I'd like to expand on this question :
Dijkstra's algorithm uses cost function $f(n) = g(n)$ whereas A* uses cost function $f(n) = g(n) + h(n)$, with $g(n)$ being the cost of the path from the start node to node $n$, and $h(n)$ is a heuristic function that estimates the cost of the cheapest path from node $n$ to the goal.
It is clear from the linked question's answer that A* needs its $g(n)$ function in the cost function. My question however is the following. Can one use the cost function :
$f(n) = \alpha g(n) + (1-\alpha)h(n)$
for some alpha $0<\alpha<1$ ?
I ask because in some cases I observed it can be much faster to prioritize (through a coefficient) estimated cost over already traversed cost. I am not sure however if this still results in an optimal path?
EDIT : multiplying the heuristic $h(n)$ by some alpha $0<\alpha<1$ is allowed, since this operation still underestimates if $h(n)$ already did (which is necessary to obtain the resulting optimal path). I am more concerned about the multiplying of $g(n)$.