camerini algorithm to find minimum bottleneck spanning tree, proof - algorithm

I'm looking for a proof for camerini algorithm for finding MBST in undirected graph .
I couldn't find anywhere and I couldn't do it myself.
Thanks.
problem explanation

Related

Finding the minimum spanning tree of a graph using Kruskal's Algorithm

Here is a Graph where I need to find the minimum spanning tree of G using Prim's and Kruskal's algorithms.
I found the minimum spanning tree using Prim's algorithm. Here is my attempt.
I am having difficulty in finding the minimum spanning tree using Kruskal's algorithm. I have seen many videos related to Kruskal's graph algorithm but I ended up getting the same graph as Prim's algorithm.
Can anyone please show me how to find the minimum spanning tree of the graph using Kruskal's algorithm?
Prims and Kruskals will always give you the same answer if all the
edges of the graph have distinct weights, as there is only a single min-spanning tree that exists. For graph having many edges with
same weights, the algorithms could give you a different answer but not
always. Depends on the way the nodes are explored in the
implementation. This graph can have many different min-spanning trees.
As your graph has all distinct edge weights, you will always get the same answer.
Prim's and Kruskal's algorithm both find minimum spanning trees. Because the graph that you gave has edge weights that are all different from each other there will be no different spanning trees that are both minimum.as long as both algorithms are correctly implemented they will find the same tree. meaning there can be no variation between the MST.

Finding shortest path in non-weighted graphs

During a course in University concerning graph theory, we were talking about finding shortest paths thus Dijkstra's algorithm came up, at that point I should mention that the edges of the graph were weighted, with weights>0. Then the professor asked how we could find the shortest path if the edges weren't weighted, I thought the same algorithm would do, since the edges had the "same" non-negative weight. But he suggested BFS. Is this true? wouldn't Dijkstra work correct? I'm not questing BFS finding the path but since it is exhaustive I thought maybe it would be better to avoid it.
Dijsktra worked fine for me even with non-wighted graphs. Every connection just has weight 1.

Cheriton-Tarjan algorithm for MST

I am searching the Cheriton-Tarjan algorithm for weighted minimum spanning trees, with O(m*loglogn). But I was not able to find it anywhere. Can someone explain me the algorithm or tell me a link as where to find it?
It's "Tarjan" not "Trajan". That may be a reason why you've had trouble finding it.
Here's pseudocode for the algorithm, taken from here (Graphcs, Algorithms, and Optimization by William Kocay and Donald Kreher):

Sollin's algorithm in graph

What's is the Sollin's algorithm in graph theory?
Can you write pseudocode of this algorithm or explain how this algorithm work?
Sollin's algorithm is an algorithm for finding the Minimum Spanning Tree (MST) on a graph. Find more information here.

Algorithm to connect all dots with the minimum total distance

I have a set of points and a distance function applicable to each pair of points. I would like to connect ALL the points together, with the minimum total distance. Do you know about an existing algorithm I could use for that ?
Each point can be linked to several points, so this is not the usual "salesman itinerary" problem :)
Thanks !
What you want is a Minimum spanning tree.
The two most common algorithms to generate one are:
Prim's algorithm
Kruskal's algorithm
As others have said, the minimum spanning tree (MST) will allow you to form a minimum distance sub-graph that connects all of your points.
You will first need to form a graph for your data set though. To efficiently form an undirected graph you could compute the Delaunay triangulation of your point set. The conversion from the triangulation to the graph is then fairly literal - any edge in the triangulation is also an edge in the graph, weighted by the length of the triangulation edge.
There are efficient algorithms for both the MST (Prim's/Kruskal's O(E*log(V))) and Delaunay triangulation (Divide and Conquer O(V*log(V))) phases, so efficient overall approaches are possible.
Hope this helps.
The algorithm you are looking for is called minimum spanning tree. It's useful to find the minimum cost for a water, telephone or electricity grid. There is Prim's algorithm or Kruskal algorithm. IMO Prim's algorithm is a bit easier to understand.
here http://en.wikipedia.org/wiki/Minimum_spanning_tree you can find more information about the minimum spanning tree, so you can adapt it to solve your problem.
Take a look at the Dijkstra's algorithm:
Dijkstra's algorithm, conceived by Dutch computer scientist Edsger Dijkstra in 1956 and published in 1959, is a graph search algorithm that solves the single-source shortest path problem for a graph with nonnegative edge path costs, producing a shortest path tree. This algorithm is often used in routing and as a subroutine in other graph algorithms.
http://en.wikipedia.org/wiki/Dijkstra's_algorithm

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