# 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.

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99 views

### MST with possibly minimal diameter

I am working with some research problem connected loosely to TSP which requires to find the Minimum Spanning Tree of a fully connected, weighted graph, where all the weights are positive and the graph ...
101 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?
48 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) + ...
390 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,...
95 views

### For which class of graphs can a minimum spanning tree always be associated to a shortest path tree?

Given a connected graph $G=(X,E)$ with positive edge weights. We assume that $G$ contains a unique min weight spanning tree $T_{\min}$ (this is true for example when for all the cuts, the edge with ...
200 views

### Updating a mst after increasing the weight of an edge in the mst

Suppose we have a weighted undirected graph $G$ and a minimum spanning tree $T$ Let $G2$ be a new graph by increasing the weight of one edge $e = (a,b)$ that is part of $T$. I'm using a common ...
70 views

### Minimum spanning tree formulation

Can any expert explain the reasoning behind the constraint in the following formulation of the minimum spanning tree? To formulate the minimum-cost spanning tree (MST) problem as an LP, we ...
19 views

### Minimum Weight Binary Spanning Tree

Let $G=(V,E)$ be a simple graph with weights $w_{ij}$ (can be assumed to be positive). Is it possible to find the minimum (or maximum) weight, rooted spanning tree that is binary? That means every ...
102 views

### Build transportation system to travel between cities

Given $n$ cities, I'm looking to build a transportation system that allows travelling between every two cities. For every two cities $i$ and $j$, a road can be paved in the cost of $c_{ij}$. Also, ...
93 views

### n'th cheapest MST in a graph, with multiple edges that can have the same weight

I'm trying to think about an algorithm for this problem. I know there is an algorithm for the second cheapest MST in a graph, but if I understood it correctly it only solves cases in which every ...
284 views

### How can I modify Kruskal's algorithm to work with limited resources?

In Kruskal algorithm, I decided to use it to construct a minimum spanning tree for a set of N = 100,000 points. Since my hardware does not allow me to keep all the distances between pairs of points, I ...
80 views

### NP-hardness of Capacitated Minimum Spanning Tree and Price Collecting Steiner Tree on dag/tree

I am thinking about the NP-completeness of two graph problems on different graph structures. For example: The Capacitated Minimum Spanning Tree for graph is NP-hard. However, is the problem still ...
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### Is there a way to solve the optimal branching / arborescence problem with path-dependent weights?

The optimal branching problem (solved by Edmond's algo or Tarjan's algo) finds the spanning arborescence for a particular graph.  I'm looking for a formulation of the problem that allows for path-...
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### Prove: If $T'$ is not a MST one of its edges may be replaced by a lighter edge in order to get a lighter spanning tree

Prove: If $T'=(V,F')$ is a spanning tree, but not a MST of $G=(V,E)$, then there are edges $e' \in F'$ and $e \in$ $E$ \ $F'$ such that $w(e)<w(e')$ and $T'$ \ $\{e'\} \cup \{e\}$ is a spanning ...
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### Minimum spanning tree with given edges

So the question is: "Given an undirected graph G=(V,E) with positive edge weights, and E'⊆E, show an algorithm that finds, from all the minimum spanning trees, the one that has more edges from E' ".
96 views

### Boruvka algorithm in Elog(log(V)) complexity

I am trying to implement Boruvka algorithm with the use of fibonacci heaps. My idea is the following: Since Boruvka's algorithm operates like this: Input is a connected, weighted and un-directed ...
39 views

### Minimum spanning tree classification problem with adjacent neighbors

I am new to minimum spanning trees. But have used the last few days on a problem I think matches MST, but cannot really figure out the connection. The problem is a minimum cost problem where N ...
131 views

### How many roots are there in an undirected root

Given an undirected tree with 7 nodes how many roots would this tree have. My intuition tells me that because the tree is undirected it would either be 7 or 0. How would I solve this?
178 views

### Boruvka MST algorithm using doubly linked lists

I'm reading Sedgewick & Wayne's Algorithms book, and one of the questions in one of the chapters is the following: Develop an implementation of Boruvka's algorithm that uses doubly-linked ...
47 views

### N-Guest Table, Graph Problem

The Queen of England wants to organize a set of tables for n guests talking different languages. The tables have to be set in a way that every guest can speak to his neighbor on the right and his ...
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### Distributed MST in $O(n \log \log n)$

I'm facing the following problem: Describe a distributed MST algorithm in time $O(n \log \log n)$ I've managed to think of the following, Run GHS(Gallager, Humblet and Spira) algorithm, till there ...
638 views

### Using Prim's for finding minimum cost arborescence in a special setting

I have a complete graph G which is directed. In essence, a node is connected to all other nodes in the graph. Also, for every pair of nodes, say ...
Given an undirected graph $G=(V,E)$ and a weight function $w:E\to\{1,2,3,\dots,10\}$: Let $U\subseteq E$ the set of all edges $e\in E$ that has a cycle in $G$ such that the cycle contains $e$ and all ...