Suppose the most recent checkpoint was executed at $t_{\text{c}\text{h}\text{e}\text{c}\text{k}}$ and the database system

crashed at $t_{\text{f}\text{a}\text{i}\text{l}}.$ There are four transactions $T_a,T_b,T_c$ and $T_d$ such that:

$T_a$ was committed before $t_{\text{c}\text{h}\text{e}\text{c}\text{k}}$;

$T_b$ was started before $t_{\text{c}\text{h}\text{e}\text{c}\text{k}}$ and committed before system crash;

$T_c$ was started after $t_{\text{c}\text{h}\text{e}\text{c}\text{k}}$ and committed before system crash;

$T_d$ was started after $t_{\text{c}\text{h}\text{e}\text{c}\text{k}}$ and active at the time of system crash.

The above situation is depicted in the following diagram:

$($a$)$ If the deferred update mechanism is used for database recovery, what actions are

taken by the recovery process for those four transactions?

$($b$)$ If the immediate update $($Steal$/$No$-$Force$)$ mechanism is used for database recov$-$

ery, what actions are taken by the recovery process for those four transactions?