Question

Consider a network of 5 nodes located at following coordinate points: sink(0,0), B(0,2), C(2,0), D(2,2) and E(4,0). Assume the transmission power necessary to reach a node at distance d is a+b*d2 (a and b are constants). What is the least common transmission power necessary to form a connected network topology: 1) star topology centered at the sink, and 2) multi-hop topology?

Answer 1

To form a connected network topology, the least common transmission power required for both (1. star topology centered at the sink, and 2. multi-hop topology) would be a + 4b.

Step-by-step explanation:

To form a connected network topology with the least common transmission power, we need to consider two scenarios: star topology centered at the sink and multi-hop topology.

In a star topology, all the nodes are connected directly to a central node, which in this case is the sink.

The transmission power required for the nodes to connect with the sink depends on their distance from the sink.

Let's calculate the transmission power required for each node:

B: Distance from sink = 2 units, Transmission power = a + b*2²= a + 4b

C: Distance from sink = 2 units, Transmission power = a + b*2² = a + 4b

D: Distance from sink = sqrt(8) units, Transmission power = a + b*(sqrt(8))² = a + 8b

E: Distance from sink = 4 units, Transmission power = a + b*4² = a + 16b

1) Therefore, to form a star topology, the least common transmission power required would be a + 4b.

In a multi-hop topology, nodes are connected to each other through intermediate nodes.

We need to find the path with the minimum total transmission power from the source (sink) to the destination (each node).

Let's calculate the total transmission power for each potential path:

Sink -> B: Total transmission power = a + 4b

Sink -> C -> D: Total transmission power = a + 4b + a + 8b = 2a + 12b

Sink -> E: Total transmission power = a + 16b

2) Therefore, in a multi-hop topology, the least common transmission power required would be a + 4b.