GATE CS 2019 Q39 — Deadlock

To find the condition that guarantees no process other than $p$ and $q$ can complete, we analyze the resource allocation state step-by-step:

1. Initial State: All instances of resource $R$ are currently in use, which means the number of currently free/available instances of $R$ is $0$.
2. Execution of $p$ and $q$: We are given that $Y_p = Y_q = 0$. This means processes $p$ and $q$ require $0$ additional resources to complete. Since they need no additional resources, they can execute to completion and release the resources they currently hold ($X_p$ and $X_q$ respectively).
3. Available Resources after $p$ and $q$ finish: Once $p$ and $q$ complete, the total available resources in the system will be: $$\text{Available} = X_p + X_q$$
4. Condition for other processes to complete: For any other process $k$ (where $k \neq p$ and $k \neq q$) to complete, the system must be able to satisfy its maximum additional request $Y_k$.
   If the total available resources $X_p + X_q$ is strictly less than the minimum additional requirement of all other processes, then no remaining process can ever satisfy its request: $$X_p + X_q < \text{Min} \{Y_k \mid 1 \leq k \leq n, k \neq p, k \neq q \}$$ Under this condition, even the process with the smallest additional requirement cannot be satisfied, ensuring that no other process can complete execution.

Thus, Option A is the correct condition.