+\subsection{The algorithm details}
+
+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 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.
+
+\begin{figure}[!t]
+ \centering
+ \includegraphics[scale=0.5]{fig/start_freq}
+ \caption{Selecting the initial frequencies}
+ \label{fig:st_freq}
+\end{figure}
+
+The nodes in a heterogeneous platform have different computing powers, thus
+while executing message passing iterative synchronous applications, fast nodes
+have to wait for the slower ones to finish their computations before being able
+to synchronously communicate with them as in Figure~\ref{fig:heter}. These
+periods are called idle or slack times. The algorithm takes into account this
+problem and tries to reduce these slack times when selecting the frequency
+scaling factors vector. At first, it selects initial frequency scaling factors
+that increase the execution times of fast nodes and minimize the differences
+between the computation times of fast and slow nodes. The value of the initial
+frequency scaling factor for each node is inversely proportional to its
+computation time that was gathered from the first iteration. These initial
+frequency scaling factors are computed as a ratio between the computation time
+of the slowest node and the computation time of the node $i$ as follows:
+\begin{equation}
+ \label{eq:Scp}
+ \Scp[i] = \frac{\max_{i=1,2,\dots,N}(\Tcp[i])}{\Tcp[i]}
+\end{equation}
+Using the initial frequency scaling factors computed in (\ref{eq:Scp}), the
+algorithm computes the initial frequencies for all nodes as a ratio between the
+maximum frequency of node $i$ and the computation scaling factor $\Scp[i]$ as
+follows:
+\begin{equation}
+ \label{eq:Fint}
+ F_{i} = \frac{\Fmax[i]}{\Scp[i]},~{i=1,2,\dots,N}
+\end{equation}
+If the computed initial frequency for a node is not available in the gears of
+that node, it is replaced by the nearest available frequency. In
+Figure~\ref{fig:st_freq}, the nodes are sorted by their computing power in
+ascending order and the frequencies of the faster nodes are scaled down
+according to the computed initial frequency scaling factors. The resulting new
+frequencies are highlighted in Figure~\ref{fig:st_freq}. This set of
+frequencies can be considered as a higher bound for the search space of the
+optimal vector of frequencies because selecting scaling factors higher
+than the higher bound will not improve the performance of the application and it
+will increase its overall energy consumption. Therefore the algorithm that
+selects the frequency scaling factors starts the search method from these
+initial frequencies and takes a downward search direction toward lower
+frequencies. The algorithm iterates on all remaining frequencies, from the higher
+bound until all nodes reach their minimum frequencies, to compute their overall
+energy consumption and performance, and select the optimal frequency scaling
+factors vector. At each iteration the algorithm determines the slowest node
+according to the equation (\ref{eq:perf}) and keeps its frequency unchanged,
+while it lowers the frequency of all other nodes by one gear. The new overall
+energy consumption and execution time are computed according to the new scaling
+factors. The optimal set of frequency scaling factors is the set that gives the
+highest distance according to the objective function (\ref{eq:max}).
+
+Figures~\ref{fig:r1} and \ref{fig:r2} illustrate the normalized performance and
+consumed energy for an application running on a homogeneous platform and a
+heterogeneous platform respectively while increasing the scaling factors. It can
+be noticed that in a homogeneous platform the search for the optimal scaling
+factor should start from the maximum frequency because the performance and the
+consumed energy decrease from the beginning of the plot. On the other hand, in
+the heterogeneous platform the performance is maintained at the beginning of the
+plot even if the frequencies of the faster nodes decrease until the computing
+power of scaled down nodes are lower than the slowest node. In other words,
+until they reach the higher bound. It can also be noticed that the higher the
+difference between the faster nodes and the slower nodes is, the bigger the
+maximum distance between the energy curve and the performance curve is while the
+scaling factors are varying which results in bigger energy savings.
+Finally, in a homogeneous platform the energy consumption is increased when the scaling factor is very high.
+Indeed, the dynamic energy saved by reducing the frequency of the processor is compensated by the significant increase of the execution time and thus the increased of the static energy. On the other hand, in a heterogeneous platform this is not the case.
+
