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\chapter{Data partitioning and load balancing}
\label{cha:lb}
As problem sizes grow, parallel computing has become an important tool
in computational science. Many large-scale computing tasks are
today run on parallel computers, with multiple processors or cores.
An important issue is then how to best divide the data and work
among processes (processors). This problem is known as \emph{data partitioning}
or \emph{load balancing}. We will use these phrases interchangably.
Partitioning or load balancing can either be performed once
(static partitioning) or multiple times (dynamic load balancing).

We wish to divide work evenly but at the same time minimize
communication among processes. Communication could either be
message passing (on distributed memory systems) or memory access
(on shared-memory systems). 

We give a brief overview of the different categories of partitioning methods,
and later explain how to use them in Zoltan.


\section{Geometric methods}
Geometric methods rely on each object having a set of coordinates.
Data objects are partitioned based on geometric locality.
We assume that it is beneficial to keep objects that are close together on 
the same processor, but there is no explicit model of communication.
Examples of geometric methods include recursive coordinate bisection (RCB)
and space-filling curves. An advantage of geometric methods is that
they are very fast to compute, but communication volume in
the application may be high.

\section{Graph partitioning}
Graph partitioning is perhaps the most popular method. This approach is
based on representing the application 
(computation) as a graph, where data objects are vertices and 
data dependencies are edges. The graph partitioning problem is then
to partition the vertices into equal-sized sets, while minimizing the
number of edges with endpoints in different sets (parts).
This is an NP-hard optimization problem, but fast multilevel 
algorithms and software produce good solutions in practice.
In general, graph partitioning produces better quality partitions
than geometric methods but also take longer to compute.


\section{Hypergraph partitioning}
The graph model has several deficiencies. First, only symmetric
relations between pairs of objects can be represented. Second, 
the communication model is inaccurate and does not model 
communication volume correctly.
Hypergraphs generalize graphs, but can represent relationships
among arbitrary sets of objects (not just pairs). Also,
communication volume is exact, so hypergraph methods 
produce very high quality partitions. The main drawback of hypergraph
methods is that they take longer to run than graph algorithms. 

\section{Methods in Zoltan}
Zoltan is a toolkit containing many load balancing methods. We explain how 
to use Zoltan in the next chapter. The methods currently available
in Zoltan (version 3.1) are:
\begin{description}
\item[Simple:] BLOCK, RANDOM. These are intended for testing, not real use.
\item[Geometric:] RCB, RIB, HSFC.
\item[Graph:] Zoltan has a native graph partitioner, and optionlly supports ParMetis and PT-Scotch.
\item[Hypergraph:] Zoltan has a native parallel hypergraph partitioner.
\end{description}