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Fill in the blanks in the following template of an algorithm to compute all pairs shortest path lengths in a directed graph $G$ with $n*n$ adjacency matrix $A$. $A[i,j]$ equals $1$ if there is an edge in $G$ from $i$ to $j$, and $0$ otherwise. Your aim in filling in the blanks is to ensure that the algorithm is correct.

INITIALIZATION: For i = 1 ... n
{For j = 1 ... n
{ if a[i,j] = 0 then P[i,j] =_______ else P[i,j] =_______;}
}

ALGORITHM: For i = 1 ... n
{For j = 1 ... n
{For k = 1 ... n
{P[__,__] = min{_______,______}; }
}
}    
1. Copy the complete line containing the blanks in the Initialization step and fill in the blanks.
2. Copy the complete line containing the blanks in the Algorithm step and fill in the blanks.
3. Fill in the blank: The running time of the Algorithm is $O$(___).

here we are not given any information about the weights between the edges, then how can we solve it. The only way I think it can be solved is when A[i,j]= Weight[i,j] if there is an edge in G from i to j.

@Bikram

P[  j , k ] = min(  p[ j , k ] ,  p[ j , i ] +   p[  i , k ] ) }

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INITIALIZATION:
For i = 1 ... n {
For j = 1 ... n {
if a[i,j] = 0 then P[i,j] = infinite
// i.e. if there is no direct path then put path length as infinite
else P[i,j] =a[i,j];
}
}

ALGORITHM:
For i = 1 ... n { //i loops over the intermediate vertices
For j = 1 ... n {
For k = 1 ... n {
P[j, k] = min( p[j,k] , p[j,i] + p[i,k]);
}
}
}              

Time complexity $O(n^3)$

This algorithm is for weighted graph but it will work for unweighted graph too because if $p[i,j]=1$, $p[i,k]=1$ and $p[k,j]=1$  then according to the algorithm $p[i,j] = \min(p[i,j] ,p[i,k] + p[k,j]) = \min(1,2) =1$

And all the other cases are also satisfied. $($like if $p[i,j]$ was $0$ in last iteration and there exist a path via $k)$

by

ALGORITHM:

For i = 1 ... n

{For j = 1 ...

{For k = 1 ...

{ P[  j , k ] = min(  p[ j , k ] ,  p[ j , i ] +   p[  i , k ] ) };

}

}

@prashant , edit the ans

what to edit?

is the last line should change by

P[  j , k ] = min(  p[ j , k ] ,  p[ j , i ] +   p[  i , k ] )

but why we take first point as j ??

it has edited ,.. now correct ..
The comment by himanshu is the standard way of implementing the all pairs shortest path.First loop gives intermediate vertex.

Although given answer also seems correct
It's wrong. This will give incorrect answers. @Himanshu1 is correct.

@Rishabh Gupta 2  Sir

Could you please explain why the above one  is wrong. In the Cormen book ,

p[i,j]=min{p[i,j],p[i,k]+p[k,j]} is given.

why are diagonal elements not initialized to zeroes?
what is wrong in that? It is same as in cormen

@Arjun Sir

Plz tell me which code is correct here. I am not getting, what is wrong in answer.

edited by

yes @Himanshu1 is correct here. First create an edge with inner most two loops and then checking an external point with outer most loop. discuss

And all the other cases are also satisfied. (like if p[i,j] was 0 in last iteration and there exist a path via k)

Can anybody make me understand this last line in the expalanation...

I'll try to answer in order to provide maximum clarity.

INITIALIZATION: For i = 1 ... n
{For j = 1 ... n
{ if a[i,j] = 0 then P[i,j] =_______ else P[i,j] =_______;}
}

We, intialise our final distance matrix initially with the Adjacency matrix of the graph. If we have a path from vertex i to j, then we define it's cost, otherwise we set it to infinite.

so

if a[i,j] = 0 (There is no direct path from vertex i to j)  then P[i,j] =0 else P[i,j]=a[i][j] (Path cost to reach vertex j from i);

Now comes the second part

ALGORITHM: For i = 1 ... n //Consider each vertex to be an intermediate vertex.
{For j = 1 ... n
{For k = 1 ... n
{
for Each pair of vertices (j,k), find the minimum path to reach k from j and this minimum path has 2 ways
(1)Either directly go from j to k or
(2)Using i as an intermediate vertex, go to vertex i from j and then go to k from i.
Find minimum of above two.
The ith loop considers for each path, the intermediate vertex that can give minimum cost path to reach k from j.
P[j,k] = min{P[j][k],p[j][i]+p[i][k]}; }
}
}    

Running time of our algorithm is $O(V^3)$ where V is the number of the vertices in the graph.

### 1 comment

As it is given 0 or 1 we donot consider the edge weights here so finally it boils down to a boolean  variable having 0 or 1

so  can this

P[j,k] = min{P[j][k],p[j][i]+p[i][k];

be not written as:

P[j,k] = min{P[j][k],p[j][i] & p[i][k]);

&=logical 'and'.

It is a Floyd Warshall algorithm(Dynamic Programming approach).

for i = 1 to N
for j = 1 to N
if there is an edge from i to j
dist[i][j] = the length of the edge from i to j
else           dist[i][j] = INFINITY
for k = 1 to N
for i = 1 to N
for j = 1 to N
dist[k][i][j] = min(dist[k-1][i][j], dist[k-1][i][k] + dist[k-1][k][j])
Time Complexity: $O(n^3)$

References:
by