GATE CSE 2016 Set 1 | Question: 38

Question

Consider the weighted undirected graph with $4$ vertices, where the weight of edge $\{i,j\}$ is given by the entry $W_{ij}$ in the matrix $W$.

 W=$\begin{bmatrix} 0&2 &8 &5 \\ 2&0 &5 &8 \\ 8&5 &0 &x \\ 5& 8 &x &0 \end{bmatrix}$

The largest possible integer value of $x$, for which at least one shortest path between some pair of vertices will contain the edge with weight $x$ is ___________.

Comments

If you see the line “at least one shortest path between some pair of vertices” in question it means that there can be more than 1 shortest path between pair of vertices and out of that n shortest path at least one of them uses edge x. In that way 12 goes as the answer. If word guaranteed  is used in question then we might had to go with 11.

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No Dipak you are not reading it in context I think, see they have said basically you choose the largest value of ‘x’, and then get shortest path between each vertex, (not in all possible ways you can get shortest path between each vertex) for that chosen value of ‘x’ , so that x will edge in some vertex shortest path, which in turn means means give me value of ‘x’ for which it is guaranteed to be used in shortest path between some pair of vertex of the graph

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Revision of the single shortest Path NPTEL
https://youtu.be/VQqVDNCJ1jA

Answer 1

Let us list down the shortest edge between each pair of vertices $x,y$ in the graph

$$\begin{array}{|c|c|c|}\hline \textbf{x} & \textbf{y} & \textbf{shortest path from x to  y} \\\hline  \text{1} & \text{2} & \text{2} \\\hline   \text{1} & \text{3} & \text{there is a direct path from 1 to 3 of weight 8.We can also choose to go}\\ && \text{ via node 4 with total path weight $5+x.$ if $5+x<8(x<3)$ then shortest}\\&&\text{path is $5+x$ otherwise shortest path is the weight of the direct path}\\ && \text{which is 8} \\\hline  \text{1} & \text{4} & \text{5} \\\hline \text{2} & \text{3} & \text{5} \\\hline  \text{2} & \text{4} & \text{there is a direct path from 2 to 4 of weight 8. We can also choose to go}\\ && \text{via node 3 with total path weight}{ 5+x}\text{. If }{ 5+x<8(x<3)}\text{ the shortest}\\ && \text{path is 5+x. otherwise shortest path is the weight of the direct path}\\ && \text{which is 8} \\\hline  \text{3} & \text{4} & \text{We can chose to go through the direct path of weight x or via nodes 2, 1 }\\ && \text{to node 4 with weight } 5+2+5=12 \text{. If }{x<12}\text{ then we will chose x to be the shortest}\\&& \text{path otherwise $12$ is the shortest path} \ \\\hline \end{array}$$ 

• Case 1: Let us say $x < 3$. Say $x = 2$.

When we put $x = 2$ the above table is modified as

$$\begin{array}{|c|c|c|}\hline \textbf{x} & \textbf{y} & \textbf{shortest path from x to y} \\\hline  \text{1} & \text{2} & \text{2} \\\hline   \text{1} & \text{3} & \text{7} \\\hline  \text{1} & \text{4} & \text{5} \\\hline \text{2} & \text{3} & \text{5} \\\hline  \text{2} & \text{4} & \text{7} \\\hline  \text{3} & \text{4} & \text{2 // Note that the shortest path between nodes $3$ and $4$ is $x =2$} \\\hline  \end{array}$$

• Case 2: $3 \leq x < 13$. Let's say $x = 12$. The table is now modified as

$$\begin{array}{|c|c|c|}\hline \textbf{x} & \textbf{y} & \textbf{shortest path from x to y} \\\hline  \text{1} & \text{2} & \text{2} \\\hline   \text{1} & \text{3} & \text{8} \\\hline  \text{1} & \text{4} & \text{5} \\\hline \text{2} & \text{3} & \text{5} \\\hline  \text{2} & \text{4} & \text{8} \\\hline  \text{3} & \text{4} & \text{12 // Note that the shortest path between nodes $3$ and $4$ is $x=12$} \\ && \text{and one of the shortest path is the direct edge $x$} \\ \hline   \end{array}$$ 

Now the question asks you to find the largest possible integer value of $x$ such that shortest path between at least one pair of nodes in the graph is equal to $x.$ For values $x = 2,3,4,\ldots,12$ the shortest path between node $3$ and $4$ is equal to $x$.

The largest among this is $x = 12$. So the answer is $12$

PS:  If the question is modified as "The largest possible integer value of $x$, for which the shortest path between some pair of vertices is guaranteed to contain the edge with weight $x$ is"
then the answer will be $11.$

Correct Answer: $12.$

Comments

best explanation

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y can't 13 be the answer ?

In the above graph of jankyMurthy..label nodes 1=A 2=B 3=C and 4=D.

Now u can take path from C to D as direct edge with weight 13 or indirect from C to A(5) and A to D(8).. which is again 13...so why not 13 can be the maximum weight ?

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A correction: 

The shortest distance between node 3 and node 4 is 12 (via node4, node1, node2, node3) or x (via direct path). So if you keep x=12, at least one shortest path between some pair of vertices will contain the edge with weight x.
 

If you consider the shortest path as 13 (5+8 or 8+5), then answer would be 13, which is not the case.

