Consider the following algorithm,

Algorithm S$($n$)$

$\{$ if$($n $=1)\{\}$ else $\{$ X $0$: S$($n $-1)$ Y$($n$)\}$

Let the time complexities of XI$)$ and Y$($n$)$ are O$(1)$ and O$(1/$ n$)$ Derive the recurrence relation for the algorithm Sin$)$ and its time complexity.

Which of the following option is correct?

T$($n$)=$constant, if n $=1=1+$ T$($n $-1),$ if n $>1$ and O$($log$($n$))$ respectively

T$($n$)=$constant. if n $=1=1+$ T$($n $-1)+$ T$(1/$n$),$if n$>1$ and theta$($log$($n$))$ respectively

T$($n$)=$constanut. if n $=1=1+$ T$($n $-1)+1/$n $,$ if n $>1$ and theta$($n$)$ respectively

T$($n$)=$constant, if n $=1=1+$ T$($n $-1),$if n$>1$ and Omega$($n$)$ respectively