Questions tagged [minimum-spanning-tree]

Use this tag whenever your question is related to minimum spanning tree (MST). An MST of a connected edge-weighted graph G is a spanning tree whose sum of edge weights is as small as possible. We usually assume $G$ is finite, simple and undirected.

Filter by
Sorted by
Tagged with
21
votes
3answers
15k views

When is the minimum spanning tree for a graph not unique

Given a weighted, undirected graph G: Which conditions must hold true so that there are multiple minimum spanning trees for G? I know that the MST is unique when all of the weights are distinct, but ...
13
votes
1answer
2k views

Are all MST minimum spanning trees reachable by Kruskal and Prim?

I believe this is true but have not been able to get a formal proof for either. But is it true that any minimum spanning tree is reachable by applying Kruskal's algorithm? Similarly, is this true for ...
8
votes
3answers
6k views

Do Kruskal's and Prim's algorithms yield the same minimum spanning tree?

Assuming the edges are undirected, have unique weight, and no negative paths, do these algorithms produce the same Minimum Spanning Trees?
8
votes
1answer
1k views

Does a graph always have a minimum spanning tree that is binary?

I have a graph and I need to find a minimum spanning tree to a given graph. What is to be done so that the output obtained is a binary tree?
7
votes
2answers
5k views

Updating an MST $T$ when the weight of an edge not in $T$ is decreased

Given an undirected, connected, weighted graph $G = (V,E,w)$ where $w$ is the weight function $w: E \to \mathbb{R}$ and a minimum spanning tree (MST) $T$ of $G$. Now we decrease the weight by $k$ of ...
6
votes
1answer
3k views

Finding MST after adding a new vertex

Let $G=(V,E)$ which is undirected and simple. We also have $T$, an MST of $G$. We add a vertex $v$ to the graph and connect it with weighted edges to some of the vertices. Find a new MST for the new ...
5
votes
2answers
461 views

Is the inverse of MST cycle property always true? Why?

I am trying to find an algorithm which would check for each edge in a given weighted undirected graph whether it belongs to any of the graph's Minimum Spanning Trees. I have found many potential ...
5
votes
2answers
538 views

MST with half the edges the maximum weight

I have been cracking my head over the following question - You are given an undirected connected graph with an even number of edges. Half of the edges have weight less than C (possibly with ...
5
votes
2answers
744 views

Find all critical edges for minimum spanning tree

This is a problem from the textbook "Algorithms, 4th edition" by Robert Sedgewick and Kevin Wayne. 4.3.26 Critical edges. An MST edge whose deletion from the graph would cause the MST weight to ...
5
votes
0answers
81 views

Minimum spanning tree and Hamiltonian path

For a graph $G(V,E)$, under what conditions is a minimum spanning tree of $G$ equal to a hamiltonian path on $G$? IS there any body of literature connecting these two?
4
votes
2answers
568 views

Diameter-constrained Minimum Spanning Tree Problem

The diameter-constrained Minimum Spanning Tree (MST) problem is as follows: you have a undirected weighted graph $G = (V,E)$ of different weights where $V$ is the set of vertices and $E$ is the set of ...
4
votes
1answer
260 views

Are all minimum spanning trees optimized for fairness?

I know by definition that a minimum spanning tree (MST) of a weighted, connected graph has the lowest global value for for sum of all edges in a path that connects all vertices. I'm curious if all ...
4
votes
1answer
66 views

Showing that all vertex degrees in MSTs of Euclidian graphs are in O(1)

There is a finite set of points $S$ in the plane with $ |S| = n$. MST is the minimal spanning tree of S. "Minimal" here refers to the Euclidean distance between the points of $S$, so the MST is the ...
4
votes
1answer
95 views

Is a perfect matching's weight less than MST of a metric graph?

This is part of a bigger proof I'm trying to solve, which eventually came down to one thing: Let $G=(V,E)$ an un-directed, complete, metric graph (maintains the triangle inequality) with an even ...
4
votes
1answer
422 views

Minimum spanning tree formulation as integer program

The minimum spanning tree problem can be solved in polynomial time via Kruskal's or Prim's algorithm. However, every integer program I have seen that corresponds to the MST problem require a ...
4
votes
3answers
192 views

Are edges in a minimum spanning tree not heavier than respective edges in another spanning tree?

