Your task: given three arrays $Y,F,$ and $V($You$,$ Friend, Vending machine$)$

each containing $n$ positive integers, decide if there exist $Y[i]+F[j]=V[k]O(n^{2}logn)Y=[3,2,1],F=[4,5,6],V=[50,8,13]Y[1]+F[2]=V[1]2+6+8YFV=[50,2123,9123]i,jkY[i]+F[j]=V[k]0i,j,k$ such that

$Y[i]+F[j]=V[k].$ You can assume that all integers $in$ all arrays are distinct $(also$

between different arrays$).$ Since you want $to$ eat soon, you want $an$ algorithm for

this task which solves your problem fast: running $in(expected)$ time $O(n^{2}logn).$

Example: $$

$Y=[3,2,1],F=[4,5,6],V=[50,8,13]$

$We$ need $to$ return true, since $Y[1]+F[2]=V[1](i.e.,2+6+8).$

For the same $Y$ and $F,$ but with $V=[50,2123,9123],$ we'd return false, $as$ there

are $noi,j,$ and $k$ such that $Y[i]+F[j]=V[k].$

$AI$ Output:

Algorithm Description: Hash Map Initialization: Start by creating a hash map $(\text{'}$map$\_$v$\text{'})$ that maps

each price in array $\text{'}$ V $\text{'}$ to its index. This allows us to quickly lookup

whether a specific price exists in $\text{'}$ V $\text{'}.$

Iterate Over Pairs:

Use two nested loops to iterate over all pairs $\text{'}(i,j$ where $\text{'}i\text{'}$ is an

index in array $\text{'}$ Y $\text{'}$ and $\text{'}$ j $\text{'}$ is an index in array $\text{'}$ F $\text{'}.$

For each pair $\text{'}(i,j{)}^{\text{'}}\text{'},$ calculate the sum 'sum $=Y[i]+F[j]\text{'}.$

Binary Search for Matching Price:

For each calculated sum 'sum', check if 'sum' exists in 'map$\_$v$\text{'}.$

If 'sum' exists, retrieve its index from 'map$\_$v$\text{'}.$

Check Validity: Ensure that the retrieved index from 'map$\_$v$\text{'}$ is different

from $\text{'}i\text{'}$ and $\text{'}j\text{'}$ to satisfy the condition that you and your friend are both

contributing at least one coin.

Return Result:

If a valid index $\text{'}$ k $\text{'}$ is found that satisfies the conditions, return 'true'.

If no such triplet $\text{'}(i,j,k\text{'}$ is found after checking all pairs, return

'false'.

Correctness Proof:

Hash Map Usage: By using a hash map for $\text{'}{V}^{\text{'}},$ 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 $\text{'}$ V $\text{'}$ ensures that each lookup operation is

efficient within the nested loop structure.

Time Complexity Analysis:

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

Nested Loops: There are $O(n^2)$ pairs $\text{'}(i,j)\text{'}$ to consider.

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

$O(logn).$

Therefore, the overall time complexity is $O(n^{2}logn),$ dominated by the nested

loops and the binary search operations.

Your task: To analyse the above response consider each of the following ques$-$

tions and briefly explain your answer to each.

a$)$ Does the AI response answer the question, i$.$e$.,$ does the algorithm solve

the stated problem and does it run in the required time?

b$)$ Is the algorithm described in sufficient detail, i$.$e$.,$ is all information present

or are any parts unclear or missing?

c$)$ Is the correctness argument complete or are there leaps or errors in the

logic?

d$)$ Is the running time analysis correct for the provided algorithm?