## 3 Study and Results on Messy Broadcasting
### Messy Broadcasting in Grid
#### 3.2.2 Model \(M_{3}\)
We ease this behavior so that, at this point, the message is first sent to the previous sender on the border, then toward \(v\), and finally, to the next vertex on the border. In this way, this path conveys the information to \(v\) in \(t_{x(v)}(v)-1\) time-units. It is as if the new behavior aims to evade informing the rest of the border as much as possible, hence the name _border-evading scheme_. It is easy to verify that this description for each vertex \(v\) has no conflict with the description for another vertex \(u\), which was the case before. In the border-evading scheme, for all interior vertices the time of all four associated border-creeping paths is reduced by precisely one time-unit compared to their maximum. Accordingly, the separating lines and regions associated with the earliest reaching border-creeping paths (detected before) remain the same. Thus far, it is deduced that for border vertices \(t_{3}(0,v)=t_{x(v)}(v)\) and for interior vertices \(t_{x(v)}(v)-1\leq t_{3}(0,v)\leq t_{x(v)}(v)\). Let \(B\) be the set of border vertices, then since \(t_{3}(0)=\max_{v\in V(G)}\{t_{3}(0,v)\}\) we have  
\[\max\biggl{\{}\max_{v\in B}\Bigl{\{}t_{x(v)}(v)\Bigr{\}},\ \max_{v\in V(G)-B} \Bigl{\{}t_{x(v)}(v)-1\Bigr{\}}\biggr{\}}\leq t_{3}(0)\leq\max_{v\in V(G)} \Bigl{\{}t_{x(v)}(v)\Bigr{\}}\]  
Let  
\[v^{*}=\operatorname*{arg\,max}_{v\in V(G)}\biggl{\{}\max_{v\in B}\Bigl{\{}t_{ x(v)}(v)\Bigr{\}},\ \max_{v\in V(G)-B}\Bigl{\{}t_{x(v)}(v)-1\Bigr{\}}\biggr{\}}\]  
with a preference to be an interior vertex. In this way, if \(v^{*}\in B\), then for any interior vertex \(v\), \(t_{3}(0,v^{*})\geq t_{3}(0,v)\); consequently, \(t_{3}(0)=t_{3}(0,v^{*})\) and the border-evading scheme is an optimum scheme. Otherwise, if \(v^{*}\in V(G)-B\), if we can modify the border-evading scheme in such a way that \(v^{*}\) in the new scheme is informed at time-unit \(t_{x(v^{*})}(v^{*})\), then the new scheme will be an optimum scheme, and \(t_{3}(0)=t_{x(v^{*})}(v^{*})\). For that, first, let us find \(v^{*}\). **Continuous view**. The previously identified regions of the grid are associated with the type of border-creeping path that reaches those areas the fastest. In the border-evading scheme, consider an edge \(vu\). If \(vu\) is entirely within one region, the time it takes from the first moment \(v\) is informed to the first moment \(u\) is informed is a constant based on the associated border-creeping path. For example, in the region associated with \(dr\), a move to the right has +4 change while in \(du\) it has +3 and in \(rll\) it has \(-4\). One can change the view from discrete movements (e.g., going up by one unit) to a continuous one. In this view, define the gradient of each point within a region based on the aforementioned uniform changes; in case a point belongs to multiple regions, choose one randomly. The vector field is conservative since it can be viewed as the gradient of a function composed of unions of linear equations. To know the first informed time of a vertex \(v\), consider a point \(p\) at the coordinate of \(v\). For any continuous path from 0 to \(v\) (due to path independence), the line integral over the vector field is the answer. This time can be associated with any other point in the grid. As an example, take a look at Figure 4 under the magnifying glass. As indicated, the change in the first informed time of the two vertical blue dots is \(4\epsilon+(-4)(1-\epsilon)\). Using the continuous view and Lemma 4, by investigating the vector field it is not hard to see that a point with maximum time is the intersection of \(L(dru,\leavevmode\
obreak\ rdl)\), and \(L(rd,\leavevmode\
obreak\ dru)\) lines; \(p^{*}=\left(\frac{\gamma_{n-2}}{8},\leavevmode\
obreak\ m-\frac{n+6/7}{8}\right)\). This is true even for the case \(n=m\), the only case in which \(L(dru,\leavevmode\
obreak\ rdl)\) comes before \(L(dr,\leavevmode\
obreak\ rd)\). **Important note:** There are two latent assumptions.