Let $G$ be an undirected connected weighted graph, and let $T$ be a minimum spanning tree of $G$ with edge weights: $w_1 \le w_2 \le ... \le w_{n-1}$. Now let $T'$ be some other spanning tree of $G$ (...
3
votes
1answer
80 views

Minimum diameter spanning tree problem

Minimum diameter spanning tree (MDST) problem is defined as following: given the connected weighted graph $G(V, E)$, weight function $w: E \rightarrow R, w(e) > 0\ \forall e \in E$, find the ...
3
votes
2answers
407 views

Minimum path - robot motion problem combined with freeze tag problem

Alright, I am not entirely sure if this is the right place to ask this, but here goes: I have a map of coordinates of robots and obstacles. The first robot is awake from the start of the problem and ...
3
votes
1answer
5k views

Why do we have different algorithm for MST when graphs are directed?

What was the reason to come up with Chu–Liu/Edmonds' algorithm when the input graph is directed instead of using the Prim's or Krushkal's method for finding Minimum spanning tree ? What cases are not ...
3
votes
2answers
158 views

Finding MST for connected graph with $|E|=|V|+87$

Given $G=(V,E)$ undirected, connected graph and weights given by $w:E \to \mathbb R$. We also know that $|E|=|V|+87$. Find Minimum spanning tree of $G$. Obviously we can use Prim in $O(|V| \cdot \...
3
votes
3answers
103 views

MST: Is there such an example of a graph with unique mst and not unique light edge?

The problem is the following: Give an example of a graph that has a unique minimum spanning tree but for every cut of the graph, there is not a unique light edge crossing the cut. I am trying to ...
3
votes
2answers
2k views

Is the Nearest Neighbor Algorithm a valid algorithm to find a Minimum Spanning Tree?

I just wrote a program that runs the Travelling Salesman Problem using the Nearest Neighbor Algorithm. Afterwards, I started looking into Minimum Spanning Trees (MST). From my understanding: The ...
3
votes
1answer
79 views

Maximal Minimum Spanning Tree by Removing $k$ Edges

The problem is as follows: Consider a connected, undirected, and weighted graph $G = (V, E, w)$ and an integer $0 < k < |E| - |V| + 1$. Describe and analyze and efficient algorithm to remove ...
3
votes
1answer
97 views

What is the formal way to prove this is still a minimum spanning tree?

I don't get the formal way of proving the following cases: Suppose I have a minimum spanning tree $T$ in a graph $G(V, E)$, with positive edge weights $w$, I want to prove the following: 1) If I ...
3
votes
1answer
113 views

Constructing a minimum spanning tree from an all-shortest path graph?

Given an $n \times n$ shortest path distance matrix $D$. And a complete graph $G(D)$ on $n$ nodes, where edge $(i, j)$ has weight $D_{ij}$. Furthermore, the distance matrix $D$ is computed for a ...
3
votes
1answer
137 views

What is the best way to merge cycles to minimise total weight?

Suppose I have a vertex-disjoint set $S$ of simple cycles in a weighted undirected graph. So no vertex $v$ is contained in more than one cycle. A cycle $c$ is a closed path with no repeated vertices: $...
3
votes
1answer
70 views

Algebraic (spectral) algorithms for the minimum spanning tree problem

Are there any algorithms that use the spectral properties of a graph to solve the minimum spanning tree problem? To clarify further what I have in mind, starting with the Laplacian matrix I want to ...
3
votes
1answer
67 views

Finding minimum spanning tree of a special form graph

I'm trying to find an efficient algorithm that will find me the minimum spanning tree of an undirected, weighted graph of this particular form: My idea was a recursive solution: Suppose the algorithm ...
3
votes
1answer
55 views

Algorithm for MST connecting a subgraph

I already know how to find the MST of a connected graph. This MST will have the least total weight and will connect all nodes in the graph. However, this is a problem I have to deal with: Given a ...
3
votes
1answer
265 views

Alternative algorithm for minimum spanning tree construction

Let $\textit{G(V,E)}$ be an undirected connected graph with distinct costs on its edges. Initialize $\textit{T}$ to be any spanning tree of $\textit{G}$. Consider an algorithm which replaces an ...
3
votes
1answer
126 views

Between every two MST's there's a series of “nearby” MST's

Given undirected connected graph $G=(V,E)$ and a weight function $w:E\to\mathbb{R}$, two MST's $T_1, T_2$ are nearby if there exists $e\in T, e'\in T'$ such that $T'=(T-\{e\})\cup\{e'\}$. Prove that ...
3
votes
1answer
379 views

Local search to find minimum degree spanning tree

Suppose for a graph $G=(V,E)$ and a spanning tree T of G, $\Delta(T)$ is the largest degree of a vertex in T, and let $\Delta^*$ be the smallest such quantity over all spanning trees of $G$. We have ...
3
votes
0answers
44 views

