Annotation of OpenXM/doc/ascm2001p/homogeneous-network.tex, Revision 1.7
1.7 ! takayama 1: % $OpenXM: OpenXM/doc/ascm2001p/homogeneous-network.tex,v 1.6 2001/06/20 03:18:21 noro Exp $
1.1 noro 2:
3: \subsection{Distributed computation with homogeneous servers}
4: \label{section:homog}
5:
6: One of the aims of OpenXM is a parallel speedup by a distributed computation
1.7 ! takayama 7: with homogeneous servers. Let us see some examples.
1.5 takayama 8: %As the current specification of OpenXM does
9: %not include communication between servers, one cannot expect
10: %the maximal parallel speedup. However it is possible to execute
11: %several types of distributed computation as follows.
1.1 noro 12:
1.3 noro 13: \subsubsection{Competitive distributed computation by various strategies}
1.1 noro 14:
1.3 noro 15: SINGULAR \cite{Singular} implements {\it MP} interface for distributed
16: computation and a competitive Gr\"obner basis computation is
1.7 ! takayama 17: illustrated as an example of distributed computation by the MP interface.
1.6 noro 18: Such a distributed computation is also possible on OpenXM.
1.3 noro 19:
20: \begin{verbatim}
1.6 noro 21: extern Proc1,Proc2$ Proc1 = -1$ Proc2 = -1$
1.3 noro 22: /* G:set of polys; V:list of variables */
23: /* Mod: the Ground field GF(Mod); O:type of order */
24: def dgr(G,V,Mod,O)
25: {
26: /* invoke servers if necessary */
27: if ( Proc1 == -1 ) Proc1 = ox_launch();
28: if ( Proc2 == -1 ) Proc2 = ox_launch();
29: P = [Proc1,Proc2];
30: map(ox_reset,P); /* reset servers */
31: /* P0 executes Buchberger algorithm over GF(Mod) */
32: ox_cmo_rpc(P[0],"dp_gr_mod_main",G,V,0,Mod,O);
33: /* P1 executes F4 algorithm over GF(Mod) */
34: ox_cmo_rpc(P[1],"dp_f4_mod_main",G,V,Mod,O);
35: map(ox_push_cmd,P,262); /* 262 = OX_popCMO */
36: F = ox_select(P); /* wait for data */
37: /* F[0] is a server's id which is ready */
38: R = ox_get(F[0]);
39: if ( F[0] == P[0] ) { Win = "Buchberger"; Lose = P[1]; }
40: else { Win = "F4"; Lose = P[0]; }
41: ox_reset(Lose); /* reset the loser */
42: return [Win,R];
43: }
44: \end{verbatim}
1.6 noro 45: In the above Asir program, the client creates two servers and it requests
1.7 ! takayama 46: Gr\"obner basis computations by the Buchberger algorithm
! 47: and the $F_4$ algorithm to the servers for the same input.
1.6 noro 48: The client watches the streams by {\tt ox\_select()}
49: and the result which is returned first is taken. Then the remaining
50: server is reset.
1.1 noro 51:
1.4 noro 52: \subsubsection{Nesting of client-server communication}
53:
1.7 ! takayama 54: %%Prog: load ("dfff"); df_demo(); enter 100.
1.4 noro 55: Under OpenXM-RFC 100 an OpenXM server can be a client of other servers.
56: Figure \ref{tree} illustrates a tree-like structure of an OpenXM
57: client-server communication.
58: \begin{figure}
59: \label{tree}
60: \begin{center}
61: \begin{picture}(200,70)(0,0)
62: \put(70,70){\framebox(40,15){client}}
63: \put(20,30){\framebox(40,15){server}}
64: \put(70,30){\framebox(40,15){server}}
65: \put(120,30){\framebox(40,15){server}}
66: \put(0,0){\framebox(40,15){server}}
67: \put(50,0){\framebox(40,15){server}}
68: \put(150,0){\framebox(40,15){server}}
69:
70: \put(90,70){\vector(-2,-1){43}}
71: \put(90,70){\vector(0,-1){21}}
72: \put(90,70){\vector(2,-1){43}}
73: \put(40,30){\vector(-2,-1){22}}
74: \put(40,30){\vector(2,-1){22}}
75: \put(140,30){\vector(2,-1){22}}
76: \end{picture}
77: \caption{Tree-like structure of client-server communication}
78: \end{center}
79: \end{figure}
80: Such a computational model is useful for parallel implementation of
81: algorithms whose task can be divided into subtasks recursively.
82:
1.1 noro 83: %A typical example is {\it quicksort}, where an array to be sorted is
84: %partitioned into two sub-arrays and the algorithm is applied to each
85: %sub-array. In each level of recursion, two subtasks are generated
86: %and one can ask other OpenXM servers to execute them.
