new

[book_gpu.git] / BookGPU / Chapters / chapter9 / ch9.tex

diff --git a/BookGPU/Chapters/chapter9/ch9.tex b/BookGPU/Chapters/chapter9/ch9.tex
index d10daa6a78e92c9934a28e485fbbc88a527735a2..442c53a03ba02139a8b7ac7f8c68b084c9723823 100644 (file)
--- a/BookGPU/Chapters/chapter9/ch9.tex
+++ b/BookGPU/Chapters/chapter9/ch9.tex
@@ -1,10 +1,10 @@
-\chapterauthor{Malika Mehdi and Ahc\`{e}ne Bendjoudi}{CERIST Research Center, DTISI, 3 rue des frères Aissou, 16030 Ben-Aknoun, Algiers, Algeria}
+\chapterauthor{Malika Mehdi and Ahc\`{e}ne Bendjoudi}{CERIST Research Center, Algiers, Algeria}

 
 \chapterauthor{Lakhdar Loukil}{University of Oran, Algeria}
 
 %\chapterauthor{Ahc\`{e}ne Bendjoudi}{CERIST Research Center, DTISI, 3 rue des frères Aissou, 16030 Ben-Aknoun, Algiers, Algeria}
 
 
 \chapterauthor{Lakhdar Loukil}{University of Oran, Algeria}
 
 %\chapterauthor{Ahc\`{e}ne Bendjoudi}{CERIST Research Center, DTISI, 3 rue des frères Aissou, 16030 Ben-Aknoun, Algiers, Algeria}
 

-\chapterauthor{Nouredine Melab}{Université Lille 1, LIFL/UMR CNRS 8022, 59655-Villeneuve d'Ascq cedex, France}
+\chapterauthor{Nouredine Melab}{University of Lille 1, CNRS/LIFL/INRIA, France}

 
 \chapter{Parallel GPU-accelerated metaheuristics}
 \label{chapter9}
 
 \chapter{Parallel GPU-accelerated metaheuristics}
 \label{chapter9}


@@ -36,7 +36,7 @@ remarks are given in Section~\ref{ch8:conclusion}
 
 Combinatorial optimization (CO) is a branch of applied and discrete mathematics.
 It consists in finding optimal configuration(s) among a finite set of possible configurations
 
 Combinatorial optimization (CO) is a branch of applied and discrete mathematics.
 It consists in finding optimal configuration(s) among a finite set of possible configurations

-(or solutions) of a given combinatorial optimization problem (COP).  The set of all possible solutions noted $S$ is called solution space or search space. Each solution  in $S$ is defined by its real cost calculated by an objective function. COPs are generally defined as follows~\cite{blumMeta}:\\  %(Definition~\ref{def:cops})
+(or solutions) of a given combinatorial optimization problem (COP).  The set of all possible solutions noted $S$ is called solution space or search space. Each solution  in $S$ is defined by its real cost calculated by an objective function. COPs are generally defined as follows~\cite{blumMeta}:  %(Definition~\ref{def:cops})

 
 
 %\begin{minipage}{0.5\linewidth}
 
 
 %\begin{minipage}{0.5\linewidth}


@@ -44,7 +44,7 @@ A combinatorial problem $P=(S,f)$ can be defined by
 \begin{itemize}
 \item a set of decision variables $X$,
 \item an objective function $f$ to optimize (minimize or maximize) over the set $S$,
 \begin{itemize}
 \item a set of decision variables $X$,
 \item an objective function $f$ to optimize (minimize or maximize) over the set $S$,

-\item subject to constraints on the decision variables.\\
+\item subject to constraints on the decision variables.

 \end{itemize}
 %\end{minipage}
 
 \end{itemize}
 %\end{minipage}
 


@@ -73,7 +73,7 @@ quality solutions in reasonable computation time compared to exact
 methods but with no guarantee to find optimal or even bounded
 solutions. This is due to the nature of the search process adopted
 by these approaches which consists of performing a
 methods but with no guarantee to find optimal or even bounded
 solutions. This is due to the nature of the search process adopted
 by these approaches which consists of performing a

-search in a subset of the whole search space.\\
+search in a subset of the whole search space.

 
 Regarding the number of solutions considered at each iteration in
 the search process, two classes of metaheuristics can be
 
 Regarding the number of solutions considered at each iteration in
 the search process, two classes of metaheuristics can be


@@ -90,7 +90,7 @@ annealing}\index{metaheuristics!simulated annealing}~\cite{Kirkpatrick1983SA},
 \emph{tabu search}~\cite{Glover1989TS}, \emph{iterated local
 search\index{metaheuristics!iterated local
 search}}~\cite{stutzle2006ILSforQAP}, and \emph{variable
 \emph{tabu search}~\cite{Glover1989TS}, \emph{iterated local
 search\index{metaheuristics!iterated local
 search}}~\cite{stutzle2006ILSforQAP}, and \emph{variable

-neighborhood search}~\cite{HansenMladenovic1997VNS}.\\
+neighborhood search}~\cite{HansenMladenovic1997VNS}.

 
 Unlike s-metaheuristics, p-metaheuristics start with a population
 of solutions and implement an iterative process that evolves the
 
 Unlike s-metaheuristics, p-metaheuristics start with a population
 of solutions and implement an iterative process that evolves the


@@ -99,13 +99,13 @@ solutions. The process is repeated until a stopping criterion is
 satisfied. \emph{Evolutionary algorithms}, \emph{swarm
 optimization}, and \emph{ant colonies} fall into this class.
 
 satisfied. \emph{Evolutionary algorithms}, \emph{swarm
 optimization}, and \emph{ant colonies} fall into this class.
 

-
+%\clearpage

 \section{Parallel models for metaheuristics}\label{ch8:sec:paraMeta}
 Optimization problems, whether real-life or academic, are more
 often NP-hard and CPU time and/or memory consuming. Metaheuristics
 allow the significant reduction of the computational time of the search
 process but remain time-consuming particularly when it comes
 \section{Parallel models for metaheuristics}\label{ch8:sec:paraMeta}
 Optimization problems, whether real-life or academic, are more
 often NP-hard and CPU time and/or memory consuming. Metaheuristics
 allow the significant reduction of the computational time of the search
 process but remain time-consuming particularly when it comes

-dealing with large-sized problems. \\
+dealing with large-sized problems. 

 
 The use of parallelism in the design of metaheuristics is a relevant
 approach that is widely adopted by the combinatorial optimization
 
 The use of parallelism in the design of metaheuristics is a relevant
 approach that is widely adopted by the combinatorial optimization


@@ -130,7 +130,7 @@ high-performance computing.
 