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@nitesh Bcz from 3 -> 2 ->1-> 4 is a path with weight 12

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Ya i got it,my bad..

thanks for pointing that path

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Does this question use the concept of Floyd Warshal's algo?

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By x=2 also gives at least one shortest path. So why not x=2?

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Because the question asks for the largest possible integer value of x.

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The answer needs editing here:

Shortest Path from 1 to 3 : 7 (1-2-3)

Shortest Path from 2 to 4 : 7 (2-1-4)

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but $12$ does not ensure that it will be included in shortest path because

$3\rightarrow2 \rightarrow1\rightarrow4$ also gives a way of getting $12$.
Taking $x=11$ ensuring it will taken why we are taking $12$?

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Hello saxena same thing perplexed me.

see this graph

Here if question is just same as our original question like "largest value of $w$ such that at least one shortest path b/w some pair of vertices will contain this edge with weight $w$"

So confusion is b/w $w=5$ or $w=6$. when $w=6$ then to reach vertex $2$ to vertex $3$ there are two shortest paths , either $2\rightarrow 1\rightarrow 3$ or $2\rightarrow 3$.

Now see questioner is asking like , 'i don't care if there are multiple shortest paths b/w some pair of vertices but at least one out of those must contain this edge with weight '$w$'. so in that way , if we have to tell maximum possible value of $w$ then it's $6$ in that case.

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very well explained.

My doubt is: how to ensure that atleast 1 is guaranteed? like for x=12 it may still go thorugh other path 3-2-1-4 which is also 12? is it like if there are multiple long path and another direct path with same cost then Floud Warshall always choses the direct path during algo application?



as per above pic, the weights are initialized to the edge cost, now they are updated only when the other intermediate is smaller (not equal) than it.

so here even 12 will guarantee that x is chosen?

am i right?

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Classic solution :)

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The answer should be 11, right ??

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precise explanation thanks;

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Please correct your ans. It is right ans with mistakes in calculations. Shortest path from 3-4 is of length 12 and not 13(path being 3-2-1-4). fix this solution someone

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While going from vertex 1 to 3 and taking path of 1-4-3 we get 5+x.Why does value of x in above mentioned ans is taken as,less than 3 and not equal to 3.

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Isn’t shortest path from vertex 2 to 4 is 7? 2->1->4 will give shortest path of 7 instead of 2->4 which will be 8.

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Yes, it’s 7

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“

 If the question is modified as "The largest possible integer value of xx, for which the shortest path between some pair of vertices is guaranteed to contain the edge with weight xx is"
then the answer will be 11. – why is this the case ??

pls help

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because you will have multiple paths if the value of x is 12, so you cant guarantee which path is selected

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@Sachin Mittal 1 Sir @Deepak Poonia Sir @Lakshman Bhaiya bhaiya Please edit the answer . 

The answer needs editing here:

Shortest Path from 1 to 3 is 7  . The path is (1-2-3)

Shortest Path from 2 to 4 is 7 . The path is (2-1-4)

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By brute force method we can get solution easily

Answer 2

Answer is $x=12$.  Here, when we read the last sentence of the question, i.e the largest possible integer value of $x$. when $x=11$, the shortest path is an edge with weight $x$ only. But when $x=12$, there are $2$ shortest paths, and we can say that the edge with weight $x$ is also the shortest path. At least $1$ shortest path contain edge $x$. So the answer will be $12$.

Comments

why not 4.after 8 then two path 5 and x.so x=4

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In question this line( at least one shortest path between some pair of vertices will contain the edge with weight xx) should not have written edges/edge instead of edge only ? Because shortest path may contain more than one edge and the weight of all edges have to be x .

 

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What if question ask for minimum value of X?

Answer 3

ans is simple....if we want to get shortest path between c and d without x its 8+5=13 so to get the shortest path between c and d we can include weight 12 which is maximum...correct me if i am wrong...

Comments

the shortest path between $c$ to $d$ from your dig is $c-b-a-d = 12.$
The ques is not that straight, and it says that for what largest value of $x$, at least one shortest path will contain $x$. If we take $x = 12$, there will be two shortest paths between $c$ and $d$, ie. $c-b-a-d$ and $c-d$ itself.so taking $x= 12$, "at least one shortest path between any pair of vertices" may belong to it.

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you are being corrected : the shortest path between c and d is not 13 but 12 (c-b-a-d) if you don't consider x

Answer 4

For the people who didn't understand the above explanations ( although going through them again will make it clear) I'm putting in simple words.

We're asked to find value$'x'$ so that shortest path between any two vertices contains the edge with cost $'x'.$

Let us verify for the vertices $3$ and $4.$

To travel from vertex $3$ to vertex $4$: we have $3$ possible paths.

$1.$ Path $3-1-4$ with cost$=8+5=13$

$2.$ Path $3-2-4$ with cost$=5+8=13$

$3.$ Edge $3-4$ with cost$=x$

So, for the edge $3-4$ to be the shortest path between vertices $3$ and $4$, $x$ should be less than $13$.

Since we're asked to find the maximum value for $x$, the answer is $12.$

Comments

why not considering 3-2-1-4 which is 12?

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 Yeah, even that path is also the shortest path. But it is given that,

The largest possible integer value of x, for which at least one shortest path between some pair of vertices will contain the edge with weight x is

Between the pair of vertices 3 & 4, Path 3-2-1-4 is the shortest path with cost 12. But this path doesn't contain the edge with weight x. That's why we considered the path 3-4.