Analysis of parallel algorithms for energy consumption

Analysis of Parallel Algorithms for Energy
Conservation in Scalable Multicore Architecture
Problem statement
An important concern for computing, as for society, is conserving energy. Moreover, energy
conservation is critical in mobile devices for practical reasons. The relation between parallel
applications and their energy requirements on scalable multicore processors are examined. It
believe this sort of analysis can provide programmers with intuitions about the energy required by
the parallel algorithms they are using–thus guiding the choice of algorithm, architecture and the
number of cores used for a particular application. Researchers have studied performance scalability
of parallel algorithms for some time [10]. However, it is important to note that performance
scalability of a parallel algorithm is not the same as its energy scalability. This difference between
performance scalability and energy scalability is due to two important factors. There is a nonlinear
relationship between power and frequency at which the cores operate in multicore processors. In
fact, the power consumed by a core is (typically) proportional to the cube of its frequency.
Executing parallel algorithms typically involves communication (or shared memory accesses) as well
as computation. The power and performance characteristics of communication and computation
may be different. For example, in many algorithms, communication time may be masked by
overlapping communication and computation (e.g., [1]). However, the power required for
communication would be unaffected whether communication overlaps with the computation or not.
Objective
(a) To analyse energy characteristics of parallel algorithms executed on scalable multicore
processors.
(b) To study the sensitivity of the analysis to changes in parameters such as the ratio of power
required for computation versus power required for communication.
(c) To determine how many cores to use in order to minimize energy consumption.

Methodology
The methodology is to evaluate energy scalability under iso-performance of parallel applications.
Partitioning Step
Find the critical path of the parallel algorithm. The critical path is the longest path through the task
dependency graph (where edges represents task serialization) of the parallel algorithm. Note that
the critical path length gives a lower bound on execution time of the parallel algorithm. In the
parallel algorithm, the critical path is easy to find. It is the execution path of the core that has the
sum of all numbers at the end. Half of the cores send the sum they compute to the other half so that
no core receives a sum from more than one core. The receiving core then add the number the local
sum they have computed. We perform the same step recursively until there is only one core left. At
the end of computation, one core will store the sum of all N numbers.
Communication and computation steps
Partition the critical path into communication and computation steps. We can see that there are log
(M) communication steps and ((N=M) � 1 + log (M)) computation steps. Consider a simple parallel
algorithm to add N numbers using M cores. Initially all N numbers are equally distributed among the
M cores; at the end of the computation, one of the cores stores their sum. Without loss of
generality, assume that the number of cores available is some power of two. The algorithm runs in
log (M) steps.
Adding N numbers using 4 actors; Left most
line represents the critical path; embarrassingly
parallel application but represents a broad
class of tree algorithms

Agglomeration step
Scale computation steps of the critical path so that the parallel performance matches the
performance requirement. Scaling the computation time of the critical path to the difference of the
required performance and the communication time of the critical path, and obtain the new reduced
frequency at which all M cores should run. Reduced frequency obtained at which all M cores should
run to complete in time T.
Where β represents number of cycles required per addition. In order to achieve energy savings, we
require 0 < X0 < F. This restriction provides a lower bound on the input size as a function of M and Kc.
Evaluate the message complexity (total number of messages processed) of the parallel algorithm.
The example algorithms discuss before show that the message complexity of some parallel
algorithms may depend only on the number of cores, while for others it depends on both the input
size and the number of cores used. It is trivial to see that number of message transfers for this
parallel algorithm when running on M cores is (M-1). In the algorithm, the message complexity is
only dependent on M and not on the input size N.
Evaluate the total idle time at all the cores assuming the frequency obtained, running at new
frequency X’. Scaling the parallel algorithm critical path may lead to an increase in idle time in other
paths at other cores. Total idle time is:
Where the first term represents the total idle time spent by idle cores while other cores are busy
computing and second term represents the total idle time spent by idle cores while other cores are
involved in message communication.
Step 6 Frame an expression for energy consumption of the parallel algorithm using the energy
model. The energy expression is the sum of the energy consumed by Computation, Ecomp,
Communication, Ecomm and Idling (static power), Eidle
Ecomp is lower if the cores run at a lower frequency, while Eidle may increase as the busy cores take
longer to finish. Ecomm may increase as more cores are used since the computation is more
distributed. Observe that (N-1) is the total number of computation steps. The energy consumed for
computation, communication and idling while the algorithm is running on M cores at reduced
frequency X’ is:

Mapping step
Analyse the equation to obtain the number of cores required for minimum energy consumption as a
function of input size. The energy expression is dependent on many variables such as N (Input Size),
M (Number of cores), (Number of instruction per addition), Kc (no of cycles executed at maximum
frequency for single message communication time), Em (energy consumed for single message
communication between cores), Ps (static power) and the maximum frequency of a core. In most
architectures, the number of cycles involved per addition is just one, so we assume = 1. Set idle
energy consumed per cycle as (Ps=F) = 1, where the cycle is at the maximum frequency F. Express all
energy values with respect to this normalized energy value.
Figure 1 plot for any input size N, initially energy decreases with increasing M and later on increases
with increasing M. Energy for computation decreases with an increase in number of cores running at
reduced frequencies, and energy for communication increases with increasing cores. We can see
that increasing the input size leads to an increase in the optimal number of cores required for
minimum energy consumption. We now consider the sensitivity of this analysis with respect to the
ratio k. Figure 2 plots the optimal number of cores required for minimum energy consumption by
varying input size and k.
Figure 2 shows that for a fixed input size, the optimal number of cores required for minimum energy
consumption decreases with increasing k. Moreover, it is evident that with increasing input size, the
trend remains the same (approximating a negative exponential curve with negative coefficient).

Analysis of parallel algorithms for energy consumption

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Analysis of parallel algorithms for energy consumption