\end{equation}
Where:\\
\begin{equation}
-\label{eq:perf}
+\label{eq:perf2}
MinTcm = \min_{i=1,2,\dots,N} (Tcm_i)
\end{equation}
where $TcpOld_i$ is the computation time of processor $i$ during the first
reduction with minimum execution time reduction.
This problem can be solved by making the optimization process for energy and
-execution time follow the same direction. Therefore, the equation of the
+execution time following the same direction. Therefore, the equation of the
normalized execution time is inverted which gives the normalized performance equation, as follows:
\begin{multline}
\label{eq:pnorm_inv}
\label{sec.optim}
\subsection{The algorithm details}
-In this section algorithm \ref{HSA} is presented. It selects the frequency scaling factors
+In this section, algorithm \ref{HSA} is presented. It selects the frequency scaling factors
vector that gives the best trade-off between minimizing the energy consumption and maximizing
the performance of a message passing synchronous iterative application executed on a heterogeneous
platform. It works online during the execution time of the iterative message passing program.
It uses information gathered during the first iteration such as the computation time and the
communication time in one iteration for each node. The algorithm is executed after the first
iteration and returns a vector of optimal frequency scaling factors that satisfies the objective
-function (\ref{eq:max}). The program apply DVFS operations to change the frequencies of the CPUs
+function (\ref{eq:max}). The program applies DVFS operations to change the frequencies of the CPUs
according to the computed scaling factors. This algorithm is called just once during the execution
of the program. Algorithm~(\ref{dvfs}) shows where and when the proposed scaling algorithm is called
in the iterative MPI program.
\subsection{The comparison of the proposed scaling algorithm }
\label{sec.compare_EDP}
-In this section, the scaling factors selection algorithm
-is compared to Spiliopoulos et al. algorithm \cite{Spiliopoulos_Green.governors.Adaptive.DVFS}.
+In this section, the scaling factors selection algorithm, called MaxDist,
+is compared to Spiliopoulos et al. algorithm \cite{Spiliopoulos_Green.governors.Adaptive.DVFS}, called EDP.
They developed a green governor that regularly applies an online frequency selecting algorithm to reduce the energy consumed by a multicore architecture without degrading much its performance. The algorithm selects the frequencies that minimize the energy and delay products, $EDP=Enegry*Delay$ using the predicted overall energy consumption and execution time delay for each frequency.
To fairly compare both algorithms, the same energy and execution time models, equations (\ref{eq:energy}) and (\ref{eq:fnew}), were used for both algorithms to predict the energy consumption and the execution times. Also Spiliopoulos et al. algorithm was adapted to start the search from the
initial frequencies computed using the equation (\ref{eq:Fint}). The resulting algorithm is an exhaustive search algorithm that minimizes the EDP and has the initial frequencies values as an upper bound.
