The Gateway to Computer Science Excellence
+20 votes
3.4k views

Consider a weighted complete graph $G$ on the vertex set $\{v_1,v_2,.....v_n\}$ such that the weight of the edge $(v_i, v_j)$ is  $2|i-j|$. The weight of a minimum spanning tree of $G$ is:

  1. $n-1$
  2. $2n-2$
  3. $\begin{pmatrix} n \\ 2 \end{pmatrix}$
  4. $n^2$
in Algorithms by Active (3.3k points)
edited by | 3.4k views
+11

for a weight of the edge (vi,vj) is 2|i-j|

The weight of MST will be 2(n-1)

for a weight of the edge (vi,vj) is |i-j|

The weight of MST will be (n-1)           // for this question, there is a typo.

+5
Option B should be 2n-2...
+3
It seems this question needs small correction.
0
what will be the case if weight of the edge v(i, j) is | i+j | ?
0
I think it would be [(n+1)(n+2)/2] - 3.

5 Answers

+29 votes
Best answer
$2(n-1)$ the spanning tree will traverse adjacent edges since they contain the least weight.
by Active (3.3k points)
edited by
+1
MST will be line graph.
0

MST will be line graph.

Chhotu  pls explain how?

0
@set2018

MST will add the edges with minimum weight, Weight of edges = 2|i−j|.

All successive number will have weights of 2. (like 1-2 weight is 2 , 2-3 weight is 2 so on) Hence it will be a line graph of weight 2 (n-1). i.e n-1 edges each of weight 2.

Take for example K4 an try.
0
Suppose I take 4 vertices as an example. Then the minimum spanning tree is a line graph with 3 edges of weight 2 each.

now both options B and C gives me the correct minimum weight as 6.
0

@Gupta731

To avoid that take a complete graph of $4$ vertices and then you will only get $\mathbf {B}$ as the answer.

+14 votes
None of the options match the Actual answer.

 For connecting every i & i+1 node we have edge of weight 2 ,therefore We get 2*(n-1).

Correct Answer: $B$
by Boss (41.9k points)
edited by
+1
which is equal to 2n-2
+1 vote

Ans is 2n-2

by Junior (737 points)
+1 vote
$\underline{\mathbf{Answer:B}}\Rightarrow$

$\underline{\mathbf{Explanation:}}\Rightarrow$

Make a square of $4$ vertices and make it as a complete graph, that is, having $6$ edges.

Now, make the spanning tree. It will be the $3$ adjacent edges and the weight will be $2+2+2=6$.

Now, check the options and substitute $\mathbf {n = 4}$, then only option $\mathbf B$ is satisfied.

$\therefore \mathbf B$ will be the answer.
by Boss (19.2k points)
edited by
0 votes

To solve the above question we can consider a complete graph with 3 vertices. 

The weight of minimum spanning tree of a Complete Graph of 3 vertices will be 4 (2 (edge 1,2)+2 (edge 2,3)). 

The only option satisfying G with n=3 is option B (i.e 2n-2). 

by (215 points)
Answer:

Related questions

Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true
50,737 questions
57,385 answers
198,542 comments
105,343 users