## 3 Study and Results on Messy Broadcasting
### Messy Broadcasting in Grid
#### 3.2.2 Model \(M_{3}\)
The placement of \(p^{*}\) was determined by investigating the vector field and not by directly investigating the line integrals. Take \(L(dr,\leavevmode\
obreak\ dru)\), for example. Its equation was derived from \(t_{dr}(v)=t_{dru}(v)\). The latent assumption in the value of \(t_{dru}(v)\) is that \(v\)'s vertical position is not \(n\), which comes from the \(+2\) applied for changing the direction upward. Consequently, if \(\frac{\gamma_{n-2}}{8}>n-1\) or \(m-\frac{n+6/7}{8}>m-1\) are true, the above position might be problematic, and additional checks are required. The first restriction is false if and only if \(n\geq 7\), and the second is false if and only if \(n\geq 8\). For \(n\geq 8\), based on the vector fields, the choice for \(v^{*}\) will be limited to three possible vertices in the vicinity of \(p^{*}\):  
1. \(v_{1}^{*}=\left(\left\lfloor\frac{\gamma_{n-2}}{8}\right\rfloor,\leavevmode \
obreak\ m-\left\lceil\frac{n+6/7}{8}\right\rceil\right)\);
2. \(v_{2}^{*}=\left(\left\lfloor\frac{\gamma_{n-2}}{8}\right\rfloor+1,\leavevmode \
obreak\ m-\left\lceil\frac{n+6/7}{8}\right\rceil\right)\);
3. \(v_{3}^{*}=\left(\left\lfloor\frac{\gamma_{n-2}}{8}\right\rfloor+1,\leavevmode \
obreak\ m-\left\lceil\frac{n+6/7}{8}\right\rceil+1\right)\). See Figure 4 under the magnifying glass. The red point is \(p^{*}\), and the possible three vertices have blue points. To understand which has the higher first informed time, the changes from \(v_{1}^{*}\) to \(v_{2}^{*}\), and then, from \(v_{2}^{*}\) to \(v_{3}^{*}\) will be calculated. Let \(\epsilon=\frac{\gamma_{n}-2}{8}-\left\lfloor\frac{\gamma_{n}-2}{8}\right\rfloor\). By the slope of the lines, it is deduced that vertices \(v_{2}^{*}\) and \(v_{3}^{*}\) are in the \(dru\) region (cyan color in Figure 4), and vertex \(v_{1}^{*}\) in the \(rd\) region (gray). Note that if \(n\) mod \(8=1\), then \(v_{1}^{*}\) is in the boundary of the \(rd\) and \(dr\) regions; in this case, we consider the \(rd\) region. As depicted in Figure 4, this positioning implies that the change from \(v_{1}^{*}\) to \(v_{2}^{*}\) is \(4\epsilon+(-4)(1-\epsilon)\) and from \(v_{2}^{*}\) to \(v_{3}^{*}\) is \(+3\). Consequently, vertex \(v_{3}^{*}\) is the sole last-informed vertex except when (1) \(n\) mod \(8=5\), where \(v_{1}^{*}\) is also the last informed; (2) \(n\) mod \(8=6\), where \(v_{1}^{*}\) is the sole last-informed vertex. On the other hand, if \(n\leq 7\), the inequality \(m-\frac{n+6/7}{8}>m-1\) is true, which means the latent assumption for the provided \(p^{*}\) is violated. In this case, by using the vector field it is not hard to see that a choice for \(v^{*}\) is coordinate \((n,\leavevmode\
obreak\ m)\). **Optimum scheme, changing the border-evading scheme**. Now that a possible candidate for \(v^{*}\) (and its time) is determined, we can describe a new similar scheme that will be completed in \(t_{3}(0,v^{*})\), and hence, is an optimum scheme. If \(n\leq 7\), no change is necessary. Otherwise, (1) if \(n\) mod \(8\
eq 6\), change the behavior at \((n,\leavevmode\
obreak\ m-\left\lceil\frac{n+6/7}{8}\right\rceil+1)\) so that at that point there is no evading of the border as described before; see Figure 4 for the modification in red. In this way, the \(dru\)-path time to \(v^{*}\) is increased by one, and since \(dru\)-path was the only path that could inform the fastest, the desired scheme is achieved. (2) If \(n\) mod \(8=6\), in this case, \(v^{*}\) resides (only) in the \(rd\) region; therefore, with a similar justification modify the behavior of \((0,\leavevmode\
obreak\ m-\left\lceil\frac{n+6/7}{8}\right\rceil)\). Adjust the timing of the vertices whose border-evading path relied on this point, and a new valid scheme is achieved. **First move revisited**. At first, it was assumed that the first move is downward, through the smaller dimension. A similar analysis can be made if the first move is toward the right. It can be seen that the new \(v^{*}\) will be informed sooner. Consequently, sending through the smaller dimension as the first move is an optimum choice.