Annotation of OpenXM/doc/issac2000/homogeneous-network.tex, Revision 1.7
1.7 ! takayama 1: % $OpenXM: OpenXM/doc/issac2000/homogeneous-network.tex,v 1.6 2000/01/15 02:24:18 takayama 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}
20: Input :\= $f_1, f_2 \in Z[x]$\\
21: \> such that $deg(f_1), deg(f_2) < 2^M$\\
22: Output : $f = f_1f_2 \bmod p$\\
23: $P \leftarrow$ \= $\{m_1,\cdots,m_N\}$ where $m_i$ is a prime, \\
24: \> $2^{M+1}|m_i-1$ and $m=\prod m_i $ is sufficiently large. \\
25: Separate $P$ into disjoint subsets $P_1, \cdots, P_L$.\\
26: for \= $j=1$ to $L$ $M_j \leftarrow \prod_{m_i\in P_j} m_i$\\
27: Compute $F_j$ such that $F_j \equiv f_1f_2 \bmod M_j$\\
28: \> and $F_j \equiv 0 \bmod m/M_j$ in parallel.\\
29: \> ($f_1, f_2$ are regarded as integral.\\
30: \> The product is computed by FFT.)\\
31: return $\phi_m(\sum F_j)$\\
32: (For $a \in Z$, $\phi_m(a) \in (-m/2,m/2)$ and $\phi_m(a)\equiv a \bmod m$)
33: \end{tabbing}
34:
35: Figure \ref{speedup}
36: shows the speedup factor under the above distributed computation
37: on {\tt Risa/Asir}. For each $n$, two polynomials of degree $n$
38: with 3000bit coefficients are generated and the product is computed.
39: The machine is Fujitsu AP3000,
40: a cluster of Sun connected with a high speed network and MPI over the
41: network is used to implement OpenXM.
42: \begin{figure}[htbp]
43: \epsfxsize=8.5cm
44: \epsffile{speedup.ps}
45: \caption{Speedup factor}
46: \label{speedup}
47: \end{figure}
48:
49: The task of a client is the generation and partition of $P$, sending
50: and receiving of polynomials and the synthesis of the result. If the
1.5 takayama 51: number of servers is $L$ and the inputs are fixed, then the cost to
52: compute $F_j$ in parallel is $O(1/L)$, whereas the cost
53: to send and receive polynomials is $O(L)$
1.3 noro 54: because we don't have the broadcast and the reduce
55: operations. Therefore the speedup is limited and the upper bound of
1.4 noro 56: the speedup factor depends on the ratio of
57: the computational cost and the communication cost.
58: Figure \ref{speedup} shows that
1.5 takayama 59: the speedup is satisfactory if the degree is large and $L$
60: is not large, say, up to 10 under the above envionment.
61: If OpenXM provides the broadcast and the reduce operations, the cost of
62: sending $f_1$, $f_2$ and gathering $F_j$ may be reduced to $O(log_2L)$
63: and we will obtain better results in such a case.
1.3 noro 64:
1.5 takayama 65: \subsubsection{Competitive distributed computation by various strategies}
1.3 noro 66:
67: Singular \cite{Singular} implements {\tt MP} interface for distributed
68: computation and a competitive Gr\"obner basis computation is
1.5 takayama 69: illustrated as an example of distributed computation.
70: Such a distributed computation is also possible on OpenXM.
71: The following {\tt Risa/Asir} function computes a Gr\"obner basis by
1.3 noro 72: starting the computations simultaneously from the homogenized input and
73: the input itself. The client watches the streams by {\tt ox\_select()}
74: and The result which is returned first is taken. Then the remaining
75: server is reset.
76:
77: \begin{verbatim}
78: /* G:set of polys; V:list of variables */
79: /* O:type of order; P0,P1: id's of servers */
80: def dgr(G,V,O,P0,P1)
81: {
82: P = [P0,P1]; /* server list */
83: map(ox_reset,P); /* reset servers */
84: /* P0 executes non-homogenized computation */
85: ox_cmo_rpc(P0,"dp_gr_main",G,V,0,1,O);
86: /* P1 executes homogenized computation */
87: ox_cmo_rpc(P1,"dp_gr_main",G,V,1,1,O);
88: map(ox_push_cmd,P,262); /* 262 = OX_popCMO */
89: F = ox_select(P); /* wait for data */
90: /* F[0] is a server's id which is ready */
91: R = ox_get(F[0]);
92: if ( F[0] == P0 ) {
93: Win = "nonhomo"; Lose = P1;
94: } else {
95: Win = "homo"; Lose = P0;
96: }
97: ox_reset(Lose); /* reset the loser */
98: return [Win,R];
99: }
100: \end{verbatim}
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