 From the granularity of parallelism point of view, three major parallel
 models for metaheuristics can be distinguished~\cite{talbi2009mfdti}: \emph{algorithmic-level}\index{metaheuristics!algorithmic-level parallelism},
 
 From the granularity of parallelism point of view, three major parallel
 models for metaheuristics can be distinguished~\cite{talbi2009mfdti}: \emph{algorithmic-level}\index{metaheuristics!algorithmic-level parallelism},

-\emph{iteration-level}\index{metaheuristics!iteration-level parallelism}, and \emph{solution-level} as illustrated in Figure~\ref{ch8:fig:paraMeta}. \\
+\emph{iteration-level}\index{metaheuristics!iteration-level parallelism}, and \emph{solution-level} as illustrated in Figure~\ref{ch8:fig:paraMeta}. 

 
 \begin{figure}[h!]
 \centerline{\includegraphics[width=0.6\textwidth]{Chapters/chapter9/figures/paraMeta.pdf}}
 
 \begin{figure}[h!]
 \centerline{\includegraphics[width=0.6\textwidth]{Chapters/chapter9/figures/paraMeta.pdf}}


@@ -187,8 +187,8 @@ consuming. Unlike the two previous parallel models, the
 solution-level\index{metaheuristics!solution-level parallelism}
 parallel model is problem-dependent.}
 \end{itemize}
 solution-level\index{metaheuristics!solution-level parallelism}
 parallel model is problem-dependent.}
 \end{itemize}

-
-\section{Challenges for the design of GPU-based metaheuristics}
+%\clearpage
+\section[Challenges for the design of GPU-based  metaheuristics]{Challenges for the design of GPU-based\hfill\break  metaheuristics}

 \label{ch8:sec:challenges}
 
 Developing GPU-based parallel
 \label{ch8:sec:challenges}
 
 Developing GPU-based parallel


@@ -226,12 +226,12 @@ threads and the corresponding metaheuristic elements (one neighbor
 per thread, one individual per thread, single population per
 threads block, one ant per thread, etc.) must be defined to ensure
 a maximum occupancy of the GPU and
 per thread, one individual per thread, single population per
 threads block, one ant per thread, etc.) must be defined to ensure
 a maximum occupancy of the GPU and

-to cover CPU/GPU communication\index{GPU!CPU/GPU communication} and memory access times.\\
+to cover CPU/GPU communication\index{GPU!CPU/GPU communication} and memory access times.

 
 According to the used metaheuristic and to the handled problem, the data
 values may have different types and different ranges of their values. The data
 types should then be chosen carefully and the ranges of the data values should
 
 According to the used metaheuristic and to the handled problem, the data
 values may have different types and different ranges of their values. The data
 types should then be chosen carefully and the ranges of the data values should

-be analyzed to minimize the amount of occupied memory space.\\
+be analyzed to minimize the amount of occupied memory space.

 
 In addition to the size and latency of GPU memory issues, the
 memory access pattern is another important issue to be dealt with
 
 In addition to the size and latency of GPU memory issues, the
 memory access pattern is another important issue to be dealt with


@@ -251,7 +251,7 @@ data (e.g., data for fitness evaluation that all threads
 concurrently access) on the constant memory, and the most accessed
 data structures (e.g., population of individuals for a CUDA thread
 block) on the shared memory.
 concurrently access) on the constant memory, and the most accessed
 data structures (e.g., population of individuals for a CUDA thread
 block) on the shared memory.

-
+\clearpage

 \subsection{Threads synchronization}
 \index{GPU!threads synchronization} The thread
 synchronization issue is caused by both the GPU architecture and
 \subsection{Threads synchronization}
 \index{GPU!threads synchronization} The thread
 synchronization issue is caused by both the GPU architecture and


@@ -294,7 +294,7 @@ the different threads degrading the performance of the application
 in terms of execution time. The challenge here is then to revisit
 the traditional irregular metaheuristic codes to eliminate these
 divergences.
 in terms of execution time. The challenge here is then to revisit
 the traditional irregular metaheuristic codes to eliminate these
 divergences.

-
+\clearpage

 \subsection{Task distribution and CPU/GPU communication}
 
 The performance of GPU-based metaheuristics in terms of execution
 \subsection{Task distribution and CPU/GPU communication}
 
 The performance of GPU-based metaheuristics in terms of execution


@@ -313,7 +313,7 @@ perform a careful analysis on the serial code of the
 metaheuristic, identify the compute-intensive tasks (e.g.,
 exploration of the neighborhood, evaluation of individuals), and
 then offload them to the GPU, while the remaining
 metaheuristic, identify the compute-intensive tasks (e.g.,
 exploration of the neighborhood, evaluation of individuals), and
 then offload them to the GPU, while the remaining

-tasks still run on the CPU in a serial way. \\
+tasks still run on the CPU in a serial way. 

 
 The CPU/GPU communication is done through the global
 memory which is a slow memory making the memory transfer between
 
 The CPU/GPU communication is done through the global
 memory which is a slow memory making the memory transfer between


@@ -328,7 +328,7 @@ therefore, the whole execution time of the metaheuristic.
 After more than two decades of research by the combinatorial optimisation
 community devoted to developing adequate parallel metaheuristics for different types of
 parallel architectures (clusters, supercomputers and grids), the actual developement
 After more than two decades of research by the combinatorial optimisation
 community devoted to developing adequate parallel metaheuristics for different types of
 parallel architectures (clusters, supercomputers and grids), the actual developement

-of General Perpose GPU (GPGPU) brings new challenges for parallel metaheuristics on SIMD architectures.\\
+of General Perpose GPU (GPGPU) brings new challenges for parallel metaheuristics on SIMD architectures.

 
 The first works on metaheuristic algorithms implemented on GPUs
 started on old graphics cards before the appearance of modern GPUs
 
 The first works on metaheuristic algorithms implemented on GPUs
 started on old graphics cards before the appearance of modern GPUs


@@ -339,7 +339,7 @@ of EAs on graphics processing cards and the work by Catala et al. in~\cite{catal
 is implemented on old GPU architectures. Yu et al.~\cite{yu2005} and
 Li et al.~\cite{li2007} proposed a full parallelization of genetic
 algorithms  on old GPU architectures using
 is implemented on old GPU architectures. Yu et al.~\cite{yu2005} and
 Li et al.~\cite{li2007} proposed a full parallelization of genetic
 algorithms  on old GPU architectures using

-shader libraries based on Direct3D and OpenGL.\\
+shader libraries based on Direct3D and OpenGL.

