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From Apache Wiki <wikidi...@apache.org>
Subject [Hama Wiki] Update of "SpMV" by Mikalai Parafeniuk
Date Wed, 27 Jun 2012 08:17:05 GMT
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The "SpMV" page has been changed by Mikalai Parafeniuk:

  == Distributed Sparse Matrix-Vector Multiplication on Hama ==
- NOTE: the most important issues are locality and load balance partitioning. But, this is
very initial stage.
+ === Introduction ===
+ In further description we will research problem in form u = Av. Most computational algoritms
spend large percent of time for solving systems of linear equations. In general, linear system
of equations can be represented in matrix form Ax = b, where A is matrix with n rows and n
columns, b - vector of size n, x - unknown solution vector which we are searching. Some approaches
for solving linear systems has iterative nature. Assume, we know the initial approximation
of x = x0. After that we represen our system in form xn = Bxn-1 + c, where c - vector of size
n. After that we have to found next approximations of x till the convergence. In real world
most of matrices contain relatively small number of non-zero items in comparison to total
number of matrix items. Such matrices are called sparse matrices, matrices which filled most
with non-zero items are called dense. So this page will describe the problem of sparse matrix
vector multiplication(SpMV) with use of Bulk Synchronous Programming(BSP) model implemented
in Apache Hama project. As shown above, SpMV can be used in different iterative solvers for
system of linear equations.
+ Bulk Synchronous model proposes it's own smart way of parallelization of programs. The input
problem is separated by peers. Peers can be a processors, threads, separate machines, different
items of cloud. BSP algorithm is divided in sequence of supersteps. Barrier synchronization
of all peers is made after each superstep. The implementation of BSP(Apache Hama) contains
primitives for defining peer number, communication with other peers with different communication
primitives, optimizations of communication between peers.
- === Distributed SpMV introduction ===
+ === Problem description ===
+ As a sequential problem SpMV is almost trivial problem. But in case of parallel version
we should think about some additional aspects:
+ 1. Partitioning of matrix and vector components. This means that we should split the input
matrix and vectors by peers, if we want to have benefits from usage of parallel algorithm.
Wise partitioning should be taken or communication time will rise very much and algorithm
will be inefficient.
+ 2. Load balancing. This means that each peer must perform nearly the same amount of work,
and none of them should idle.
+ 3. We should keep communication in bounds. In case of paralel SpMV we should take partitioning
wise to keep communication in appropriate bounds independently of sparsity patterns of input
matrix and vector.
- ...
+ === Implementation tips ===
+ 1. Order of distribution and representation. We have two choices in this aspect: represent
matrix first and distribute later, or distribute matrix first and represent later. In first
case  (represent first, distribute later) all simple operations will be non-local and will
bring some unnecessary overhead. In other case (distribute first, represent later) all local
operations on processor remain local: algorithm first determines responsible processor and
it performs operation locally. Thats why I prefer distribution first representation later
+ 2. Data transmission direction. Here we also have two choices: delivery vector component
to processor which possesses non-zero matrix component or vice versa. In most cases a number
of non-zero items in matrix is much larger than vector length, thats why we prefer transmission
of vector.
- === BSP Implementation ===
+ === Algorithm description ===
+ The generic algorithm will be divided in three supersteps:
+ 0. Matrix and vector distribution.
+ 1. Fanout.
+ 2. Local computation.
+ 3. Fanin.
+ In Fanout phase all processors gets needed v components. In local computation phase local
contribution to result vector is calculated. In Fanin phase all local contributions are sent
to an owner of u. Most of efforts should be taken to choose right matrix and vector distribution
which will improve the comunication volume of Fanout and Fanin phase. As base implementation
of distribution I propose to create Cartesian (column mappings are not dependent of row mappings
and vice versa) cyclic-block distribution with cyclical distribution of matrix diagonal. Also
I assume that distr(u) != distr(v), which gives us more freedom in optimising vector distribution.
This type of distribution has such advantages: it is simple, in fanin only communication with
processor column is needed, in fanout only communication with processor row is needed, we
can easily predict the productivity of algorithm. After matrix distribution we perform vector
distribution in greedy way for each processor: processor grabs items until he reaches it's
optimum state. After that stage some vector components can be unassigned (nearly 10%). We
well distribute them in greedy fashion to. To support efficient local computation used data
structure should provide indeces of rows and columns which have the non-zero item in them.
Local computation must be performed with local indeces.
-  * Task partitioning to multi BSP processors
-  * Algorithm pseudo code
+ === Possible improvements ===
+ 1. Implementation of Mondrian distribution. In most cases it gives better results in comparison
with cyclic-block Cartesian scheme.
+ 2. Implement algorithm for detecting matrix sparsity patterns. This will give us a possibility
to define, for example, if matrix is random sparse matrix, or Laplacian matrix. In case of
random sparse matrices we can use distribution patterns which are independent of matrix sparsity
pattern. In case of Laplacian matrices we diamond distribution can give better result.
+ 3. In phase of vector distribution when some vectors remain unassigned we can use graph
algoritms to determine the owner of vector component.
- ...
+ === Literature ===
+ 1. Rob H. Bisseling - Parallel Scientific computation. (chapter 4).
+ 2. Steve Rennich - Block SpMV on GPU.
- === Ideas to improve the performance ===
- NOTE: list ideas you found.
-  * Reordering and Partitioning
-  * ..., etc.

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