# Questions tagged [prims-algorithm]

Prim's algorithm is a greedy algorithm which forms minimum spanning tree for the given graph. During execution, Prim's algorithm grows single tree by adding edges to it, unlike Kruskal's algorithm in which there can be multiple minimum spanning trees during execution.

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### Lightest Paths Tree that is 10 times heavier than an MST of the same graph

Something I was asked to solve and tried to come up with a formula or some method to solve it after I did and couldn't. Given is a graph G=(V,E) that is undirected and weighted. Say we want to find ...
1 vote
331 views

### Find MST with bounded edge weight in O(|V|+|E|)

I need to describe (even in words) an algorithm that gets a connected, undirected, weighted graph so that the weight of each edge is either 1,2 or 3; and returns MST of that graph is time complexity O(...
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1 vote
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### Why Prim algorithm may fail to return minimum spanning forest on disconnected undirected graph

I am trying to address the following claim- "running prim algorithm on a disconnected undirected graph returns minimum spanning forest". I thing that the claim may be false (i.e there might ...
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1 vote
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### MST with weights in {1, 2, 3, 4, 5}

I am given an undirected connected graph with $n$ nodes, average degree $\sqrt{\log n}$, and each edge having integer weight in $\{1,..,5\}$. I am asked to describe MST algorithm which is as efficient ...
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1 vote
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### How to find if there's an MST where vertex $v$ has degree 2 in it? [duplicate]

I've faced this question and I hope that someone can help with it. Question: We're given an undirected graph $G=(V,E,w)$ where $w\colon E\rightarrow \mathbb{Q}$ and vertex $v$. We want to find if ...
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### Why do basic graph algorithms (BFS, DFS, Prim, Kruskal) have a similar structure?

This is my first post on CS Stack Exchange. For some time, I have been studying basic graph algorithms, mainly BFS, DFS, minimum spanning trees and their basic algorithms (Kruskal and Prim). One thing ...
74 views

### Minimum unrooted binary spanning tree

Given a graph $G$ with $n$ tip vertices, $n-2$ internal vertices, and a cost on each edge $C(v)$, find a minimum spanning tree subject to degree constraints: tips have a degree of $1$ internal ...
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560 views

### The Cut Lemma for graphs with non-distinct edges

In my introductory algorithms class I recently learned about the Cut Lemma and how it can be used to prove correctness for many Minimum Spanning Tree algorithms like Kruskal's and Prim's. In class, to ...
1 vote
319 views

### Find MST on grid graph with only weight of 1 and 2 in $O(|V|+|E|)$

Given a grid graph $G=(V,E)$ which has only two different integer costs/weights of 1 and 2. Find Minimum Spanning Tree in $O(|V|+|E|)$. I tried the following: Changing Kruskal using a counting Sort ...
772 views

### Best algorithm (Time Complexity) to find Minimum spanning tree of an complete, positive weighted, undirected, graph

Suppose that we have a complete undirected positive weighted graph $G = \langle V, E\rangle$. What is the most efficient algorithm, in terms of time complexity, to find an MST for $G$? The best prime ...
102 views

### Improve Prim's algorithm runtime

Assume we run Prim’s algorithm when we know all the weights are integers in the range {1, ...W} for W, which is logarithmic in |V|. Can you improve Prim’s running time? When saying 'Improving', it ...
1 vote
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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 ...
1 vote
493 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 ...
2k 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 ...
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1 vote
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### 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 ...
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