There exists a very slowly growing function $\backslash$alpha such that m union$-$find operations on n elements can be done in time O$($m $\backslash$alpha $($n$)).$ That function $\backslash$alpha can be defined in terms of the Ackermann function, by $\backslash$alpha $($k$)=$ smallest natural number n such that Ackermann$($n$,$ n$)>=$ k$.$

a$.$ What is $\backslash$alpha $(5)?$

b$.$ What is $\backslash$alpha $(50)?$

c$.$ What is $\backslash$alpha $(500)?$

d$.$ What is $\backslash$alpha $(500,000)?$

e$.$ What is $\backslash$alpha $(500,000,000)?$