 
 Such architectures are based on preconfigured pipelined stages
 used to accelerate the transformation of 3D geometric primitives
 
 Such architectures are based on preconfigured pipelined stages
 used to accelerate the transformation of 3D geometric primitives


@@ -377,7 +377,7 @@ Nevertheless, small neighborhoods may lead to  nonoptimal
 occupation of the CUDA threads which may lead in turn to an
 overhead due to the communication and memory latencies. Therefore,
 large neighborhoods are necessary for efficient implementation of
 occupation of the CUDA threads which may lead in turn to an
 overhead due to the communication and memory latencies. Therefore,
 large neighborhoods are necessary for efficient implementation of

-local searches on GPUs.\\
+local searches on GPUs.

 
 Luong et al.~\cite{luong2010large} proposed efficient
 mappings for large neighborhood structures on GPUs. In this work,
 
 Luong et al.~\cite{luong2010large} proposed efficient
 mappings for large neighborhood structures on GPUs. In this work,


@@ -393,10 +393,10 @@ efficiently map the different neighborhoods on the device memory,
 more explicitly, how to calculate the memory index of each
 solution associated to each CUDA thread's \textit{id}.
 %For 1-Hamming neighborhoods, as there is exactly n solutions in the neighborhood, the mapping of this neighborhood to CUDA threads is obvious: the CPU host offloads to GPU exactly $n$ threads, and each thread id is associated to one index in the binary vector. In the case of 2-Hamming and 3-Hamming neighborhoods, each thread id should be mapped respectively to two and three indexes  in the candidate vector.
 more explicitly, how to calculate the memory index of each
 solution associated to each CUDA thread's \textit{id}.
 %For 1-Hamming neighborhoods, as there is exactly n solutions in the neighborhood, the mapping of this neighborhood to CUDA threads is obvious: the CPU host offloads to GPU exactly $n$ threads, and each thread id is associated to one index in the binary vector. In the case of 2-Hamming and 3-Hamming neighborhoods, each thread id should be mapped respectively to two and three indexes  in the candidate vector.

-The three neighborhoods are  implemented and experimented on the Permuted Perceptron Problem (PPP) using a tabu search\index{metaheuristics!tabu search} algorithm (TS). Accelerations from $9.9 \times$ to $18.5 \times$ are obtained on different problem sizes.\\ % The experiments are performed on an Intel Xeon 8 cores 3GHz coupled with an NVIDIA GTX 280 card.\\
+The three neighborhoods are  implemented and experimented on the Permuted Perceptron Problem (PPP) using a tabu search\index{metaheuristics!tabu search} algorithm (TS). Accelerations from $9.9 \times$ to $18.5 \times$ are obtained on different problem sizes. % The experiments are performed on an Intel Xeon 8 cores 3GHz coupled with an NVIDIA GTX 280 card.\\

 
 In the same context, Deevacq et al.~\cite{audreyANT}
 
 In the same context, Deevacq et al.~\cite{audreyANT}

-proposed two parallelization strategies inspired by the multi-walk
+proposed two parallelization strategies inspired by the multiwalk

 parallelization strategy, of a 3-opt iterated local
 search algorithm (ILS)
 over a CPU/GPU architecture. In the first strategy, each Local
 parallelization strategy, of a 3-opt iterated local
 search algorithm (ILS)
 over a CPU/GPU architecture. In the first strategy, each Local


@@ -410,7 +410,7 @@ be stored on the global memory, while the exploration of a single
 solution at a time in the second strategy allows the use of the
 shared memory to store the related data structures. The two
 strategies have been experimented on standard benchmarks of
 solution at a time in the second strategy allows the use of the
 shared memory to store the related data structures. The two
 strategies have been experimented on standard benchmarks of

-the Traveling Salesman Problem (TSP) with sizes varying from $100$ to $3038$ cities. The results indicate that increasing the number of solutions to be explored simultaneously improves the speedup in the two strategies, but at the same time it decreases final solution quality. The greater speedup factor reached by the second strategy is $6 \times$.\\
+the Traveling Salesman Problem (TSP) with sizes varying from $100$ to $3038$ cities. The results indicate that increasing the number of solutions to be explored simultaneously improves the speedup in the two strategies, but at the same time it decreases final solution quality. The greater speedup factor reached by the second strategy is $6 \times$.

 
 The same strategy is used by Luong et al.
 in~\cite{luongMultiStart} to implement multistart parallel local
 
 The same strategy is used by Luong et al.
 in~\cite{luongMultiStart} to implement multistart parallel local


@@ -437,7 +437,7 @@ implemented on the quadratic assignment problem (QAP) using CUDA.
 The acceleration rates obtained for the
 algorithmic-level with usage of texture memory rise from $7.8\times$ to
 $12\times$ (for different
 The acceleration rates obtained for the
 algorithmic-level with usage of texture memory rise from $7.8\times$ to
 $12\times$ (for different

-QAP benchmark sizes). \\
+QAP benchmark sizes). 

 
 Janiak et al.~\cite{Janiak_et_al_2008} implemented two
 algorithms for TSP and the flow-shop scheduling problem (FSP).
 
 Janiak et al.~\cite{Janiak_et_al_2008} implemented two
 algorithms for TSP and the flow-shop scheduling problem (FSP).


@@ -450,7 +450,7 @@ However, even though their experiments report that the use of GPU
 speedups the serial execution almost $16 \times$, the mapping of
 one thread with one tabu search
 requires a large number of local search algorithms to
 speedups the serial execution almost $16 \times$, the mapping of
 one thread with one tabu search
 requires a large number of local search algorithms to

-cover the memory access latency. The same mapping policy is adopted by Zhu et al. in~\cite{zhu_et_al_2008} (one thread is associated to one local search) solving the quadratic assignment problem but using the CUDA toolkit instead of shader libraries.\\
+cover the memory access latency. The same mapping policy is adopted by Zhu et al. in~\cite{zhu_et_al_2008} (one thread is associated to one local search) solving the quadratic assignment problem but using the CUDA toolkit instead of shader libraries.

