$1.$ Hash Map Initialization: Start by creating a hash map $($map$\_$v$)$ that maps

each price in array $$V$$ to its index. This allows us to quickly lookup

whether a specific price exists in $$V$.$

$2.$ Iterate Over Pairs:

$$ Use two nested loops to iterate over all pairs $($i$,$ j$)$ where $$i$$ is an

index in array $$Y$$ and $$j$$ is an index in array $$F$.$

$$ For each pair $($i$,$ j$),$ calculate the sum $$sum $=$ Y$[$i$]+$ F$[$j$].$

$3.$ Binary Search for Matching Price:

$$ For each calculated sum $$sum$,$ check if $$sum$$ exists in $$map$\_$v$.$

$$ If $$sum$$ exists, retrieve its index from $$map$\_$v$.$

$4.$ Check Validity: Ensure that the retrieved index from $$map$\_$v$$ is different

from $$i$$ and $$j$$ to satisfy the condition that you and your friend are both

contributing at least one coin.

$5.$ Return Result:

$$ If a valid index $$k$$ is found that satisfies the conditions, return $$true$.$

$$ If no such triplet $($i$,$ j$,$ k$)$ is found after checking all pairs, return

$$false$.$

Correctness Proof:

$$ Hash Map Usage: By using a hash map for $$V$,$ we achieve O$(1)$ average$-$

time complexity for lookups, ensuring efficiency in checking whether a

sum exists.

$$ Binary Search: The binary search operation $($using $$lower$\_$bound$$ or sim$-$

ilar$)$ for finding the index in $$V$$ ensures that each lookup operation is

efficient within the nested loop structure.

Time Complexity Analysis:

$$ Hash Map Construction: Building $$map$\_$v$$ takes O$($n$)$ time.

$$ Nested Loops: There are O$($n$2)$ pairs $($i$,$ j$)$ to consider.

$$ Binary Search: Each binary search operation within the nested loop is