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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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1answer
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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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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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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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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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196 views

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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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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126 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 ...
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1answer
54 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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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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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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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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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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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
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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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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
49 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 ...
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1answer
113 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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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 edges are not in any MST?

This is a homework question. I do not want the solution - I'm offering the solution I've been thinking of and wish to know whether is it good or why is it flawed. Consider a weighted undirected graph....
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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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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
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
60 views

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
554 views

Finding missing edge weights in the context of minimum spanning tree

I came across following problem: Problem 1 Suppose that minimum spanning tree of the following edge weighted graph contains the edges with weights $x$ and $z$, then what are $x$ and $z$? The ...
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152 views

Two criteria for an edge to belong to all MSTs

Let $G$ be a connected undirected graph, with integral positive weights on the edges, and let $e_1$ be an edge of $G$. As part of an assignment I proved the following Lemma 1: The edge $e_1$ ...
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Minimum spanning tree of a connected induced subgraph

I'm doing an online course in which I'm struggling with the following multiple-choice question: Suppose $ T $ is a minimum spanning tree of the connected graph $ G $. Let $ H $ be a connected ...
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How to restore a broken minimal spanning tree?

We're given $T$ a minimal spanning tree (MST) of a non-directed, connected graph $G=(V,E)$ with non-negative weights for each edge $e \in E$. Let $e^* \in T$ be an edge in $T$ and let $G'=(V,E')$ be ...
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338 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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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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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
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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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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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250 views

MST for a finite number of weights

Let $F$ be a finite set of real numbers say $\{w_1,\dots,w_k\}$. Let $G=(V,E)$ be an undirected connected graph and let $w\colon E\to F$ be a weight function. Describe a linear time algorithm that ...
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1answer
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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,...
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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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179 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
813 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
100 views

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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197 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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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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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 ...
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1answer
89 views

Minimum sub-tree of a graph that covers each color at least once

I have a connected graph $G$ with $k$ different colors assigned to $n$ nodes where $k<n$. All edges have unit weight. I want to figure out an algorithm to find a minimum sub-tree of $G$ that ...
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Is a spanning tree an MST if its weight can't be decreased by adding an edge and removing one? [duplicate]

My gut says it's true and I have tested it on a few examples. However, I can't prove it. I thought of using contradiction; suppose there exists another tree T' with smaller weight which has m edges ...