 
 Luong et al.~\cite{luong2012ppsn} proposed a GPU-based
 implementation of hybrid metaheuristics on heterogeneous parallel
 
 Luong et al.~\cite{luong2012ppsn} proposed a GPU-based
 implementation of hybrid metaheuristics on heterogeneous parallel


@@ -470,7 +470,7 @@ associated to one CPU core and  used to accelerate the
 neighborhood calculation of several S-metaheuristics at the same
 time.
  In order to efficiently exploit the remaining CPU cores, a load-balancing heuristic is also proposed in order to decide on the number of additional
 neighborhood calculation of several S-metaheuristics at the same
 time.
  In order to efficiently exploit the remaining CPU cores, a load-balancing heuristic is also proposed in order to decide on the number of additional

-S-metaheuristics to launch on the remaining CPU cores relative to the efficiency of the GPU calculations. The proposed approach is applied to the QAP using several instances of the Fast Ant Colony Algorithm (FANT)~\cite{taillardFant}. \\
+S-metaheuristics to launch on the remaining CPU cores relative to the efficiency of the GPU calculations. The proposed approach is applied to the QAP using several instances of the Fast Ant Colony Algorithm (FANT)~\cite{taillardFant}. 

 
 All the previously noted works  exploit the same parallel models
 used on CPUs based on the task parallelism. A different
 
 All the previously noted works  exploit the same parallel models
 used on CPUs based on the task parallelism. A different


@@ -499,9 +499,9 @@ involved in the evaluation of a single move. Experimentations are
 done on standard QAP instances from the
 QAPLIB~\cite{burkard1991qaplib}. Speedups up to $10 \times$ are
 achieved by the GPU implementation compared
 done on standard QAP instances from the
 QAPLIB~\cite{burkard1991qaplib}. Speedups up to $10 \times$ are
 achieved by the GPU implementation compared

-to the same sequential implementation on CPU using SA-matrix.\\
+to the same sequential implementation on CPU using SA-matrix.

 
 

-\subsection{Implementing population-based metaheuristics on GPUs}
+\subsection[Implementing population-based metaheuristics on GPUs]{Implementing population-based metaheuristics on GPUs}

 
 State-of-the-art works dealing with the implementation of
 p-metaheuristics on GPUs generally rely on parallel models and
 
 State-of-the-art works dealing with the implementation of
 p-metaheuristics on GPUs generally rely on parallel models and


@@ -515,7 +515,7 @@ colony optimization
 (ACO), and particle
 swarm optimization\index{metaheuristics!particle swarm
 optimization} (PSO).
 (ACO), and particle
 swarm optimization\index{metaheuristics!particle swarm
 optimization} (PSO).

-
+\clearpage

 \subsubsection*{Evolutionary algorithms}
 
 Traditional parallel models for EAs are classified in three main
 \subsubsection*{Evolutionary algorithms}
 
 Traditional parallel models for EAs are classified in three main


@@ -526,7 +526,7 @@ models based on the use of one population disposed (generally) on
 a toroidal grid.
 
 The three traditional models have been implemented on GPUs by several researchers for different
 a toroidal grid.
 
 The three traditional models have been implemented on GPUs by several researchers for different

-optimization problems. The main chalenges to be raised when implementing the traditional models on GPUs concern (1) the saturation of the GPU in order to cover memory latency by calculations, and (2) efficent usage of the hierarchical GPU  memories.\\
+optimization problems. The main chalenges to be raised when implementing the traditional models on GPUs concern (1) the saturation of the GPU in order to cover memory latency by calculations, and (2) efficent usage of the hierarchical GPU  memories.

 
 In~\cite{kannan}, Kannan and Ganji present a CUDA implementation
 of the drug discovery application Autodock (molecular docking
 
 In~\cite{kannan}, Kannan and Ganji present a CUDA implementation
 of the drug discovery application Autodock (molecular docking


@@ -550,7 +550,7 @@ advantage of the per block approach resides in the use of shared
 memory instead of global memory to store all the information
 related to each individual. The obtained speedups range from  $10
 \times$ to $47 \times$ for population sizes
 memory instead of global memory to store all the information
 related to each individual. The obtained speedups range from  $10
 \times$ to $47 \times$ for population sizes

-ranging from $50$ to $10000$. \\
+ranging from $50$ to $10000$. 

 
 Maitre et al.~\cite{maitre2009} also exploited the
 master-slave parallelism of EAs on GPUs using EASEA. EASEA is a
 
 Maitre et al.~\cite{maitre2009} also exploited the
 master-slave parallelism of EAs on GPUs using EASEA. EASEA is a


@@ -571,7 +571,7 @@ during the experiments: a benchmark mathematical function and a
 real problem (molecular structure prediction). In order to
 maximize the GPU occupation, very large populations are used (from
 $2000$ to $20000$). Even though transferring such large
 real problem (molecular structure prediction). In order to
 maximize the GPU occupation, very large populations are used (from
 $2000$ to $20000$). Even though transferring such large

-populations from the CPU to the GPU device memory at every generation is very costly, the authors report important speedups on the two problems on a GTX260 card:  $105 \times$ is reported for the benchmark function while for the real problem the reported speedup is $60 \times$. This may be best explained by the complexity of the fitness functions. Nevertheless, there is no indication in the paper about the memory management of the populations on GPU.\\
+populations from the CPU to the GPU device memory at every generation is very costly, the authors report important speedups on the two problems on a GTX260 card:  $105 \times$ is reported for the benchmark function while for the real problem the reported speedup is $60 \times$. This may be best explained by the complexity of the fitness functions. Nevertheless, there is no indication in the paper about the memory management of the populations on GPU.

 
 The master-slave  model is efficient when the fitness function is
 highly time intensive. Nevertheless, it requires the use of
 
 The master-slave  model is efficient when the fitness function is
 highly time intensive. Nevertheless, it requires the use of


@@ -580,7 +580,7 @@ per-block is used (one individual perblock) when the acceleration
 of the fitness function itself is possible. The use of many
 subpopulations of medium sizes is another way to obtain a maximum
 occupation of the GPU. This is coarse-grained parallelism (island
 of the fitness function itself is possible. The use of many
 subpopulations of medium sizes is another way to obtain a maximum
 occupation of the GPU. This is coarse-grained parallelism (island

-model).\\
+model).

