union of two sets and checking the membership of an element.

When an edge is removed from the queue this data structure makes it possible to easily check if two endpoints of the edge belong to the same tree. If this is the case, the trees, which are presented as sets, can be easily merged. The algorithm may be described step-by-step. First is the initialization part:. Next, the algorithm iterates over the queue. In each round, the edge in front of the queue is extracted.

Then the two endpoints are compared. If their IDs are not equal they belong to different treesthen the trees which contain these endpoint are merged. A Union-Find data structure merges two trees by assigning a new ID to all vertices contained in both trees. In this case the edge is also added to the resulting tree. Otherwise, the endpoints are in the same tree, so adding the edge would result in a circle in the tree.

The above iteration continues until no more edges are included in the queue, or all vertices are contained in the same tree their IDs are equal. When two endpoints of an edge are checked during the algorithm, this edge is added to the result if it is connecting two different trees. The resulting minimum spanning forest may be represented as the union of all such edges. The Speed of an algorithm is the total number of individual steps which are performed during the execution.

These steps are for example:. Since it can be very difficult to count all individual steps, it is desirable to only count the approximate magnitude of the number of steps. This is also called the running time of an algorithm.

Particularly, it is intersting to know the running time of an algorithm based on the size of the input in this case the number of the vertices and the edges of the graph. Even a simple disjoint-set data structure can perform operations proportional to log size. In each iteration, two find-set operation and possibly one union operation are performed. Provided that edges are already sorted or can be sorted in linear time i.

The proof consists of two parts. First, it is proved that the algorithm produces a spanning tree. Second, it is proved that the constructed spanning tree is of minimal weight. T cannot have a cycle because the algorithm only chooses those edges which are connecting two different trees. T cannot be disconnected, since the first encountered edge that joins two components of T would have been added by the algorithm. Thus, T is a spanning tree of G. We show that the following proposition P is true by induction: If E1 is the set of edges chosen at any stage of the algorithm, then there is some minimum spanning tree that contains E1.

Furthermore there is an interesting book about shortest paths: The algorithms presented on the pages at hand are very basic examples for methods of discrete mathematics the daily research conducted at the chair reaches far beyond that point.

## Kruskal’s algorithm

These pages shall provide pupils and students with the possibility to better understand and fully comprehend the algorithms, which are often of importance in daily life. Therefore, the presentation concentrates on the algorithms’ ideas, and often explains them with just minimal or no mathematical notation at all.

Please be advised that the pages presented here have been created within the scope of student theses, supervised by Chair M9.

The code and corresponding presentation could only be tested selectively, which is why we cannot guarantee the complete correctness of the pages and the implemented algorithms. Naturally, we are looking forward to your feedback concerning the page as well as possible inaccuracies or errors. Please use the suggestions link also found in the footer.

What do you want to do first? Read Detailed Description of the Kruskwl. Legend Node Edge with weight Which graph do you want to execute the algorithm on? Start with an example graph: To create a node, make a double-click in the drawing area. To create an edge, first click on the output node and then click on the destination node. The edge algoriyhme can be changed by double clicking on the edge.

Right-clicking deletes edges and nodes. A occured when reading from file: Algorithm status rewind prev next fast forward pause. Explanation Pseudocode Algorithm status will appear here. V make-tree v.

## The Minimum Spanning Tree Algorithm

If you switch tabs, the execution will be terminated. What is the cheapest way to connect all vertices? An edge may be connecting two vertices in the algkrithme tree or in different trees. If the later is the case, then the trees are merged. Create a graph and play through the algorithm Try algorithm after creating krusmal graph Try algorithm on an example graph. Test your knowledge in the exercises Exercise: How does the algorithm decide?

### Kruskal Algorithm – File Exchange – MATLAB Central

What is the pseudocode of the algorithm? Speed of the algorithms The Speed of an algorithm is the total number of individual steps which are performed during the execution. These re are for example: Assignments — Set distance of a node to Comparisons — Is 20 greater than 23? Minimality We show that the following proposition P is true by induction: Clearly P is true at the beginning, when E1 is empty: Now assume P is true for some non-final edge set E1 and let T1 be a minimum spanning tree that contains E1.

If the next chosen edge e is also in T1, then P is true for the union of E1 and e. If there were no such edge f, then e could not have been added to E1, since doing so would have created the cycle C. Therefore, by the principle of induction, P holds when Fe has become a spanning tree, which is only possible if E1 alvorithme a minimum spanning tree itself.

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