Question

A stage of an \(n \times n\) banyan network consists of \((n / 2) 2 \times 2\) switching elements. The first stage directs packets to the correct half of the network, the next stage to the correct quarter, and so on, until the packet is routed to the correct output. Derive an expression for the number of \(2 \times 2\) switching elements needed to make an \(n \times n\) banyan network. Verify your answer for \(n=8\).

Short answer

The number of 2x2 switching elements is \( \frac{n}{2} \times \text{log}_2(n) \). For \( n = 8 \), 12 switching elements are required.

Step-by-step solution

- Understanding the Banyan Network

A banyan network is a specific type of data routing network designed to direct data packets to specific outputs in multiple stages. Each stage consists of \( \frac{n}{2} \) 2x2 switching elements. The number of stages required is determined by the number of times the network needs to be halved to reach a single output line.

- Determine the Number of Stages

To find the number of stages, recognize that the routing continues until each packet reaches a single output. Since each stage divides the network into halves, there will be \( \text{log}_2(n) \) stages, where \( n \) is the total number of outputs.

- Number of 2x2 Switching Elements per Stage

Each stage consists of \( \frac{n}{2} \) switching elements. With \( \text{log}_2(n) \) being the number of stages, we compute the number of 2x2 switching elements required for each stage.

- Derive the Total Number of Switching Elements

To determine the total number of 2x2 switching elements in the entire network, multiply the number of switching elements per stage by the total number of stages: \[ \text{Total switching elements} = \frac{n}{2} \times \text{log}_2(n) \]

- Verify for n=8

Substitute \( n = 8 \) into the derived expression: \[ \frac{8}{2} \times \text{log}_2(8) = 4 \times 3 = 12 \]. Therefore, a banyan network with 8 inputs and 8 outputs needs 12 switching elements.

Key concepts

Switching Elements

Switching elements play a crucial role in data routing by directing data packets to their correct destinations. In a banyan network, we use 2x2 switching elements, which means each element has 2 input ports and 2 output ports. These elements can switch data packets from any of the two inputs to any of the two outputs.

The key functions of switching elements are:

• Directing packets to the appropriate output based on pre-established rules.
• Ensuring smooth transfer of packets without collision.

The number of switching elements needed in each stage of a banyan network is essential for determining the efficiency and capacity of the network.

Network Stages

A banyan network consists of multiple stages, where each stage helps in progressively routing the data packets closer to their final output. Each stage halves the packet paths, directing them to the appropriate segment of the next stage.

Starting from the first stage:

• The first stage directs packets to the correct half of the network.
• The second stage directs packets to the correct quarter.
• This process continues until the packets reach their designated output ports.

These stages play a vital role in systematically breaking down the routing process, ensuring that each packet reaches its correct destination through a series of logical steps.

Logarithmic Stages

The term 'logarithmic stages' refers to the number of stages required in a banyan network. The number of stages needed is determined by how many times the network needs to be halved, represented mathematically as \(\text{log}_2(n)\), where \( n \) is the total number of outputs.

For example:

• If \( n = 8 \), then \( \text{log}_2(8) = 3 \), so the network will have 3 stages.
• If \( n = 16 \), then \( \text{log}_2(16) = 4 \), so the network will have 4 stages.

This logarithmic relationship ensures that the network remains efficient, as each additional stage exponentially increases the number of possible outputs.

Data Routing

In a banyan network, data routing is the process of directing data packets from their input to the desired output. This is achieved by passing the packets through a series of switching elements and stages.

The routing strategy involves:

• Using switching elements to decide the direction of each data packet at every stage.
• Ensuring each packet is routed towards its correct output by successive stage halving.

Effective data routing in a banyan network means that each packet must take the shortest possible path without any conflicts or delays, ensuring high efficiency and accurate delivery of data.