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\vskip 1cm 
\vskip 1cm 
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31 
\begin{enumerate} 
\begin{enumerate} 
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\item {\bf fractional vs. fractional with threshold models} 
\item {\bf ``Fractional'' vs. ``Fractional with threshold'' models} 
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34 
\noindent 
\noindent 
35 
There are two points to be stressed here: first, there are no acknowledgements (i.e. the sender does not know which message was delivered), and 
There are two points to be stressed here: first, there are no acknowledgements (i.e. the sender does not know which message was delivered), and 
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second, we are dealing with a distributed scenario. This is a different setting from the one from paper [22] in references where the messages were 
second, we are dealing with a distributed scenario. This is a different setting from the one from paper [22] in references where the messages were 
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apriori sent only to uninformed vertices (which corresponds to either a centralized scenario, or some 
apriori sent only to uninformed vertices (which corresponds to either a centralized scenario, or some 
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system of acknowledgemets). 
system of acknowledgements). 
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40 
In particular, it is not true that if $\lfloor\alpha m\rfloor\le c(G)1$ where $m$ is the number of messages 
In particular, it is not true that if $\lfloor\alpha m\rfloor\le c(G)1$ where $m$ is the number of messages 
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sent in this particular step, then at least one {\em new vertex} is reached (as stated in the review 
sent in this particular step, then at least one {\em new vertex} is reached (as stated in the review 
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45 
As for the comparison of the fractional, and fractional with threshold models, there are no known results. On one hand, if the topology is known to the vertices, 
As for the comparison of the fractional, and fractional with threshold models, there are no known results. On one hand, if the topology is known to the vertices, 
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one can easily 
one can easily 
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construct an example of a graph (e.g. a caterpilar) where optimal broadcast in the fractional model would send a single message in every of the 
construct an example of a graph (e.g. a caterpillar) where optimal broadcast in the fractional model would send a single message in every of the 
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first few steps in order to inform a ''cirtical path'' (recall that one message is guaranteed to be delivered); this is not possible in 
first few steps in order to inform a ''critical path'' (recall that one message is guaranteed to be delivered); this is not possible in 
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the fractional model with threshold which leads to a slower broadcast. On the other hand, one might possibly argue that 
the fractional model with threshold which leads to a slower broadcast. On the other hand, one might possibly argue that 
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if the topology is not known to the vertices, no such strategy is possible, and the best strategy in the fractional model 
if the topology is not known to the vertices, no such strategy is possible, and the best strategy in the fractional model 
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requires sending many messages; the models would be moreorless the same then. 
requires sending many messages; the models would be moreorless the same then. 
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53 
Even if the latter 
Even if the latter 
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was true, there are no results concerning broadcasting in fractional model in the distributed setting without acknowledgemets. 
was true, there are no results concerning broadcasting in fractional model in the distributed setting without acknowledgements. 
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56 
To help readers to settle this issue, we stressed more the difference in settings between paper [22] and this, and the lack of acknowledgements. 
To help readers to settle this issue, we stressed more the difference in settings between paper [22] and this, and the lack of acknowledgements. 
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58 


59 
\item {\bf lack of motivation} 
\item {\bf Lack of motivation} 
60 


61 
\noindent 
\noindent 
62 
We included a more detailed discussion about the motivation. 
We included a more detailed discussion about the motivation. 
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64 
\item {\bf use of $T$ in the definition of the model} 
\item {\bf Use of $T$ in the definition of the model} 
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66 
\noindent 
\noindent 
67 
We changed the wording of abstract. In the definition it is explicitly stated that $T$ is always assumed to be $c(G)1$. 
We changed the wording of abstract. In the definition it is explicitly stated that $T$ is always assumed to be $c(G)1$. 
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69 
\item {\bf statement of Lemma 1} 
\item {\bf Statement of Lemma 1} 
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71 
\noindent 
\noindent 
72 
The statement of Lemma 1 holds for any (integer) $n>0$. This has been added to the 
The statement of Lemma 1 holds for any (integer) $n>0$. This has been added to the 
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claim of the Lemma. 
claim of the Lemma. 
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75 
\item {\bf p.7, l7: derivation of the limit} 
\item {\bf P.7, l7: derivation of the limit} 
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77 
\noindent 
\noindent 
78 
The derivation of the limit is straightforward, using wellknown standard 
The derivation of the limit is straightforward, using wellknown standard 
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$$ \lim_{n\to\infty}1/2\frac{4X(12/n)}{1+\sqrt{14X(1/n2/n^2)}} = X$$ 
$$ \lim_{n\to\infty}1/2\frac{4X(12/n)}{1+\sqrt{14X(1/n2/n^2)}} = X$$ 
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We opted not to include these standard steps in the paper. 
We opted not to include these standard steps in the paper. 
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87 
\item {\bf numbers $\ell$ and $r$ in Lemmata 1 and 2, respectively} 
\item {\bf Numbers $\ell$ and $r$ in Lemmata 1 and 2, respectively} 
88 


89 
\noindent 
\noindent 
90 
Formally speaking, $r$ in Lemma 2 denotes the number of delivered messages, whereas $\ell$ in Lemma 1 denotes the number 
Formally speaking, $r$ in Lemma 2 denotes the number of delivered messages, whereas $\ell$ in Lemma 1 denotes the number 
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of informed vertices. However, given the context of their usage, the quantities are almost identical, so we changed $\ell$ to $r$ 
of informed vertices. However, given the context of their usage, the quantities are almost identical, so we changed $\ell$ to $r$ 
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in Lemma 1 
in Lemma 1. 
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94 
\item {\bf Lemma 4 is trivial} 
\item {\bf Lemma 4 is trivial} 
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