 
 The coarse-grained model is  used by Pospichal et al.
 in~\cite{pospichal10} to implement a parallel genetic
 
 The coarse-grained model is  used by Pospichal et al.
 in~\cite{pospichal10} to implement a parallel genetic


@@ -596,7 +596,7 @@ stored on shared memory of each block. Nevertheless, because
 interblock communications are not possible on the CUDA
 architecture, the islands evolve independently in each block, and
 migrations are performed
 interblock communications are not possible on the CUDA
 architecture, the islands evolve independently in each block, and
 migrations are performed

-asynchronously through the global memory. That is, after a given number of generations, selected individuals for migration from each island are copied to the GPU global memory part of the neighbor island and then to its shared memory to replace the worst individuals in the local population. The experiments are performed on three benchmark mathematical functions. During these experiments, the island sizes are varied from $2$ to $256$ individuals and island numbers from $1$ to $1024$. The maximum performance is achieved for high number of islands and increasing population sizes.\\
+asynchronously through the global memory. That is, after a given number of generations, selected individuals for migration from each island are copied to the GPU global memory part of the neighbor island and then to its shared memory to replace the worst individuals in the local population. The experiments are performed on three benchmark mathematical functions. During these experiments, the island sizes are varied from $2$ to $256$ individuals and island numbers from $1$ to $1024$. The maximum performance is achieved for high number of islands and increasing population sizes.

 
 The same strategy is also adopted by Tsutsui and Fujimoto
 in~\cite{tsutsuiGAQAP}  to implement a coarse-grained genetic
 
 The same strategy is also adopted by Tsutsui and Fujimoto
 in~\cite{tsutsuiGAQAP}  to implement a coarse-grained genetic


@@ -614,7 +614,7 @@ QAP benchmarks from the QAPLIB~\cite{burkard1991qaplib}. The GPU
 implementation reached speedups of $2.9\times$ to $12.6 \times$
 compared to a single core implementation of a coarse-grained
 genetic algorithm on a
 implementation reached speedups of $2.9\times$ to $12.6 \times$
 compared to a single core implementation of a coarse-grained
 genetic algorithm on a

-Intel i7 processor.\\
+Intel i7 processor.

 
 Nowotniak and Kucharski~\cite{nowotniak} proposed a GPU-based
 implementation of a Quantum Inspired Genetic Algorithm (QIGA). The 
 
 Nowotniak and Kucharski~\cite{nowotniak} proposed a GPU-based
 implementation of a Quantum Inspired Genetic Algorithm (QIGA). The 


@@ -629,7 +629,7 @@ same algorithm in order to assess statistical efficiency. The
 populations are stored in the shared memory, the data matrix used
 for fitness evaluation is placed in read only constant memory, and
 finally seeds for random numbers generated on the GPU are stored
 populations are stored in the shared memory, the data matrix used
 for fitness evaluation is placed in read only constant memory, and
 finally seeds for random numbers generated on the GPU are stored

-in the global memory.\\
+in the global memory.

 
 In coarse-grained parallelism, the use of the per-block approach
 to implement the islands (one subpopulation per thread block) is
 
 In coarse-grained parallelism, the use of the per-block approach
 to implement the islands (one subpopulation per thread block) is


@@ -637,7 +637,7 @@ almost natural  and it allows the use of shared memory to store
 the subpopulations. Fine-grained parallel models for EAs (cellular
 EAs) have been used by many authors to implement parallel EAs on
 GPUs. Indeed, the fine-grained parallelism of EAs fits perfectly
 the subpopulations. Fine-grained parallel models for EAs (cellular
 EAs) have been used by many authors to implement parallel EAs on
 GPUs. Indeed, the fine-grained parallelism of EAs fits perfectly

-to the SIMD architecture of the GPU. \\
+to the SIMD architecture of the GPU. 

 
 Pinel et al. in~\cite{pinel2012JPDC} developed a highly
 parallel synchronous cellular genetic
 
 Pinel et al. in~\cite{pinel2012JPDC} developed a highly
 parallel synchronous cellular genetic


@@ -663,7 +663,7 @@ threads used to accelerate the recombination operators especially
 when dealing with large instances of the problem. In addition to
 the recombination operators, the rest of the CGA steps are  also
 parallelized on GPU (fitness evaluation, mutation, and
 when dealing with large instances of the problem. In addition to
 the recombination operators, the rest of the CGA steps are  also
 parallelized on GPU (fitness evaluation, mutation, and

-replacement).\\
+replacement).

 
 A similar work is proposed by Vidal and Alba  in~\cite{albaCGAGPU}
 where a  CGA using a toroidal grid is completely implemented on
 
 A similar work is proposed by Vidal and Alba  in~\cite{albaCGAGPU}
 where a  CGA using a toroidal grid is completely implemented on


@@ -687,7 +687,7 @@ communications,
 synchronization, and access to global memory. Finally, an
 interesting review on GPU parallel computation in bio-inspired
 algorithms is proposed by Arenas et al.
 synchronization, and access to global memory. Finally, an
 interesting review on GPU parallel computation in bio-inspired
 algorithms is proposed by Arenas et al.

-in~\cite{arenas2011}. \\
+in~\cite{arenas2011}. 

 
 \subsubsection*{Ant colony optimization}
 
 
 \subsubsection*{Ant colony optimization}
 


@@ -698,7 +698,7 @@ State-of-the-art works on parallelizing
 ACO focus on
 accelerating the tour construction step performed by each ant by
 taking a task-based parallelism approach, with pheromone
 ACO focus on
 accelerating the tour construction step performed by each ant by
 taking a task-based parallelism approach, with pheromone

-deposition on the CPU.\\
+deposition on the CPU.

 
 In~\cite{cecilia}, Cecilia et al.  present a GPU-based
 implementation of
 
 In~\cite{cecilia}, Cecilia et al.  present a GPU-based
 implementation of


@@ -720,7 +720,7 @@ update, several GPU techniques are also used to increase the data
 bandwidth of the application mainly by the use of precalculated
 matrices that are easily updated by several threads (one thread
 per matrix entry). The achieved speedups are $21 \times$ for tour
 bandwidth of the application mainly by the use of precalculated
 matrices that are easily updated by several threads (one thread
 per matrix entry). The achieved speedups are $21 \times$ for tour

-construction and  $20 \times$ for pheromone updates.\\
+construction and  $20 \times$ for pheromone updates.

 
 In another work, Tsutsui and Fujimoto~\cite{tsutsui} propose  a
 hybrid algorithm combining
 
 In another work, Tsutsui and Fujimoto~\cite{tsutsui} propose  a
 hybrid algorithm combining


@@ -753,7 +753,7 @@ performed on standard QAP benchmarks showed that the GPU
 implementation using MATA obtained a  speedup of $19 \times$
 compared to the CPU implementation,  compared with a speedup of
 only $5 \times$
 implementation using MATA obtained a  speedup of $19 \times$
 compared to the CPU implementation,  compared with a speedup of
 only $5 \times$

-when MATA is not used. \\
+when MATA is not used. 

