$($I am getting Vertex $0$ and $1$ wrong for Vertex $0$ i need to ger $4$ and Vertex $1$ i need to get $3$ but for both of thoses vertexes i keep getting no path$)$ This is my code below: #include

#include

#include

#include

using namespace std;

vector dijkstra$($int V$,$ vector$>>$ adj, int S$,$ vector& dist$)\{$

dist.resize$($V$,$ INT$\_$MAX$)$; $//$ dist$[$i$]$ will hold the shortest distance from S to i

vector prev$($V$,-1)$; $//$ prev$[$i$]$ will hold the penultimate vertex in the shortest path from S to i

$//$ Priority queue to store vertices that are being processed

priority$\_$queue, vector$>,$ greater$>>$ pq;

$//$ Initialize source vertex distance as $0$ and push it to the priority queue

dist$[$S$]=0$; $//$ Adjusted: Set distance from source to itself as $0$

pq$.$push$(\{0,$ S$\})$;

$//$ Dijkstra's algorithm

while $(!$pq$.$empty$())\{$

int u $=$ pq$.$top$().$second;

pq$.$pop$()$;

$//$ Visit each neighbor of u

for $($auto& edge : adj$[$u$])\{$

int v $=$ edge.first; $//$ neighbor vertex

int weight $=$ edge.second; $//$ weight of the edge from u to v

$//$ If a shorter path is found, update distance and penultimate vertex

if $($dist$[$u$]+$ weight $<$ dist$[$v$])\{$

dist$[$v$]=$ dist$[$u$]+$ weight;

prev$[$v$]=$ u;

pq$.$push$(\{$dist$[$v$],$ v$\})$;

$\}$

$\}$

$\}$

return prev; $//$ Return the penultimate vertex array

$\}$

int main$()\{$

int v$,$ e$,$ source;

cout $<<$ "Enter the number of vertices and edges: $"$;

cin $>>$ v $>>$ e;

$//$ Initialize the graph with n vertices

vector$>>$ adj$($v$)$;

$//$ Read edges and weights

cout $<<$ "Enter edges in the format $($from to weight$)$:$"<<$ endl;

for $($int i $=0$; i $<$ e; $++$i$)\{$

int from, to$,$ weight;

cin $>>$ from $>>$ to $>>$ weight;

adj$[$from$].$push$\_$back$(\{$to$,$ weight$\})$;

$\}$

cout $<<$ "Enter the source vertex: $"$;

cin $>>$ source;

$//$ Find shortest paths from source to all vertices

vector dist;

vector prev$\_$vertices $=$ dijkstra$($v$,$ adj, source, dist$)$;

$//$ Print shortest paths

cout $<<$ "Shortest paths from source vertex $"<<$ source $<<"$:

$"$;

for $($int i $=0$; i $<$ v; $++$i$)\{$

cout $<<$ "Vertex $"<<$ i $<<"$: $"$;

if $($dist$[$i$]==$ INT$\_$MAX$)\{$

cout $<<"$No path

$"$;

$\}$ else $\{$

cout $<<$ dist$[$i$]<<$ endl;

$\}$

$\}$

return $0$;

$\}$