GATE CS 2019 Q49 — Data Link Layer: Framing, Error Detection, MAC, Ethernet Bridging

To find the minimum number of switches needed, we can analyze the relationship between the number of switches, the ports used for interconnecting them, and the ports available for the machines.

Let $n$ be the number of $8$-port switches used.

• Each switch has $8$ ports. For $n$ switches, the total number of ports is $8n$.
• To connect $n$ switches together in a single network (without loops), we need at least $n - 1$ links between them.
• Since there are no dedicated uplink ports, each inter-switch link connects a standard port on one switch to a standard port on another switch. Thus, each link consumes $2$ ports.
• Therefore, $n - 1$ links will consume $2(n - 1)$ ports.

The number of ports remaining to connect the $15$ machines is:

Since we need to connect $15$ machines, the number of available ports must be at least $15$:

Since the number of switches $n$ must be an integer, we round up to the next whole number:

Thus, the minimum number of switches needed is 3.