Banker's Algorithm Solution

Step 1: Need Matrix

The Need Matrix is calculated as:

Need = Max - Allocation

Step 2: Initial Available Resources

The Available resources at the start are:

• R1 = 4
• R2 = 4
• R3 = 5

Step 3: Evaluate Each Sequence

Sequence (A): P2 → P4 → P1 → P3

• P2: Need = [1, 0, 1], Available = [4, 4, 5]. Allocate resources. New Available = [7, 5, 7].
• P4: Need = [0, 1, 0], Available = [7, 5, 7]. Allocate resources. New Available = [10, 7, 9].
• P1: Need = [4, 5, 2], Available = [10, 7, 9]. Allocate resources. New Available = [14, 8, 11].
• P3: Need = [6, 0, 0], Available = [14, 8, 11]. Allocate resources. New Available = [17, 12, 14].

Result: Sequence (A) is valid.

Sequence (B): P2 → P1 → P3 → P4

• P2: Need = [1, 0, 1], Available = [4, 4, 5]. Allocate resources. New Available = [7, 5, 7].
• P1: Need = [4, 5, 2], Available = [7, 5, 7]. Allocate resources. New Available = [11, 6, 9].
• P3: Need = [6, 0, 0], Available = [11, 6, 9]. Allocate resources. New Available = [14, 10, 12].
• P4: Need = [0, 1, 0], Available = [14, 10, 12]. Allocate resources. New Available = [17, 12, 14].

Result: Sequence (B) is valid.

Sequence (C): P4 → P2 → P3 → P1

• P4: Need = [0, 1, 0], Available = [4, 4, 5]. Allocate resources. New Available = [7, 6, 7].
• P2: Need = [1, 0, 1], Available = [7, 6, 7]. Allocate resources. New Available = [10, 7, 9].
• P3: Need = [6, 0, 0], Available = [10, 7, 9]. Allocate resources. New Available = [14, 8, 11].
• P1: Need = [4, 5, 2], Available = [14, 8, 11]. Allocate resources. New Available = [17, 12, 14].

Result: Sequence (C) is valid.

Sequence (D): P1 → P4 → P2 → P3

• P1: Need = [4, 5, 2], Available = [4, 4, 5]. Cannot proceed as Need > Available.

Result: Sequence (D) is invalid.

Final Answer:

Safe Sequences: (A), (B), (C)

Invalid Sequence: (D)