 
 \subsubsection*{Particle swarm optimization}
 In~\cite{zhou2009} Zhou and Tan propose a full GPU implementation
 
 \subsubsection*{Particle swarm optimization}
 In~\cite{zhou2009} Zhou and Tan propose a full GPU implementation


@@ -776,7 +776,7 @@ best of each particle), the GPU threads are mapped to the \emph{N}
 particles of the swarm one to one. Experiments done on four
 benchmark functions show speedups ranging from $3.7 \times$ to
 $9.0 \times$ for swarm sizes
 particles of the swarm one to one. Experiments done on four
 benchmark functions show speedups ranging from $3.7 \times$ to
 $9.0 \times$ for swarm sizes

-ranging from $400$ to $2800$.\\
+ranging from $400$ to $2800$.

 
 \subsection{Synthesis of the implementation strategies}
 \label{ch8:sec:synthesis} After reviewing some works dealing with
 
 \subsection{Synthesis of the implementation strategies}
 \label{ch8:sec:synthesis} After reviewing some works dealing with


@@ -790,7 +790,7 @@ Indeed, even though the parallelization models used in most works
 for GPUs are derived from the traditional parallel models of each
 metaheuristic (on CPU), their implementation could take a
 different way and sometimes it may result in new parallel models
 for GPUs are derived from the traditional parallel models of each
 metaheuristic (on CPU), their implementation could take a
 different way and sometimes it may result in new parallel models

-customized for GPUs.\\
+customized for GPUs.

 
 Traditional parallel models for metaheuristics are based on an
 intuitive task parallelism: the independent tasks of the
 
 Traditional parallel models for metaheuristics are based on an
 intuitive task parallelism: the independent tasks of the


@@ -802,7 +802,7 @@ the evaluation step. Nevertheless, because of the particularity of
 the GPU architecture, some authors have used new implementation
 techniques to enhance the data parallelism in the sequential
 algorithms in order to increase the data throughput of the
 the GPU architecture, some authors have used new implementation
 techniques to enhance the data parallelism in the sequential
 algorithms in order to increase the data throughput of the

-application.\\
+application.

 
 From this observation we propose the following classification
 based on 2 levels: design level and implementation level as
 
 From this observation we propose the following classification
 based on 2 levels: design level and implementation level as


@@ -820,7 +820,7 @@ models are implemented on GPU. This classification focuses only on
 the mapping strategies between the GPU threads and the
 parallelized tasks (neighborhood evaluation, solution
 construction, and so on). The different implementation strategies
 the mapping strategies between the GPU threads and the
 parallelized tasks (neighborhood evaluation, solution
 construction, and so on). The different implementation strategies

-are explained in the following sections.\\
+are explained in the following sections.

 
 \begin{figure}[h]
 \centerline{\includegraphics[width=1\textwidth]{Chapters/chapter9/figures/classification.pdf}}
 
 \begin{figure}[h]
 \centerline{\includegraphics[width=1\textwidth]{Chapters/chapter9/figures/classification.pdf}}


@@ -843,7 +843,7 @@ associated to a block of threads. A second level of parallelism is
 used inside each block to parallelize the fitness evaluation of a
 single solution. This kind of mapping allows the use of shared
 memory to store the data structures of the solution and does not
 used inside each block to parallelize the fitness evaluation of a
 single solution. This kind of mapping allows the use of shared
 memory to store the data structures of the solution and does not

-require the use of very large neighborhoods or populations.\\
+require the use of very large neighborhoods or populations.

 %data parallelism in SA-matrix to parallelize
 
 \subsubsection*{GPU thread mapping for iteration-level parallelism}
 %data parallelism in SA-matrix to parallelize
 
 \subsubsection*{GPU thread mapping for iteration-level parallelism}


@@ -851,7 +851,7 @@ require the use of very large neighborhoods or populations.\\
 Iteration-level parallelism consists of parallelizing the tasks
 performed independently on different solutions. Different mapping
 strategies are adopted in the reviewed works to implement these
 Iteration-level parallelism consists of parallelizing the tasks
 performed independently on different solutions. Different mapping
 strategies are adopted in the reviewed works to implement these

-models.\\
+models.

 
 In Figure \ref{ch8:fig:classification}, the first example of
 iteration-level
 
 In Figure \ref{ch8:fig:classification}, the first example of
 iteration-level


@@ -875,7 +875,7 @@ memory.
 %pheromone update data parallelism matrices
 
 \subsubsection*{GPU thread mapping  for algorithmic-level parallelism}
 %pheromone update data parallelism matrices
 
 \subsubsection*{GPU thread mapping  for algorithmic-level parallelism}

-Algorithmic-level parallelism consists of launching several self-contained algorithms in parallel. In the previously reviewed works two algorithmic-level models have been used: the multistart model and the island model (parallel EAs).\\
+Algorithmic-level parallelism consists of launching several self-contained algorithms in parallel. In the previously reviewed works two algorithmic-level models have been used: the multistart model and the island model (parallel EAs).

 
 The implementation of the multistart model is based on two
 different mapping strategies~\cite{luongMultiStart, audreyANT}:
 
 The implementation of the multistart model is based on two
 different mapping strategies~\cite{luongMultiStart, audreyANT}:


@@ -893,7 +893,7 @@ parallelized on GPU using per-thread mapping strategy (one thread
 per solution). This consists of a hierarchical parallel model
 combining algorithmic-level parallelism (multistart) with
 iteration-level
 per solution). This consists of a hierarchical parallel model
 combining algorithmic-level parallelism (multistart) with
 iteration-level

-parallelism (master-worker).\\
+parallelism (master-worker).

 
 In the island model, the same mapping is used in all the reviewed
 works~\cite{tsutsuiGAQAP, nowotniak, maitre2009}: each
 
 In the island model, the same mapping is used in all the reviewed
 works~\cite{tsutsuiGAQAP, nowotniak, maitre2009}: each


@@ -906,7 +906,7 @@ advantage of this hierarchical implementation is that it allows
 the occupation of a large number of threads even for medium
 population sizes. The second advantage consists of using shared
 memory to store subpopulations to benefit from the low latency of
 the occupation of a large number of threads even for medium
 population sizes. The second advantage consists of using shared
 memory to store subpopulations to benefit from the low latency of

-shared memory.\\
+shared memory.

