Consider the shortest paths problem from a single source s$.$ The key idea underlying Dijkstra's algorithm is the following:

For every q$,$ let W$($q$)$ denote our current best upper bound on the weight of a shortest$-$weight path from s to q$.$ Consider a neighbor v of u$.$

If W$($u$)+$w$($u$,$v$)<$ W$($v$)$ we should set W$($v$)$ to W$($u$)+$w$($u$,$v$)$ # Here w$($u$,$v$)$ is the weight on edge $\{$u$,$ v$\}$

Now consider a variant of the single$-$source shortest$-$paths problem in which paths containing more than k edges do not qualify. Consider the following variant of the aforementioned idea around which we hope to develop an optimal algorithm for this new algorithm.

For every q$,$ let W$($q$)$ denote our current best upper bound on the weight of a shortest path from s to q under the added constraint that it contains no more than k edges and let j$($q$)$ denote the actual number of edges in a path from which W$($q$)$ was obtained. Consider a neighbor v of u$.$

If W$($u$)+$w$($u$,$v$)<$ W$($v$)$ and j$($u$)<$ k we should set W$($v$)$ to W$($u$)+$w$($u$,$v$)$ and j$($v$)$ to j$($u$)+1.$

Critique this new idea from the perspective of whether or not we can build an optimal algorithm for the new problem around it$.$