Annotation of OpenXM/doc/issac2000/homogeneous-network.tex, Revision 1.9
1.9 ! noro 1: % $OpenXM: OpenXM/doc/issac2000/homogeneous-network.tex,v 1.8 2000/01/16 03:15:49 noro Exp $
1.2 takayama 2:
1.3 noro 3: \subsection{Distributed computation with homogeneous servers}
1.6 takayama 4: \label{section:homog}
1.2 takayama 5:
1.5 takayama 6: One of the aims of OpenXM is a parallel speedup by a distributed computation
1.3 noro 7: with homogeneous servers. As the current specification of OpenXM does
8: not include communication between servers, one cannot expect
9: the maximal parallel speedup. However it is possible to execute
10: several types of distributed computation as follows.
11:
12: \subsubsection{Product of univariate polynomials}
13:
14: Shoup \cite{Shoup} showed that the product of univariate polynomials
15: with large degrees and large coefficients can be computed efficiently
16: by FFT over small finite fields and Chinese remainder theorem.
17: It can be easily parallelized:
18:
19: \begin{tabbing}
1.8 noro 20: Input :\= $f_1, f_2 \in {\bf Z}[x]$ such that $deg(f_1), deg(f_2) < 2^M$\\
21: Output : $f = f_1f_2$ \\
22: $P \leftarrow$ \= $\{m_1,\cdots,m_N\}$ where $m_i$ is an odd prime, \\
1.3 noro 23: \> $2^{M+1}|m_i-1$ and $m=\prod m_i $ is sufficiently large. \\
24: Separate $P$ into disjoint subsets $P_1, \cdots, P_L$.\\
25: for \= $j=1$ to $L$ $M_j \leftarrow \prod_{m_i\in P_j} m_i$\\
26: Compute $F_j$ such that $F_j \equiv f_1f_2 \bmod M_j$\\
27: \> and $F_j \equiv 0 \bmod m/M_j$ in parallel.\\
1.8 noro 28: \> (The product is computed by FFT.)\\
1.3 noro 29: return $\phi_m(\sum F_j)$\\
1.8 noro 30: (For $a \in {\bf Z}$, $\phi_m(a) \in (-m/2,m/2)$ and $\phi_m(a)\equiv a \bmod m$)
1.3 noro 31: \end{tabbing}
32:
33: Figure \ref{speedup}
34: shows the speedup factor under the above distributed computation
1.8 noro 35: on Risa/Asir. For each $n$, two polynomials of degree $n$
1.3 noro 36: with 3000bit coefficients are generated and the product is computed.
37: The machine is Fujitsu AP3000,
38: a cluster of Sun connected with a high speed network and MPI over the
39: network is used to implement OpenXM.
40: \begin{figure}[htbp]
41: \epsfxsize=8.5cm
42: \epsffile{speedup.ps}
43: \caption{Speedup factor}
44: \label{speedup}
45: \end{figure}
46:
1.8 noro 47: If the number of servers is $L$ and the inputs are fixed, then the cost to
1.5 takayama 48: compute $F_j$ in parallel is $O(1/L)$, whereas the cost
1.8 noro 49: to send and receive polynomials is $O(L)$ if {\tt ox\_push\_cmo()} and
50: {\tt ox\_pop\_cmo()} are repeatedly applied on the client.
51: Therefore the speedup is limited and the upper bound of
1.4 noro 52: the speedup factor depends on the ratio of
1.8 noro 53: the computational cost and the communication cost for each unit operation.
1.4 noro 54: Figure \ref{speedup} shows that
1.5 takayama 55: the speedup is satisfactory if the degree is large and $L$
56: is not large, say, up to 10 under the above envionment.
57: If OpenXM provides the broadcast and the reduce operations, the cost of
58: sending $f_1$, $f_2$ and gathering $F_j$ may be reduced to $O(log_2L)$
1.8 noro 59: and we can expect better results in such a case.
1.3 noro 60:
1.5 takayama 61: \subsubsection{Competitive distributed computation by various strategies}
1.3 noro 62:
1.9 ! noro 63: SINGULAR \cite{Singular} implements {\tt MP} interface for distributed
1.3 noro 64: computation and a competitive Gr\"obner basis computation is
1.5 takayama 65: illustrated as an example of distributed computation.
66: Such a distributed computation is also possible on OpenXM.
1.8 noro 67: The following Risa/Asir function computes a Gr\"obner basis by
1.3 noro 68: starting the computations simultaneously from the homogenized input and
69: the input itself. The client watches the streams by {\tt ox\_select()}
1.9 ! noro 70: and the result which is returned first is taken. Then the remaining
1.3 noro 71: server is reset.
72:
73: \begin{verbatim}
74: /* G:set of polys; V:list of variables */
75: /* O:type of order; P0,P1: id's of servers */
76: def dgr(G,V,O,P0,P1)
77: {
78: P = [P0,P1]; /* server list */
79: map(ox_reset,P); /* reset servers */
80: /* P0 executes non-homogenized computation */
81: ox_cmo_rpc(P0,"dp_gr_main",G,V,0,1,O);
82: /* P1 executes homogenized computation */
83: ox_cmo_rpc(P1,"dp_gr_main",G,V,1,1,O);
84: map(ox_push_cmd,P,262); /* 262 = OX_popCMO */
85: F = ox_select(P); /* wait for data */
86: /* F[0] is a server's id which is ready */
87: R = ox_get(F[0]);
88: if ( F[0] == P0 ) {
89: Win = "nonhomo"; Lose = P1;
90: } else {
91: Win = "homo"; Lose = P0;
92: }
93: ox_reset(Lose); /* reset the loser */
94: return [Win,R];
95: }
96: \end{verbatim}
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