Computer Science Stack Exchange
Computer Science Stack Exchange

Questions tagged [spanning-trees]

The spanning tree of a connected undirected graph G is a tree having all the vertices and some number of edges of G.

147 questions
73
votes
10 answers

Minimum spanning tree vs Shortest path

What is the difference between minimum spanning tree algorithm and a shortest path algorithm? In my data structures class we covered two minimum spanning tree algorithms (Prim's and Kruskal's) and one shortest path algorithm (Dijkstra's). Minimum…
flashburn
  • 1,233
  • 1
  • 12
  • 22
29
votes
3 answers

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 you can't reverse this statement. If there are…
Keiwan
  • 487
  • 1
  • 5
  • 12
26
votes
7 answers

Do any two spanning trees of a simple graph always have some common edges?

I tried few cases and found any two spanning tree of a simple graph has some common edges. I mean I couldn't find any counter example so far. But I couldn't prove or disprove this either. How to prove or disprove this conjecture?
Mr. Sigma.
  • 1,301
  • 1
  • 16
  • 38
22
votes
2 answers

Do the minimum spanning trees of a weighted graph have the same number of edges with a given weight?

If a weighted graph $G$ has two different minimum spanning trees $T_1 = (V_1, E_1)$ and $T_2 = (V_2, E_2)$, then is it true that for any edge $e$ in $E_1$, the number of edges in $E_1$ with the same weight as $e$ (including $e$ itself) is the same…
Aden Dong
  • 1,151
  • 2
  • 11
  • 24
15
votes
4 answers

Graph Has Two / Three Different Minimal Spanning Trees?

I'm trying to find an efficient method of detecting whether a given graph G has two different minimal spanning trees. I'm also trying to find a method to check whether it has 3 different minimal spanning trees. The naive solution that I've though…
itamar
  • 293
  • 1
  • 3
  • 6
13
votes
2 answers

Why is the k-bounded spanning tree problem NP-complete?

The $k$-bounded spanning tree problem is where you have an undirected graph $G(V,E)$ and you have to decide whether or not it has a spanning tree such that each vertex has a degree of at most $k$. I realize that for the case $k=2$, this is the…
user17199
  • 131
  • 1
  • 1
  • 5
12
votes
2 answers

Minimal Spanning Tree With Double Weight Parameters

Consider a graph $G(V,E)$. Each edge $e$ has two weights $A_e$ and $B_e$. Find a spanning tree that minimizes the product $\left(\sum_{e \in T}{A_e}\right)\left(\sum_{e \in T}{B_e}\right)$. The algorithm should run in polynomial time with respect to…
Strin
  • 1,515
  • 1
  • 11
  • 16
11
votes
1 answer

Tighter analysis of modified Borůvka's algorithm

Borůvka's algorithm is one of the standard algorithms for calculating the minimum spanning tree for a graph $G = (V,E)$, with $|V| = n, |E| = m$. The pseudo-code is: MST T = empty tree Begin with each vertex as a component While number of components…
Joe
  • 4,105
  • 1
  • 21
  • 38
10
votes
2 answers

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 graph in $O(|V|\cdot \log |V|)$. Basically, the…
LiorGolan
  • 213
  • 2
  • 6
9
votes
1 answer

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?
Aditya.M
  • 103
  • 3
9
votes
3 answers

If all edges are of equal weight, can one use BFS to obtain a minimal spanning tree?

If given that all edges in a graph $G$ are of equal weight $c$, can one use breadth-first search (BFS) in order to produce a minimal spanning tree in linear time? Intuitively this sounds correct, as BFS does not visit a node twice, and it only…
TheNotMe
  • 571
  • 1
  • 6
  • 18
9
votes
2 answers

MST: Prim's algorithm complexity, why not $O(EV \lg V)$?

According to CLRS, the Prim's algorithms is implemented as below -- $\mathtt{\text{MST-PRIM}}(G,w,r)$ for each $u \in V[G]$ do $\mathtt{\text{key}}[u] \leftarrow \infty$ $\pi[u] \leftarrow \mathtt{\text{NIL}}$ $\mathtt{\text{key}}[r]…
ramgorur
  • 541
  • 1
  • 5
  • 16
8
votes
1 answer

Necessary and sufficient condition for unique minimum spanning tree

This is an exercise problem (Ex.3) from the excellent lecture note by Jeff Erickson Lecture 20: Minimum Spanning Trees [Fa’13] . Prove that an edge-weighted graph $G$ has a unique minimum spanning tree if and only if the following conditions…
hengxin
  • 9,671
  • 3
  • 37
  • 75
8
votes
0 answers

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 is undirected (just like in Christofides…
I_Really_Want_To_Heal_Myself
  • 131
  • 4
7
votes
1 answer

Need a hint! Karger's algorithm versus Kruskal, spanning tree distribution

Let G = (V,E) be a unit-capacity graph with n vertices and m edges. Let T denote all the spanning trees in G. If we run Karger's algorithm, we will get a random spanning tree in T formed by the contracted edges, we denote this distribution of…
MMP
  • 315
  • 1
  • 4
1
2 3
…
9 10