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RCS file: /home/cvs/OpenXM/doc/issac2000/homogeneous-network.tex,v
retrieving revision 1.3
retrieving revision 1.6
diff -u -p -r1.3 -r1.6
--- OpenXM/doc/issac2000/homogeneous-network.tex	2000/01/07 06:27:55	1.3
+++ OpenXM/doc/issac2000/homogeneous-network.tex	2000/01/15 02:24:18	1.6
@@ -1,9 +1,10 @@
-% $OpenXM: OpenXM/doc/issac2000/homogeneous-network.tex,v 1.2 2000/01/02 07:32:12 takayama Exp $
+% $OpenXM: OpenXM/doc/issac2000/homogeneous-network.tex,v 1.5 2000/01/15 00:20:45 takayama Exp $
 
 \section{Applications}
 \subsection{Distributed computation with homogeneous servers}
+\label{section:homog}
 
-OpenXM also aims at speedup by a distributed computation
+One of the aims of OpenXM is a parallel speedup by a distributed computation
 with homogeneous servers. As the current specification of OpenXM does
 not include communication between servers, one cannot expect
 the maximal parallel speedup. However it is possible to execute
@@ -48,29 +49,27 @@ network is used to implement OpenXM.
 
 The task of a client is the generation and partition of $P$, sending
 and receiving of polynomials and the synthesis of the result. If the
-number of servers is $L$ and the inputs are fixed, then the time to
-compute $F_j$ in parallel is proportional to $1/L$, whereas the time
-for sending and receiving of polynomials is proportional to $L$
+number of servers is $L$ and the inputs are fixed, then the cost to
+compute $F_j$ in parallel is $O(1/L)$, whereas the cost
+to send and receive polynomials is $O(L)$
 because we don't have the broadcast and the reduce
 operations. Therefore the speedup is limited and the upper bound of
-the speedup factor depends on the communication cost and the degree
-of inputs. Figure \ref{speedup} shows that 
-the speedup is satisfactory if the degree is large and the number of
-servers is not large, say, up to 10.
+the speedup factor depends on the ratio of 
+the computational cost and the communication cost.
+Figure \ref{speedup} shows that 
+the speedup is satisfactory if the degree is large and $L$
+is not large, say, up to 10 under the above envionment.
+If OpenXM provides the broadcast and the reduce operations, the cost of 
+sending $f_1$, $f_2$ and gathering $F_j$ may be reduced to $O(log_2L)$
+and we will obtain better results in such a case.
 
-\subsubsection{Order counting of an elliptic curve}
+\subsubsection{Competitive distributed computation by various strategies}
 
-\subsubsection{Gr\"obner basis computation by various methods}
-
 Singular \cite{Singular} implements {\tt MP} interface for distributed
 computation and a competitive Gr\"obner basis computation is
-illustrated as an example of distributed computation.  However,
-interruption has not implemented yet and the looser process have to be
-killed explicitly. As stated in Section \ref{secsession} OpenXM
-provides such a function and one can safely reset the server and
-continue to use it.  Furthermore, if a client provides synchronous I/O
-multiplexing by {\tt select()}, then a polling is not necessary.  The
-following {\tt Risa/Asir} function computes a Gr\"obner basis by
+illustrated as an example of distributed computation.
+Such a distributed computation is also possible on OpenXM.
+The following {\tt Risa/Asir} function computes a Gr\"obner basis by
 starting the computations simultaneously from the homogenized input and
 the input itself.  The client watches the streams by {\tt ox\_select()}
 and The result which is returned first is taken. Then the remaining