 
 \section{Frameworks for metaheuristics on GPUs}
 \label{ch8:sec:frameworks}
 
 \section{Frameworks for metaheuristics on GPUs}
 \label{ch8:sec:frameworks}


@@ -918,7 +918,7 @@ on GPUs. Although the works on this subject are not yet mature and
 do not cover the main metaheuristic algorithms, we will present
 the only three works to our knowledge, which propose open source
 frameworks for developing
 do not cover the main metaheuristic algorithms, we will present
 the only three works to our knowledge, which propose open source
 frameworks for developing

-metaheuristics on GPUs.\\
+metaheuristics on GPUs.

 
 The three frameworks reviewed in this section are
 PUGACE\index{GPU-based frameworks!PUGACE}~\cite{pugace},
 
 The three frameworks reviewed in this section are
 PUGACE\index{GPU-based frameworks!PUGACE}~\cite{pugace},


@@ -950,7 +950,7 @@ etc.). Different problem encoding is also supported. The framework
 is organized as a set of modules in which the different
 functionalities  are separated as much as possible in order to
 facilitate the extension of the framework. All
 is organized as a set of modules in which the different
 functionalities  are separated as much as possible in order to
 facilitate the extension of the framework. All

-the functions and procedures that execute on GPU are implemented in the same file kernel.cu. \\
+the functions and procedures that execute on GPU are implemented in the same file kernel.cu. 

 
 The implementation strategy adopted on the GPU is as follows.
 Population initialization is done on the CPU side and the
 
 The implementation strategy adopted on the GPU is as follows.
 Population initialization is done on the CPU side and the


@@ -964,7 +964,7 @@ probability of application which may give different results from
 one thread to the other), the decision of whether to apply a
 crossover is taken at the block level and applied to all the
 individuals within the block. It is the decision on the choose
 one thread to the other), the decision of whether to apply a
 crossover is taken at the block level and applied to all the
 individuals within the block. It is the decision on the choose

-of the cutting point for the crossover.\\
+of the cutting point for the crossover.

 
 The framework is validated on standard benchmarks of the QAP. Speedups of $15.44 \times$  to $18.41
 \times$ are achieved compared to a CPU implementation of a cEA
 
 The framework is validated on standard benchmarks of the QAP. Speedups of $15.44 \times$  to $18.41
 \times$ are achieved compared to a CPU implementation of a cEA


@@ -1002,7 +1002,7 @@ parallelization techniques and deployment on GPUs. The framework
 allows one to efficiently manage the hierarchical organization of
 the memories (latencies and sizes) of the GPU and its
 communication with the CPU as well as the minimizing of the user
 allows one to efficiently manage the hierarchical organization of
 the memories (latencies and sizes) of the GPU and its
 communication with the CPU as well as the minimizing of the user

-involvement in its management.\\
+involvement in its management.

 
 \begin{figure}[h]
 \centerline{\includegraphics[width=0.8\textwidth]{Chapters/chapter9/figures/paradiseoGPU.pdf}}
 
 \begin{figure}[h]
 \centerline{\includegraphics[width=0.8\textwidth]{Chapters/chapter9/figures/paradiseoGPU.pdf}}


@@ -1026,7 +1026,7 @@ it is sent back to the CPU which selects the best solution (See
 Figure~\ref{ch1:fig:paradiseoGPU}).
 
 \subsection{libCUDAOptimize: an open source library of GPU-based metaheuristics}
 Figure~\ref{ch1:fig:paradiseoGPU}).
 
 \subsection{libCUDAOptimize: an open source library of GPU-based metaheuristics}

-\index{GPU-based frameworks!libCudaOptimize}
+\index{GPU-based frameworks!libCUDAOptimize}

 LibCUDAOptimize~\cite{libcuda} is a C++/CUDA open source library
 for the design and implementation of metaheuristics on GPUs. Until
 now, the metaheuristics supported by LibCUDAOptimize are: scatter
 LibCUDAOptimize~\cite{libcuda} is a C++/CUDA open source library
 for the design and implementation of metaheuristics on GPUs. Until
 now, the metaheuristics supported by LibCUDAOptimize are: scatter


@@ -1054,7 +1054,7 @@ iteration which starts from a disrupted local optima reached by
 the previous local search process. The disruption heuristic is a
 performance parameter of an ILS algorithm and should be
 judiciously defined. A template of an
 the previous local search process. The disruption heuristic is a
 performance parameter of an ILS algorithm and should be
 judiciously defined. A template of an

-ILS algorithm is given by the Algorithm~\ref{ch8:ils_algorithm_template}.\\
+ILS algorithm is given by the Algorithm~\ref{ch8:ils_algorithm_template}.

 
 \begin{algorithm}[H]
     \SetAlgoLined
 
 \begin{algorithm}[H]
     \SetAlgoLined


@@ -1101,7 +1101,7 @@ optimization community due to its wide range of practical
 applications (site planning, schedule problem, computer-aided
 design, etc.) and to its computational challenges since it is
 considered as one of the most computationally difficult
 applications (site planning, schedule problem, computer-aided
 design, etc.) and to its computational challenges since it is
 considered as one of the most computationally difficult

-optimization problems.\\
+optimization problems.

 
 The Q3AP was first introduced by William P. Pierskalla in
 1967~\cite{Pierskalla1967Q3AP} and, unlike the QAP, the Q3AP is a
 
 The Q3AP was first introduced by William P. Pierskalla in
 1967~\cite{Pierskalla1967Q3AP} and, unlike the QAP, the Q3AP is a


@@ -1155,7 +1155,7 @@ where $\pi_1$ and $\pi_2$ are permutations over the set
 $\left\lbrace 0,1,\ldots,n-1\right\rbrace$. According to this
 formulation, minimizing the Q3AP consists of finding a double
 permutation $(\pi_1,\pi_2)$ which minimizes the objective function
 $\left\lbrace 0,1,\ldots,n-1\right\rbrace$. According to this
 formulation, minimizing the Q3AP consists of finding a double
 permutation $(\pi_1,\pi_2)$ which minimizes the objective function

-(\ref{Q3APPermutationBasedFormulation}).\\
+(\ref{Q3APPermutationBasedFormulation}).

