Define the $\backslash$alpha number of an n by n matrix A$[1...$n$,1...$n$]$ of integers, n$>=2^$k$,$k$>=0,$ to be the number of element pairs that satisfy the condition $|$A$[$i$,$j$]-$A$[$i$,$j$+1]|>=\backslash$alpha $,1<=$i$<=$n$,1<=$j$<=$n$-1,$ where $\backslash$alpha is a given integer.

a$)$ Design an efficient divide and conquer algorithm to calculate the $\backslash$alpha number of a given n by n matrix.

b$)$ What is the basic operation for your algorithm?

c$)$ Show the recurrence for the number of times your algorithm performs the basic operation in the worst case.

d$)$ Solve the above recurrence using the Master Theorem.

Explain your answer by showing the values of a$,$ b$,$ and d$,$ and indicating which case of the theorem you applied.