Q$4.$ Given an $mn$ boolean matrix B$,$ design an algorithm to find its largest square submatrix whose elements are all zeros.

Hint $1$: Your solution should consider the $0$ cells in the matrix, and look at the optimal sub$-$solutions in three directions.

Hint $2$: Your solution may contain a min function.

$($a$)$ Design a dynamic programming algorithm. Provide the recursive relation of your algorithm.

$($b$)$ Provide the time complexity and compare it to the exhaustive search approach.

$($c$)$ Apply the algorithm to the matrix example below show your steps $($Note: your final answer should be the length of the diagonal of the largest square sub$-$matrix, which is $3.$

$\backslash$table$[[1,2,0,0,0],[4,5,0,0,0],[0,9,0,0,0]]$