 
 The Q3AP is proven to be an NP-hard problem since it is an
 extension of the quadratic assignment problem and of the axial
 
 The Q3AP is proven to be an NP-hard problem since it is an
 extension of the quadratic assignment problem and of the axial


@@ -1176,7 +1176,7 @@ escape local optima. However, this strategy may generate cycles,
 i.e., previous moves can be selected again. To avoid cycles, the
 TS manages a short-term memory that contains the moves that have
 been recently performed. A TS
 i.e., previous moves can be selected again. To avoid cycles, the
 TS manages a short-term memory that contains the moves that have
 been recently performed. A TS

-template is given by Algorithm \ref{TS_pseudo_code}.\\
+template is given by Algorithm \ref{TS_pseudo_code}.

 %
 %  \begin{algorithm}
 %  \label{TS_pseudo_code}
 %
 %  \begin{algorithm}
 %  \label{TS_pseudo_code}


@@ -1220,7 +1220,7 @@ The neighborhood function is a crucial parameter in any local
 search algorithm. Indeed, if the neighborhood function is not
 adequate to the problem and/or does not consider the targeted
 computing framework, any local search algorithm will fail to reach
 search algorithm. Indeed, if the neighborhood function is not
 adequate to the problem and/or does not consider the targeted
 computing framework, any local search algorithm will fail to reach

-good quality solutions of the search space.\\
+good quality solutions of the search space.

 
 Regarding the Q3AP, many neighborhood structures can be
 considered. A basic neighborhood was proposed and investigated in
 
 Regarding the Q3AP, many neighborhood structures can be
 considered. A basic neighborhood was proposed and investigated in


@@ -1262,7 +1262,7 @@ algorithms~\cite{Ahuja:2007:VLN:1528422.1528438}. However, for
 such a neighborhood, the generation/evaluation step of an LS
 becomes a time-consuming task and may dramatically increase the
 computational time of the LS process. This
 such a neighborhood, the generation/evaluation step of an LS
 becomes a time-consuming task and may dramatically increase the
 computational time of the LS process. This

-justifies the use of intensive data-parallelism provided by GPUs where all neighboring solutions may be concurrently evaluated. \\
+justifies the use of intensive data-parallelism provided by GPUs where all neighboring solutions may be concurrently evaluated. 

 
 The considered large-sized neighborhood consists of swapping two
 positions in both permutations $\pi_1$ and $\pi_2$. This
 
 The considered large-sized neighborhood consists of swapping two
 positions in both permutations $\pi_1$ and $\pi_2$. This


@@ -1299,7 +1299,7 @@ management\index{GPU!memory management} and the
 capacity constraints of GPU memories, and the thread
 synchronization. All these issues must be regarded when designing
 parallel LS models to allow
 capacity constraints of GPU memories, and the thread
 synchronization. All these issues must be regarded when designing
 parallel LS models to allow

-solving of large scale optimization problems on GPU architectures.\\
+solving of large scale optimization problems on GPU architectures.

 
 To go back to our problem (i.e., Q3AP), we propose in
 Algorithm~\ref{ch8:algoITS} an iterated tabu
 
 To go back to our problem (i.e., Q3AP), we propose in
 Algorithm~\ref{ch8:algoITS} an iterated tabu


@@ -1403,7 +1403,7 @@ optima reached by the current TS process and reconsidering it as
 initial solution of the following TS process. Regarding our
 algorithm, the applied perturbation is a random number $\mu $ of
 swaps in either the first or the second permutation where $\mu \in
 initial solution of the following TS process. Regarding our
 algorithm, the applied perturbation is a random number $\mu $ of
 swaps in either the first or the second permutation where $\mu \in

-[2:n]$ ($n$ is the instance size).\\
+[2:n]$ ($n$ is the instance size).

 
 Experiments have been carried out on a node of the Chirloute
 cluster of the Lille site. This is one of the 10 sites that
 
 Experiments have been carried out on a node of the Chirloute
 cluster of the Lille site. This is one of the 10 sites that


@@ -1413,7 +1413,7 @@ of an Intel Xeon E5620 CPU and a NVIDIA Tesla Fermi M2050 (448
 cores) GPU type. The number of ILS iterations and the number of TS
 iterations were set respectively to 20 and 500. The tabu list size
 has been initalized to $\frac{m}{4}$, $m$ being the size of the
 cores) GPU type. The number of ILS iterations and the number of TS
 iterations were set respectively to 20 and 500. The tabu list size
 has been initalized to $\frac{m}{4}$, $m$ being the size of the

-neighborhood set.\\
+neighborhood set.

 
 Table \ref{ch8:ITSQ3APResults} reports the obtained results for
 the GPU-ITS using our large-sized neighborhood structure. The
 
 Table \ref{ch8:ITSQ3APResults} reports the obtained results for
 the GPU-ITS using our large-sized neighborhood structure. The


@@ -1428,7 +1428,7 @@ evaluation function of the parallel GPU version have been
 measured. The number of successful tries (hits) and the average
 number of ILS iterations to converge to the optimal/best known
 value are also represented. The associated standard deviation for
 measured. The number of successful tries (hits) and the average
 number of ILS iterations to converge to the optimal/best known
 value are also represented. The associated standard deviation for

-each average measurement is shown in small type characters.\\
+each average measurement is shown in small type characters.

 
 \begin{table}
 %\tiny
 
 \begin{table}
 %\tiny


@@ -1449,7 +1449,7 @@ Nug22-d & $42476$ & $43282.1$ & $44140$ & $25$\% & $4506.5$ & $32.7$ & $137.8 \t
 & & \tiny{$529.6$} & & & \tiny{$341.1$} & \tiny{$6.6$} & & \tiny{$4.0$} \\ \hline
 \end{tabular}
 \caption{Results of the GPU-based iterated tabu search for
 & & \tiny{$529.6$} & & & \tiny{$341.1$} & \tiny{$6.6$} & & \tiny{$4.0$} \\ \hline
 \end{tabular}
 \caption{Results of the GPU-based iterated tabu search for

-different Q3AP instances.} \label{ch8:ITSQ3APResults} \center
+different Q3AP instances.} \label{ch8:ITSQ3APResults} % \center %Shashi

 \end{table}
 
 %\begin{table}
 \end{table}
 
 %\begin{table}


@@ -1491,7 +1491,7 @@ This chapter has presented state-of-the-art GPU-based parallel
 metaheuristics and
 a case study on implementing large neighborhood local search
 methods on GPUs for solving large benchmarks of the quadratic
 metaheuristics and
 a case study on implementing large neighborhood local search
 methods on GPUs for solving large benchmarks of the quadratic

-three-dimensional assignment problem (Q3AP). \\
+three-dimensional assignment problem (Q3AP). 

 
 Implementing parallel
 metaheuristics on
 
 Implementing parallel
 metaheuristics on