Talk:Minimum spanning tree

Directed graphs
My textbook (Discrete Mathematics and Its Applications) doesn't say anything about the supplied graph has to be either directed or undirected - just that it has to be a weighted, connected graph. However, this article states it has to be undirected. Why is that so? A directed, weighted graph will only limit the possible number of forests. Thanks. Anders Sørensen


 * That's probably the "minimum branching"/"optimum branching" problem: . In many textbooks, "graph" means "undirected graph" (without multiple edges, halfedges, ...) unless stated otherwise. --Lukas Mach 20:00, 7 April 2007 (UTC)
 * it is a good ideal — Preceding unsigned comment added by 117.191.16.85 (talk) 14:58, 4 November 2015 (UTC)
 * This article concerns undirected graphs because finding the MST of a digraph is a different (harder) problem. Consider that neither Prim's nor Kruskal's algorithms will even give the correct answer if their initial "arbitrary" vertex selection is not the root of a DAG.  Does your textbook say that an the graph has to be strongly connected, unilaterally connected, or weakly connected in order for an MST to exist?  See the Related problems section of this article for the digraph case.  AHMartin (talk) 16:59, 12 July 2019 (UTC)

Negative weights?
"A minimum spanning tree is in fact the minimum-cost subgraph connecting all vertices, since subgraphs containing cycles necessarily have more total weight."

Correct me if I'm wrong, but this is not true for a graph with negative weights - for example, in a complete graph with all weights equal -1, every spanning tree is a minimum spanning tree, but the minimum-cost subgraph connecting all vertices is the complete graph itself.

trees are good —Preceding unsigned comment added by 69.124.89.74 (talk) 22:26, 5 December 2007 (UTC)

Missing citations?
Someone thinks this article is or was missing citations. In the current state I disagree. I do also miss any explanation here, which I would expect accompanying with a tag so dominantly placed on an article. Therefore, I have decided to remove the tag. KKoolstra (talk) 10:01, 9 April 2009 (UTC)

Mathematical problem
It seems to me that there is a mistake in the proof given under "Uniqueness". In the last line where it says "If the weight of e1 is larger than that of e2, a similar argument involving tree {e2} $$\cup$$ A - {e1} also leads to a contradiction. Thus, we conclude that the assumption that there can be a second MST was false.", it assumes that {e2} $$\cup$$ A - {e1} is a spanning tree. This is not necessarily the case because it could be that the cycle C contains several edges which are in B but not in A. This means that attaching the edge e2 to A and subtracting e1 does not necessarily create a cycle. — Preceding unsigned comment added by 70.112.206.154 (talk) 23:21, 9 March 2012 (UTC)

I agree with that. But also there is a mistake. Attaching the edge e2 to A surely creates a cycle, however, this cycle does not necessarily contains e1. So {e2} $$\cup$$ A - {e1} may not be a spanning tree. The original proof in the reference is correct. The e1 is not picked casually. It's the minimum weight edge which is in one but not both MST. So the weight of e1 is lower than that of e2 definitely, and the original proof needs to be revised.

I don't know if this section is out-of-date and all the corrections have been made, but the proof in its current form looks logically correct to me. One thing I'd change though is in the fourth step, just reorganizing the points to make it easier to grasp:
 * Then C has an edge e2 whose weight is greater than the weight of e1, since all edges in B with less weight are in A by the choice of e1, and C must have at least one edge that is not in A because otherwise A would contain a cycle in contradiction with its being an MST.

I would change it to say: Jeffxtreme (talk) 13:49, 1 October 2014 (UTC)
 * C must contain at least one edge e2 that is not in A because otherwise A would contain a cycle in contradiction with its being an MST. Thus, edge e2 is only in B.
 * From our particular choice of e1, we that the edge e2 has a greater weight than e1.
 * Then C has an edge whose weight is greater than the weight of e1.

Incorrect diagram
A diagram's text states "T is the only MST of the given graph", but the given graph actually has 2 MSTs. This could be fixed by changing the weight of EC to be at least 5. — Preceding unsigned comment added by 2620:4F:0:101:15A4:738E:AB92:D953 (talk) 00:40, 9 January 2014 (UTC)

Figure title may be incorrent
The figure showing the Cut Property has as its first sentence "This figure shows the cut property of MSP." Should MSP be changed to MST? I cannot find a definition for MSP on this page. Rellims2012 (talk) 14:19, 17 March 2015 (UTC)

Request
Can anybody knowing this stuff take a look at random minimal spanning tree? I find that one not so clear. Thanks. Oleg Alexandrov 03:23, 25 Jun 2005 (UTC)

Does anyone know how to solve the problem for directed graphs? Ali Hassan — Preceding unsigned comment added by 39.42.63.81 (talk) 14:50, 11 September 2016 (UTC)
 * Did you search for "directed graphs" in this article? Because if you did you would have found an answer. —David Eppstein (talk) 17:55, 11 September 2016 (UTC)

Tarjan's criterion
I believe there are Russian articles with Tarjan's criterion:


 * Some spanning tree is minimal if and only if the weight of any other edge (x, y) (not from spanning tree) is not less than the weight of the heaviest edge on the path from x to y in spanning tree.

Wumbolo (talk) 22:09, 6 August 2017 (UTC)

External links modified (February 2018)
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Proving correctness
It will be helpful to include the general Selection-Rejection rules from Ottman, Widmayer -- Algorithmen und Datastrukturen (pg.576). They can be used for proving Kruskal's algorithm, Jarnik's algorithm, Boruvka's algorithm. Cheater no1 (talk) 10:55, 1 August 2019 (UTC)


 * You mean the cycle property and cut property, already listed in separate subsections of the "properties" section? —David Eppstein (talk) 19:18, 1 August 2019 (UTC)

MST Cut example
Why on Earth does the diagram showing the cut remove some, but not all, of the edges? That's very odd. Doesn't really hurt anything other than adding to confusion. Hobit (talk) 15:04, 3 February 2021 (UTC)

Convergence of Kruskal's and Boruvka's algorithms

 * Specific text to be added in the section on "Parallel and distributed algorithms":

show how parallelizations of Kruskal's and Boruvka's algorithms can converge to identical implementations.


 * Reason for the addition:

A connection between these two classical MST algorithms that has not yet been mentioned


 * References supporting change:

Included in text to be added

Burtscher (talk) 19:14, 17 December 2023 (UTC) Burtscher (talk) 19:14, 17 December 2023 (UTC)


 * ❌. We would need secondary sourcing to indicate that this primary source is worth writing about. - MrOllie (talk) 19:31, 17 December 2023 (UTC)
 * If it's parallel, it's not Kruskal's algorithm, but something else. Why should it be surprising that a paper modifying classical algorithms to produce a different parallel algorithm produces only one algorithm? In what sense can implementations be said to converge? —David Eppstein (talk) 20:01, 17 December 2023 (UTC)