modification of dijkstras based on cost and quality - r

Question

I have a graph with each edge having cost and quality. I need to modify the dijkstras to find the path with the highest quality - but if the quality of two path is the same, then the path with the least cost should be chosen.
Initially, I was using dijkstras to find the path with minimum cost (code is pasted below).
Is it possible to modify those dijkstras in the above mentioned way?
If not, then please suggest another way to achieve this.
R code:
dijs<-function(n,v,cost,dist)
{
dist<-numeric(n)
flag<- numeric(n)
prev<-numeric(n)
for(i in 1:n)
prev[i] = -1
for(i in 1:n)
dist[i]<-cost[v,i]
count=2
while(count <= n)
{
min=999
for(w in 1:n)
{
if(dist[w] < min && !flag[w])
{
min=dist[w]
u=w
}
}
flag[u]=1
count<-count+1
for(w in 1:n)
{
if((dist[u]+cost[u,w] < dist[w]) && !flag[w])
{
dist[w]=dist[u]+cost[u,w]
prev[w]=u
}
}
}
printmin(v,dist,n)
return(prev)
}
main<-function()
{
cat('Enter no of nodes:', '\n')
n<-scan("",n=1,quiet=TRUE)
cat('Enter cost matrix','\n')
cost<-matrix(0,n,n)
for(i in 1:n) for(j in 1:n)
{
if(i == j)
cost[i,j]<-999
if(i != j && cost[i,j] == 0)
{
cat(sprintf("enter the cost from node %d to %d",i,j))
cost[i,j]<-scan("",n=1,quiet=TRUE)
if(cost[i,j] == 0)
cost[i,j]=999
cost[j,i]<-cost[i,j]
}
}
print(cost)
print('Enter the source:',quote=FALSE)
v<-scan("",n=1,quiet=TRUE)
prev<-digs(n,v,cost,dest)
print("the shortest distance")
for(i in 1:n)
{
cat(sprintf("path to %d ->",i))
printpath(i,prev)
cat('\n')
}
}
printmin<-function(v,mindist,n)
{
for(i in 1:n)
{
if(i != v)
{
cat(sprintf("%d -> %d, cost =%f",v,i,mindist[i]))
cat('\n')
}
}
}
printpath<-function(dest,prev)
{
if(prev[dest] != -1)
printpath(prev[dest],prev)
cat(sprintf("%d ",dest))
}

Answer 1

Assuming the quality that you use as the cost does not lead to negative cycles, I think the easiest way to do so is to compute all shortest path between source and dest and then rank all paths according to your second objective. You can use the default Dijkstra implementation to compute all equivalent shortest paths according to the first objective, just keep exploring the queue (instead of stopping when you reach the goal) until the next path taken from the queue is bigger than the optimal one. Then, compute the costs for each min path using your second objective and rank them using a sorting algorithm.
If you want to modify the original Dijkstra to compare the cost using prioritized objectives (first quality then cost or vice-versa) then you must prove that your modified version is optimal and complete