This question is on alpha$-$beta pruning. Here are a few reminders:

alpha is the best value so far for MAX. It starts at $-$oo and can increase as the search progresses

beta is the best value so far for MIN. It starts at $+$oo and can decrease as the search progresses

At a MAX node

if the value v of a child is v $>=$ beta

then prune and return the value v

else alpha $=$max$($ alpha, v$)$

At a MIN node

if the value v of a child is v alpha

then prune and return the value v

else beta $=$min$($ beta, v$)$

Examine the branches in the tree from left to right, show the backed$-$up values and the branches pruned. For

each branch pruned, write the condition used to do the pruning.

A drawback of A$*$ is its memory requirement since the frontier might get very large. Suppose you modify A$*$ as follows: You obtain by some method a path to a goal node and its associated cost f$($Goal$)=$C$.$ This cost is not necessarily minimal but it gives an upper bound on the minimal cost. $$Now use A$*$ with an admissible h function and discard immediately any frontier nodes reached whose f values are greater than C$.$ Answer these questions, explaining your reasoning.

Is this modified A$*$ algorithm guaranteed to find an optimal solution if one exists or not?$$

Since the algorithm might discard nodes in the frontier, does it mean that fewer nodes are expanded?$$

Does this modification reduce the total storage requirements?