Please help me $..$ Consider the algorithm f which takes a positive integer and in turn calls functions f$\_0($n$),$ f$\_1($n$)$ where the running times of these correspond to $\backslash$Theta $($n$)$ and $\backslash$Theta $($n$^2),$ respectively. $[1$ def f$($x s$)$:; $2$ n$=$ len $($x s$)$; $3$ ; $4$ i$=$x s$[$n$-1]$; $5$ if i $\%2=0$:; $6$ return f$\_-0($n$)$; $7$ return f$\_-1($n$)$; $8($n$)]$ In the above function, i $\%2$ denotes the remainder after dividing n by $2.$ It is $0$ when i is even and $1$ when i is odd. $($a$)$ What is the best case asymptotic running time for the above function? Note that since I am asking for best case, you may assume n is restricted to the class of input that makes the above algorithm run the fastest. For this class of input the algorithm runs in time $\backslash$Theta $($something$).$ Explain your answer. $($b$)$ Analogously, what is the worst case asymptotic running time? $($c$)$ Give the best possible upper bound on the asymptotic running time for arbitrary n$.$ This should be denoted with O$,\backslash$Omega $,$ or $\backslash$Theta $($as appropriate$).$ Explain your answer. $($d$)$ Analogously, give the best possible lower bound on the asymptotic running time for arbitrary n$.($e$)$ Suppose, instead, that all you know is that the running time of f$\_0($n$)$ is in $\backslash$Omega $($n$),$ while the running time of f$\_1($n$)$ remains $\backslash$Theta $($n$^2).$ What are the best lower and upper bounds that you can give on the asymptotic running time for best case, worst case, and general n $?$