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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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Spanning trees on disconnected graphs

Can anyone please help me out with my query: can disconnected graphs have minimum spanning trees?
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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?
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1answer
42 views

Distributed MST Construction in O(log log n) Rounds in a Clique

I'm reading the paper MST Construction in O(log log n) Communication Rounds in a Clique and trying to understand the correctness analysis, in page 5. It shows by induction on k (phase number), that ...
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34 views

Define the time complexity of Kruskal's algorithm as function

I am trying to define the time complexity of Kruskal's algorithm as function dependant on: the number of vertices V the number of edges ...
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proof that multiple minimum spanning trees (MSTs) for a given edge- weighted graph have same edge weight multi sets [duplicate]

How do we prove this? I thought we could use safe edge property and say if the edges were different, the safe edge property would choose only the minimum weight edge, but since there are multiple ...
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Do the minimum spanning trees of a graph have the same number of edges with a given weight?

I'm asking about the answer here: Do the minimum spanning trees of a weighted graph have the same number of edges with a given weight? I didn't understand the best answer here Choose edge $e \in ...
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1answer
56 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 ...
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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) + ...
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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 $T$. ...
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1answer
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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 ...
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1answer
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Distance function such that we visit every “color region” once [closed]

Consider the following image: Starting at (0,0) top left, the objective is to find a dijikistra path to the bottom right. We must go through each color exactly once, and once we go outside a color, ...
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1answer
103 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 ...
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1answer
56 views

Combinatorial Optimization: Shortest distance given sets of drivers and riders

Problem: I have 2 sets, one of drivers and one of riders. All my participants need to reach one common destination. I wish to calculate the shortest combined distance in order for all participant to ...
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1answer
33 views

Minimum Spanning Tree with one particular edge minimised(continued)

I have recently encountered a coding problem, specifically, the CCC problem S4. In the problem, it states that you are given a spanning tree, or otherwise a "valid plan of pipes", that connect each ...
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2answers
54 views

Minimum spanning tree such that one edge can be minimised

During a computer coding exam, I have encountered such a problem. Given a list of vertexes and edges between the vertexes,and a positive number, D, what is the minimum spanning tree between the ...
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1answer
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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 ...
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Understanding connection between minimum spanning tree, shortest path, breadth first and depth first traversal

In CLRS, in the later part of breadth first search topic, for unweighted graphs, it says: At the beginning of this section, we claimed that breadth-first search finds the distance to each reachable ...
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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 ...
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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 ...
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1answer
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What's the usage of $S$ in Dijkstra shortest path algorithm in the book Introduction to Algorithms?

I don't understand how the $S$ is needed in dijkstra shortest path algorithm. For each node $v$ in $G.V$, the $v.\pi = previous\_node$ is used to denote it previous node in the shortest path to the ...
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1answer
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Will MST find the shortest path for each pair $(r,v)$?

Will local best choice will lead to global best choice? In other words, I'm thinking about whether it's possible that the MST has to put its branch location in the middle of two far nodes ...
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1answer
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Generating a random minimum spanning tree

I am tring to find the simplest method of generating a random minimum spanning tree. My intention is to randomly generate a Level in a game where there are n amount of fixed sized rooms existing on a ...
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1answer
114 views

Optimal Loading of a Hopping Airplane Minimum Cost Flow Problem

I have a Problem with an Optimal Loading a Hopping Airplane example . This is the Part of Minimum Cost Flow Problem.. .. I dont understand a Picture at all. I should to make one example with numbers ...
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Minimum Distance Spanning Tree Dijkstra

I would like to construct a Minimum Distance Spanning Tree (Dijkstra) for the graph below: MDST: {(a,c), (c,h), (c,f), (a,d), (h,g), (a,b), (d,e), (h,j), (h,i), (j,k), (e,m), (i,l)} Is my ...
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348 views

DFS & BFS Spanning Trees

I want to construct a DFS and a BFS spanning trees for the graph below. The root is node a. At each step the next edge to be traversed should be the cheapest one. DFS: My understanding that to the ...
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2answers
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MST Proof (Kleinburg & Tordos)

Consider the Minimum Spanning Tree Problem on an undirected graph G = (V, E), with a cost ≥ 0 on each edge, where the costs may not all be different. If the costs are not all distinct, there can in ...
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1answer
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Proving that a spanning tree of graph is not a minimum

Let $G$ be an undirected and connected graph. Let $T$ be a spanning tree of $G$ with edges weights: $w_1 \le, w_2 \le ... \le w_{n-1}$ which are responing to the edges. $e_1,e_2,...,e_{n-1}$. Now I ...
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3answers
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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 $G$ (...
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1answer
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Cost of the MST of the graph [closed]

Studying for a test in a computer science class and cannot figure out the answer to this question. Any help would be appreciated! Although the picture shows a directed graph, please treat it as ...
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1answer
31 views

Why does the cut $(V_C,V-V_C)$ respect $A$?

Corollary 23.2 Let $G = (V,E)$ be a connected, undirected graph with a real-valued weight function $w$ defined on $E$. Let $A$ be a subset of $E$ that is included in some minimum spanning tree ...
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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: ...
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180 views

MST Of An Almost Tree

A graph $G = (V,E)$ is called an almost tree if it is connected and has most $n + c$ edges where $n = |V|$ and $c$ is a small constant number. How would I go about designing an algorithm for a given ...
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Proving equivalent definitions for MSTs

I am working on the following homework exercise: Let $G = (V,E)$ be an undirected graph and $c: E \rightarrow \mathbb{R}$ it's cost function. Further let $T = (V,E')$ be a spanning tree in G. I need ...
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1answer
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Finding Euclidean Minimum Spanning Tree

Given a set of point $P$. Find the euclidean minimum spanning tree where each points is equally distributed on the plane using randomization. We can solve this problem with Prim's algorithm in $O(n^2)...
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1answer
79 views

Minimum spanning subgraph of even degree

I have an unusual problem that I am struggling to solve. I have a set of nodes (with positive distances between them) that I want to connect into a single component. In particular, I want to form a ...
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1answer
201 views

Time Complexity of the Kruskal Algorithm after sorting

In case I have sorted edges already, What is the best time complexity of Kruskal Algorithm?
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1answer
132 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: $...
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1answer
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Do we always get the same set of edges after running Kruskal's algorithm on a single graph?

I think it should be false because there may be more than one edge with the same weight.
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1answer
72 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 ...
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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?
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1answer
818 views

Time complexity of kruskal using array data structure

I was going through MST(minimum spanning tree) algorithms in a given undirected graph. By using the disjoint data structure It is fairly easy. All I have to do follow these steps: Sort the edges as ...
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1answer
393 views

How to understand the complexity of Kruskal implemented with Quick-Union by rank and path compression?

I'm trying to understand the complexity of the Kruskal algorithm implemented with the Quick-Union by rank and with the path compression. Now there is a theorem for the last structure above: The ...
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2answers
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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 ...
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1answer
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If a graph has a unique MST, then its edge weights are distinct. True or false ? Justify your answer. [closed]

If a graph has a unique MST, then its edge weights are distinct. True or false ? Justify your answer.
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1answer
92 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 ...
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1answer
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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 ...
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1answer
109 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, ...
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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, ...
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1answer
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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 ...
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1answer
251 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 ...