Chapter 16.4, Problem 15WE

To determine

To find: the matrix that represents the number of ways a message can be sent from one point to another using exactly two relays.

Answer to Problem 15WE

  $\begin{array}{c}A\\ B\\ C\\ D\end{array}\left[\begin{array}{cccc}0& 2& 0& 2\\ 2& 0& 0& 0\\ 0& 2& 0& 2\\ 2& 0& 0& 0\end{array}\right]$

Explanation of Solution

Given information:

Use the communication network at the right for figure,

  

Calculation:

Write the matrix that illustrates the given network first. Let $X$ be the matrix.

The arrows indicate the directions in which data can be transmitted and received. In the given network, see that point $A$ can transmit directly to points $B$ and $D$ . Point $A$ can receive directly from points $B$ and $D$ . Point $B$ can transmit directly to point $A$ only.

Point $B$ can receive directly from stations $A$ and $C$ . Point $C$ can transmit directly to points $B$ and D, whereas it cannot receive directly from any point. Point $D$ can transmit directly to point $A$ only, but it can receive directly from $C$ and $A$ .

The rows of matrix $X$ should represent the directions in which data can be transmitted and the columns should represent the sources from which data may be received. Use a “1” if direct transmission is possible and use a “0” if transmission is not possible.

Thus, the matrix that illustrates the network is,

  $X=\begin{array}{c}A\\ B\\ C\\ D\end{array}\left[\begin{array}{cccc}0& 1& 0& 1\\ 1& 0& 0& 0\\ 0& 1& 0& 1\\ 1& 0& 0& 0\end{array}\right]$

Know that matrix $X^2$ represents the second stage dominances for the tournament now,

Evaluate $X^2$ .

  $\begin{array}{c}{X}^2=\left[\begin{array}{cccc}0& 1& 0& 1\\ 1& 0& 0& 0\\ 0& 1& 0& 1\\ 1& 0& 0& 0\end{array}\right]\left[\begin{array}{cccc}0& 1& 0& 1\\ 1& 0& 0& 0\\ 0& 1& 0& 1\\ 1& 0& 0& 0\end{array}\right]\\ =\left[\begin{array}{cccc}0+1+0+1& 0+0+0+0& 0+0+0+0& 0+0+0+0\\ 0+0+0+0& 1+0+0+0& 0+0+0+0& 1+0+0+0\\ 0+1+0+1& 0+0+0+0& 0+0+0+0& 0+0+0+0\\ 0+0+0+0& 1+0+0+0& 0+0+0+0& 1+0+0+0\end{array}\right]\\ =\left[\begin{array}{cccc}2& 0& 0& 0\\ 0& 1& 0& 1\\ 2& 0& 0& 0\\ 0& 1& 0& 1\end{array}\right]\end{array}$

Now, evaluate $X^3$ by multiplying $X^2$ by $X$

  $\begin{array}{c}{X}^3=\left[\begin{array}{cccc}2& 0& 0& 0\\ 0& 1& 0& 1\\ 2& 0& 0& 0\\ 0& 1& 0& 1\end{array}\right]\left[\begin{array}{cccc}0& 1& 0& 1\\ 1& 0& 0& 0\\ 0& 1& 0& 1\\ 1& 0& 0& 0\end{array}\right]\\ =\left[\begin{array}{cccc}0& 2& 0& 2\\ 2& 0& 0& 0\\ 0& 2& 0& 2\\ 2& 0& 0& 0\end{array}\right]\end{array}$

Therefore, the matrix that represents the number of ways a message can be sent from one point to another using exactly two relay is,

  $\begin{array}{c}A\\ B\\ C\\ D\end{array}\left[\begin{array}{cccc}0& 2& 0& 2\\ 2& 0& 0& 0\\ 0& 2& 0& 2\\ 2& 0& 0& 0\end{array}\right]$