Table of Contents

## How many edges does a minimum spanning tree have?

four edges

## Can two minimum spanning trees for the same graph have different edge weights?

There may be several minimum spanning trees of the same weight; in particular, if all the edge weights of a given graph are the same, then every spanning tree of that graph is minimum.

## Is there only one minimum spanning tree?

�.� The Only Minimum Spanning Tree Algorithm At all times, F satisfies the following invariant: F is a subgraph of the minimum spanning tree of G. Initially, F consists of V one-vertex trees. The generic algorithm connects trees in F by adding certain edges between them.

## When finding a minimum spanning tree which edge is chosen second?

b. Since any spanning tree has exactly jV j 1 edges, any second-best minimum spanning tree must have at least one edge that is not in the (best) minimum spanning tree. If a second-best minimum spanning tree has exactly one edge, say .

## What is second best minimum spanning tree?

A second best MST T′ is a spanning tree, that has the second minimum weight sum of all the edges, from all the possible spanning trees of the graph G.

## How can we find the second best minimum spanning tree of a graph?

Algorithm to find the second best minimum spanning tree from a weighted graph

- Sort all the graph edges.
- Compute the MST using Kruskal.
- Get the minimum weight edge from the graph that is not in the first MST and add it to the MST (now the MST has a cycle)
- Remove the maximum weight edge in the new formed cycle.

## Is minimum spanning tree of a graph is unique?

Any undirected, connected graph has a spanning tree. If the graph has more than one connected component, each component will have a spanning tree (and the union of these trees will form a spanning forest for the graph). The spanning tree of G is not unique. This is called the minimum spanning tree (MST) of G.

## How do you find the minimum spanning tree?

Find the cheapest unmarked (uncoloured) edge in the graph that doesn’t close a coloured or red circuit. Mark this edge red. Repeat Step 2 until you reach out to every vertex of the graph (or you have N ; 1 coloured edges, where N is the number of Vertices.) The red edges form the desired minimum spanning tree.

## What do you mean by the minimum spanning tree?

Definition of Minimum Spanning Tree. A spanning tree of a graph is a collection of connected edges that include every vertex in the graph, but that do not form a cycle. The Minimum Spanning Tree is the one whose cumulative edge weights have the smallest value, however.

## What is maximum spanning tree?

A maximum spanning tree is a spanning tree of a weighted graph having maximum weight. It can be computed by negating the weights for each edge and applying Kruskal’s algorithm (Pemmaraju and Skiena, 2003, p. 336). A maximum spanning tree can be found in the Wolfram Language using the command FindSpanningTree[g].

## What is the use of minimum spanning tree?

Minimum spanning trees are used for network designs (i.e. telephone or cable networks). They are also used to find approximate solutions for complex mathematical problems like the Traveling Salesman Problem. Other, diverse applications include: Cluster Analysis.

## Does minimum spanning tree give shortest path?

Like we talked about above, this means that the minimum spanning tree is guaranteed to have the lowest possible total cost of all the edges. Given this it seems reasonable to assume that it is also the shortest path between any two nodes. So, in general, the minimum spanning tree will hold some of the shortest paths.

## What is minimum cost spanning tree give its applications?

Suppose you want to construct highways or railroads spanning several cities then we can use the concept of minimum spanning trees. Designing Local Area Networks. Laying pipelines connecting offshore drilling sites, refineries and consumer markets. To reduce cost, you can connect houses with minimum cost spanning trees.

## Which is better Prims or Kruskal?

Prim’s algorithm is significantly faster in the limit when you’ve got a really dense graph with many more edges than vertices. Kruskal performs better in typical situations (sparse graphs) because it uses simpler data structures.

## Is Prims faster than Kruskal?

Prim’s algorithm gives connected component as well as it works only on connected graph. Prim’s algorithm runs faster in dense graphs. Kruskal’s algorithm runs faster in sparse graphs.

## What is the other name of Dijkstra algorithm?

Dijkstra’s algorithm (or Dijkstra’s Shortest Path First algorithm, SPF algorithm) is an algorithm for finding the shortest paths between nodes in a graph, which may represent, for example, road networks.

## What is the difference between shortest path and minimum spanning tree?

They are based on two different properties. Minimum spanning tree is based on cut property whereas Shortest path is based on the edge relaxing property. A cut splits a graph into two components. It may involve multiple edges.

## Does Dijkstra give minimum spanning tree?

Dijkstra’s algorithm is very similar to Prim’s algorithm for minimum spanning tree. Like Prim’s MST, we generate a SPT (shortest path tree) with given source as root. At every step of the algorithm, we find a vertex which is in the other set (set of not yet included) and has a minimum distance from the source.

## Can Kruskal be used to find shortest path?

The shortest path can be calculated using graph theory. The Kruskal algorithm is one of the algorithms in graph theory in finding the minimum spanning tree to connect each tree in a forest [1].

## What is shortest path spanning tree?

From Wikipedia, the free encyclopedia. Given a connected, undirected graph G, a shortest-path tree rooted at vertex v is a spanning tree T of G, such that the path distance from root v to any other vertex u in T is the shortest path distance from v to u in G.

## Which of the following is true Prims algorithm?

1. Which of the following is true? Explanation: Steps in Prim’s algorithm: (I) Select any vertex of given graph and add it to MST (II) Add the edge of minimum weight from a vertex not in MST to the vertex in MST; (III) It MST is complete the stop, otherwise go to step (II).

## What is the time complexity of Dijkstra algorithm?

Time Complexity of Dijkstra’s Algorithm is O ( V 2 ) but with min-priority queue it drops down to O ( V + E l o g V ) .

## What is the time complexity of Kruskal algorithm?

4. What is the time complexity of Kruskal’s algorithm? Explanation: Kruskal’s algorithm involves sorting of the edges, which takes O(E logE) time, where E is a number of edges in graph and V is the number of vertices.

## Which of the following is true for Dijkstra algorithm?

Explanation: Dijkstra’s Algorithm is used for solving single source shortest path problems. Explanation: Dijkstra’s Algorithm cannot be applied on graphs having negative weight function because calculation of cost to reach a destination node from the source node becomes complex.

## Which of the following is are true S1 Dijkstra algorithm is not affected by negative edge?

S1: Dijkstra’s algorithm is not affected by negative edge weight cycles in the graph and gives correct shortest path. S2: Bellman ford algorithm finds all negative edge weight cycles present in the graph.

## Which of the following is not an advantage of trees?

Which of the following is not an advantage of trees? Explanation: Undo/Redo operations in a notepad is an application of stack. Hierarchical structure, Faster search, Router algorithms are advantages of trees. 7.

## How many steps are required to prove that a decision problem is NP complete?

13. How many steps are required to prove that a decision problem is NP complete? Explanation: First, the problem should be NP. Next, it should be proved that every problem in NP is reducible to the problem in question in polynomial time.

## How do you prove a problem is NP-complete?

We can solve Y in polynomial time: reduce it to X. Therefore, every problem in NP has a polytime algorithm and P = NP. then X is NP-complete. In other words, we can prove a new problem is NP-complete by reducing some other NP-complete problem to it.

## How do you prove a problem is NP?

To prove that a problem X is NP-Complete, you need to show that it is both in NP and that it is NP-Hard. Steps 2 through 5 seek to accomplish the latter. Step 1: Show that X is in NP. We want to argue that there is a polytime verifier for X.