Consider the three transactions T$1,$ T$2,$ and T$3,$ and the schedules S$1$ and S$1$ given below. Draw the serializability $($precedence$)$ graphs for S$1$ and S$2,$ and state whether each schedule in serializable or not. If a schedule is serializable, write down the equivalent serial schedule$(5)$

$10$ marks

T$1$: rl $($P$)$; rl $($R$)$; wl $($P$)$; T$2$: $12($R$)$; r$2($Q$)$; w$2($R$)$; w$2($R$)$; T$3$: r$3($P$)$; r$3($Q$)$; w$3($Q$)$; S$1$: rl $($P$)$; r$2($R$)$; r$1($R$)$; r$3($P$)$; r$3($Q$)$; w$1($P$)$; w$3($Q$)$; r$2($Q$)$; w$2($R$)$; w$2($Q$)$: S$2$: rl $($P$)$; r$2($R$)$; r$3($P$)$; rl $($R$)$; r$2($Q$)$; r$3($Q$)$; w$1($P$)$; w$2($R$)$; w$3($Q$)$; w$2$

a$.$ Draw the serializability $($precedence$)$ graphs for S$1$ and S$2.4$ marks

b$.$ State whether each of the schedules S$1$ and S$2$ is serializable or not and why? $4$ marks

c$.$ If a schedule is serializable, write down the equivalent serial schedule$($s$).2$ marks