A schedule of three database transactions $T_1$, $T_2$, and $T_3$ is shown. $R_i(A)$ and $W_i(A)$ denote read and write of data item $A$ by transaction $T_i, i = 1,2,3$. The transaction $T_1$ aborts at the end. Which other transaction(s) will be required to be rolled back? $$R_1(X) \quad W_1(Y) \quad R_2(X) \quad R_2(Y) \quad R_3(Y) \quad ABORT(T_1)$$

To determine which transactions must be rolled back, we identify the data dependencies (dirty reads) caused by uncommitted updates:

1. Transaction operations in order:

   • $R_1(X)$: $T_1$ reads $X$.

   • $W_1(Y)$: $T_1$ writes item $Y$.

   • $R_2(X)$: $T_2$ reads item $X$.

   • $R_2(Y)$: $T_2$ reads item $Y$, which was updated by $T_1$. Since $T_1$ has not yet committed, this is a dirty read (a dependency $T_1 \to T_2$).

   • $R_3(Y)$: $T_3$ reads item $Y$, which was updated by $T_1$. This is also a dirty read (a dependency $T_1 \to T_3$).

   • $ABORT(T_1)$: $T_1$ fails and aborts.

2. Cascading Rollback:

   • Since $T_1$ aborts, all transactions that have read uncommitted data written by $T_1$ must also be rolled back to maintain consistency.

   • Both $T_2$ and $T_3$ performed dirty reads on data item $Y$ written by $T_1$.

   • Therefore, both $T_2$ and $T_3$ must be rolled back (cascading abort).

Hence, the correct option is C (Both $T_2$ and $T_3$).