87: %Though it makes little contribution to the efficiency in the case of
88: %quicksort, we present an Asir program of this distributed quicksort
89: %to demonstrate that OpenXM gives an easy way to test this algorithm.
90: %In the program, a predefined constant {\tt LevelMax} determines
91: %whether new servers are launched or whole subtasks are done on the server.
92: %
93: %\begin{verbatim}
94: %#define LevelMax 2
95: %extern Proc1, Proc2;
96: %Proc1 = -1$ Proc2 = -1$
97: %
98: %/* sort [A[P],...,A[Q]] by quicksort */
99: %def quickSort(A,P,Q,Level) {
100: % if (Q-P < 1) return A;
101: % Mp = idiv(P+Q,2); M = A[Mp]; B = P; E = Q;
102: % while (1) {
103: % while (A[B] < M) B++;
104: % while (A[E] > M && B <= E) E--;
105: % if (B >= E) break;
106: % else { T = A[B]; A[B] = A[E]; A[E] = T; E--; }
107: % }
108: % if (E < P) E = P;
109: % if (Level < LevelMax) {
110: % /* launch new servers if necessary */
111: % if (Proc1 == -1) Proc1 = ox_launch(0);
112: % if (Proc2 == -1) Proc2 = ox_launch(0);
113: % /* send the requests to the servers */
114: % ox_rpc(Proc1,"quickSort",A,P,E,Level+1);
115: % ox_rpc(Proc2,"quickSort",A,E+1,Q,Level+1);
116: % if (E-P < Q-E) {
117: % A1 = ox_pop_local(Proc1);
118: % A2 = ox_pop_local(Proc2);
119: % }else{
120: % A2 = ox_pop_local(Proc2);
121: % A1 = ox_pop_local(Proc1);
122: % }
123: % for (I=P; I<=E; I++) A[I] = A1[I];
124: % for (I=E+1; I<=Q; I++) A[I] = A2[I];
125: % return(A);
126: % }else{
127: % /* everything is done on this server */
128: % quickSort(A,P,E,Level+1);
129: % quickSort(A,E+1,Q,Level+1);
130: % return(A);
131: % }
132: %}
133: %\end{verbatim}
1.3 noro 134: %
1.4 noro 135: A typical example is a parallelization of the Cantor-Zassenhaus
1.7 ! takayama 136: algorithm for polynomial factorization over finite fields,
1.4 noro 137: which is a recursive algorithm.
138: At each level of the recursion, a given polynomial can be
139: divided into two non-trivial factors with some probability by using
140: a randomly generated polynomial as a {\it separator}.
141: We can apply the following simple parallelization:
142: When two non-trivial factors are generated on a server,
143: one is sent to another server and the other factor is factorized on the server
144: itself.
1.1 noro 145: %\begin{verbatim}
146: %/* factorization of F */
147: %/* E = degree of irreducible factors in F */
148: %def c_z(F,E,Level)
149: %{
150: % V = var(F); N = deg(F,V);
151: % if ( N == E ) return [F];
152: % M = field_order_ff(); K = idiv(N,E); L = [F];
153: % while ( 1 ) {
1.7 ! takayama 154: % /* generate a random polynomial */
1.1 noro 155: % W = monic_randpoly_ff(2*E,V);
156: % /* compute a power of the random polynomial */
157: % T = generic_pwrmod_ff(W,F,idiv(M^E-1,2));
158: % if ( !(W = T-1) ) continue;
159: % /* G = GCD(F,W^((M^E-1)/2)) mod F) */
160: % G = ugcd(F,W);
161: % if ( deg(G,V) && deg(G,V) < N ) {
162: % /* G is a non-trivial factor of F */
163: % if ( Level >= LevelMax ) {
164: % /* everything is done on this server */
165: % L1 = c_z(G,E,Level+1);
166: % L2 = c_z(sdiv(F,G),E,Level+1);
167: % } else {
168: % /* launch a server if necessary */
169: % if ( Proc1 < 0 ) Proc1 = ox_launch();
170: % /* send a request with Level = Level+1 */
171: % /* ox_c_z is a wrapper of c_z on the server */
172: % ox_cmo_rpc(Proc1,"ox_c_z",lmptop(G),E,
173: % setmod_ff(),Level+1);
174: % /* the rest is done on this server */
175: % L2 = c_z(sdiv(F,G),E,Level+1);
176: % L1 = map(simp_ff,ox_pop_cmo(Proc1));
177: % }
178: % return append(L1,L2);
179: % }
180: % }
181: %}
182: %\end{verbatim}
183: %
184: %
185: %
186: %
187: %
188: %
189: %
1.2 noro 190:
191: \subsubsection{Product of univariate polynomials}
192:
193: Shoup \cite{Shoup} showed that the product of univariate polynomials
194: with large degrees and large coefficients can be computed efficiently
195: by FFT over small finite fields and Chinese remainder theorem.