Worst Case Analysis of a Multivariate Recurrence of a Graph Algorithm

I have a graph algorithm that runs in: $$ T(n, m) = \begin{cases} c_1 & n \leq 2 \lor m = 1\\ T(n - i,\ m - j - k) + T(i, k) + c_2 m + c_3 n & m \leq (n-i)i\\ T(n - i,\ m) + T(i, m) + ...
3
votes
0answers
46 views

Edge that is not a light edge in a MST [duplicate]

Given the following statement: For a graph $G$, consider its minimum spanning tree $T$ and let $e = (a,b)$ be an edge that is not a light edge for a given cut $C$. Then $e$ never belongs to $T$. ...
3
votes
0answers
381 views

Learning automata for degree constrained minimum spanning tree problem

I'm trying to understand the algorithm described in "Degree constrained minimum spanning tree problem: a learning automata approach" (Javad Akbari Torkestani, The Journal of Supercomputing; April 2013,...
3
votes
1answer
831 views

Shortest paths in weighted graphs, and minimum spanning trees

I stuck in one challenging question, I read on my notes. An undirected, weighted, connected graph $G$, (with no negative weights and with all weights distinct) is given. We know that, in this ...
2
votes
2answers
37 views

Uniqueness of minimum spanning tree

If G has a unique minimum spanning tree, does that mean the edge weights in G are also unique? if yes why and if no why?
2
votes
3answers
124 views

Need Help Understanding MST Cutset Formulation

I just started learning about linear programming in my class, and I'm having some trouble understanding the MST Formulation Integer Linear Programming (Cutset Formulation). This is the definition: ...
2
votes
2answers
460 views

Does MST exclude the edge belongs to the cycle of the graph?

For any simple, connected, undirected graph $G(V,E,w)$ with each carrying distinct positive edge weights, is it true that MST will contain all the edges which do not lie in any cycle of $G$? I think ...
2
votes
1answer
117 views

Is a set of acyclic |V| - 1 light edges always a Minimum Spanning Tree?

I am trying to prove the algorithm for Question 5 in this practice exam. I am trying to prove this algorithm with the following three claims: Suppose we have a graph G, a minimum spanning tree T, ...
2
votes
1answer
2k views

safe edge for Minimum spanning tree

I know this question is posted many times, but i am still posting this because i have my own doubt which pulls me out to move forward to kruskal and prim's algorithm.So Please do help me out $-:$ ...
2
votes
2answers
347 views

MSTs in Christofides Algorithm

I am implementing the Christofides algorithm for TSP problem and currently I have one issue with the MST result. Let's assume that I have one graph $G$, whereas I can get $MST_1$ and $MST_2$ where ...
2
votes
1answer
294 views

Understanding the Diameter Constrained MST Problem

I am interested in understanding the DCMST problem, explained in this paper (http://www.dcc.ic.uff.br/~celso/artigos/cpdcmst.pdf). I don't think I understand it as I should. Here is how I understand ...
2
votes
2answers
1k views

Counting Minimum Spanning Trees

I understand how Kruskal's algorithm works. However, I am not sure how to determine the number of minimum spanning trees that a given graph has. For example say graph $G=(V,E)$ given by When running ...
2
votes
1answer
347 views

When is the output of shortest path $\subset$ MST?

I was wondering if the output of an algorithm like Dijkstra was always contained in the minimal spanning tree, however, a counter example to this claim are cyclic graphs like: The shortest path $B \...
2
votes
1answer
1k views

Proof for variation of Prim's and Kruskal's to find maximum-weight acyclic subgraph

I have been scratching my head to find good counter examples to the following problem: Suppose we are given a directed graph G=(V,E) in which every edge has a distinct positive edge weight. A ...
2
votes
1answer
81 views

Mimimum spanning tree with a constraint on number of certain types of edges

I have the the following problem. Say we have a graph $G = (V,E)$ where all $e \in E$ have positive weight, and $E$ can be separated in to two disjoint sets $E = A \cup B$. We have to find a spanning ...
2
votes
1answer
309 views

If all edges are unweighted AND DIRECTED, can one use BFS to obtain a MST?

I've looked through CS SE and I've found a page that said you could find use BFS to find a MST if the edges are unweighted, but what if the edges are directed? Given a directed graph between V ...
2
votes
1answer
291 views

NP-hardness of existence of spanning tree with given maximal degree

I am trying to solve the following exercise: Let $G = (V,E)$ be a graph. Show that the following two problems are NP-hard: $G$ has a spanning tree where every node has at most $k$ neighbors,...
2
votes
1answer
177 views

Expected weight of euclidean minimum spanning tree on a unit square

Suppose I randomly generate $n$ points from the unit square $[0,1]^2$, form a complete graph in which the weight of each edge is just the Euclidean distance between its endpoints, and compute the ...