196: It can be easily parallelized:
197:
198: \begin{tabbing}
199: Input :\= $f_1, f_2 \in {\bf Z}[x]$ such that $deg(f_1), deg(f_2) < 2^M$\\
200: Output : $f = f_1f_2$ \\
201: $P \leftarrow$ \= $\{m_1,\cdots,m_N\}$ where $m_i$ is an odd prime, \\
202: \> $2^{M+1}|m_i-1$ and $m=\prod m_i $ is sufficiently large. \\
203: Separate $P$ into disjoint subsets $P_1, \cdots, P_L$.\\
204: for \= $j=1$ to $L$ $M_j \leftarrow \prod_{m_i\in P_j} m_i$\\
205: Compute $F_j$ such that $F_j \equiv f_1f_2 \bmod M_j$\\
206: \> and $F_j \equiv 0 \bmod m/M_j$ in parallel.\\
207: \> (The product is computed by FFT.)\\
208: return $\phi_m(\sum F_j)$\\
209: (For $a \in {\bf Z}$, $\phi_m(a) \in (-m/2,m/2)$ and $\phi_m(a)\equiv a \bmod m$)
210: \end{tabbing}
211:
212: Figure \ref{speedup}
213: shows the speedup factor under the above distributed computation
214: on Risa/Asir. For each $n$, two polynomials of degree $n$
215: with 3000bit coefficients are generated and the product is computed.
216: The machine is FUJITSU AP3000,
217: a cluster of Sun workstations connected with a high speed network
218: and MPI over the network is used to implement OpenXM.
219: \begin{figure}[htbp]
220: \epsfxsize=8.5cm
221: \epsffile{speedup.ps}
222: \caption{Speedup factor}
223: \label{speedup}
224: \end{figure}
225:
226: If the number of servers is $L$ and the inputs are fixed, then the cost to
227: compute $F_j$ in parallel is $O(1/L)$, whereas the cost
228: to send and receive polynomials is $O(L)$ if {\tt ox\_push\_cmo()} and
229: {\tt ox\_pop\_cmo()} are repeatedly applied on the client.
230: Therefore the speedup is limited and the upper bound of
231: the speedup factor depends on the ratio of
232: the computational cost and the communication cost for each unit operation.
233: Figure \ref{speedup} shows that
234: the speedup is satisfactory if the degree is large and $L$
235: is not large, say, up to 10 under the above environment.
236: If OpenXM provides collective operations for broadcast and reduction
237: such as {\tt MPI\_Bcast} and {\tt MPI\_Reduce} respectively, the cost of
238: sending $f_1$, $f_2$ and gathering $F_j$ may be reduced to $O(\log_2L)$
239: and we can expect better results in such a case. In order to implement
240: such operations we need new specifications for inter-sever communication
241: and the session management, which will be proposed as OpenXM-RFC 102.
242: We note that preliminary experiments show the collective operations
243: work well on OpenXM.
244:
245: %\subsubsection{Competitive distributed computation by various strategies}
246: %
247: %SINGULAR \cite{Singular} implements {\it MP} interface for distributed
248: %computation and a competitive Gr\"obner basis computation is
249: %illustrated as an example of distributed computation.
250: %Such a distributed computation is also possible on OpenXM as follows:
251: %
252: %The client creates two servers and it requests
1.7 ! takayama 253: %Gr\"obner basis computations from the homogenized input and the input itself
1.2 noro 254: %to the servers.
255: %The client watches the streams by {\tt ox\_select()}
256: %and the result which is returned first is taken. Then the remaining
257: %server is reset.
258: %
259: %\begin{verbatim}
260: %/* G:set of polys; V:list of variables */
261: %/* O:type of order; P0,P1: id's of servers */
262: %def dgr(G,V,O,P0,P1)
263: %{
264: % P = [P0,P1]; /* server list */
265: % map(ox_reset,P); /* reset servers */
266: % /* P0 executes non-homogenized computation */
267: % ox_cmo_rpc(P0,"dp_gr_main",G,V,0,1,O);
268: % /* P1 executes homogenized computation */
269: % ox_cmo_rpc(P1,"dp_gr_main",G,V,1,1,O);
270: % map(ox_push_cmd,P,262); /* 262 = OX_popCMO */
271: % F = ox_select(P); /* wait for data */
272: % /* F[0] is a server's id which is ready */
273: % R = ox_get(F[0]);
274: % if ( F[0] == P0 ) {
275: % Win = "nonhomo"; Lose = P1;
276: % } else {
277: % Win = "homo"; Lose = P0;
278: % }
279: % ox_reset(Lose); /* reset the loser */
280: % return [Win,R];
281: %}
282: %\end{verbatim}
283:
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