Annotation of OpenXM/doc/ascm2001p/homogeneous-network.tex, Revision 1.5
1.5 ! takayama 1: % $OpenXM: OpenXM/doc/ascm2001p/homogeneous-network.tex,v 1.4 2001/06/20 02:50:16 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
1.5 ! takayama 7: with homogeneous servers.
! 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
17: illustrated as an example of distributed computation.
18: Such a distributed computation is also possible on OpenXM as follows:
19:
20: The client creates two servers and it requests
21: Gr\"obner basis comutations by the Buchberger algorithm the $F_4$ algorithm
22: to the servers for the same input.
23: The client watches the streams by {\tt ox\_select()}
24: and the result which is returned first is taken. Then the remaining
25: server is reset.
26:
27: \begin{verbatim}
28: extern Proc1,Proc2$
29: Proc1 = -1$ Proc2 = -1$
30: /* G:set of polys; V:list of variables */
31: /* Mod: the Ground field GF(Mod); O:type of order */
32: def dgr(G,V,Mod,O)
33: {
34: /* invoke servers if necessary */
35: if ( Proc1 == -1 ) Proc1 = ox_launch();
36: if ( Proc2 == -1 ) Proc2 = ox_launch();
37: P = [Proc1,Proc2];
38: map(ox_reset,P); /* reset servers */
39: /* P0 executes Buchberger algorithm over GF(Mod) */
40: ox_cmo_rpc(P[0],"dp_gr_mod_main",G,V,0,Mod,O);
41: /* P1 executes F4 algorithm over GF(Mod) */
42: ox_cmo_rpc(P[1],"dp_f4_mod_main",G,V,Mod,O);
43: map(ox_push_cmd,P,262); /* 262 = OX_popCMO */
44: F = ox_select(P); /* wait for data */
45: /* F[0] is a server's id which is ready */
46: R = ox_get(F[0]);
47: if ( F[0] == P[0] ) { Win = "Buchberger"; Lose = P[1]; }
48: else { Win = "F4"; Lose = P[0]; }
49: ox_reset(Lose); /* reset the loser */
50: return [Win,R];
51: }
52: \end{verbatim}
1.1 noro 53:
1.4 noro 54: \subsubsection{Nesting of client-server communication}
55:
56: Under OpenXM-RFC 100 an OpenXM server can be a client of other servers.
57: Figure \ref{tree} illustrates a tree-like structure of an OpenXM
58: client-server communication.
59: \begin{figure}
60: \label{tree}
61: \begin{center}
62: \begin{picture}(200,70)(0,0)
63: \put(70,70){\framebox(40,15){client}}
64: \put(20,30){\framebox(40,15){server}}
65: \put(70,30){\framebox(40,15){server}}
66: \put(120,30){\framebox(40,15){server}}
67: \put(0,0){\framebox(40,15){server}}
68: \put(50,0){\framebox(40,15){server}}
69: \put(150,0){\framebox(40,15){server}}
70:
71: \put(90,70){\vector(-2,-1){43}}
72: \put(90,70){\vector(0,-1){21}}
73: \put(90,70){\vector(2,-1){43}}
74: \put(40,30){\vector(-2,-1){22}}
75: \put(40,30){\vector(2,-1){22}}
76: \put(140,30){\vector(2,-1){22}}
77: \end{picture}
78: \caption{Tree-like structure of client-server communication}
79: \end{center}
80: \end{figure}
81: Such a computational model is useful for parallel implementation of
82: algorithms whose task can be divided into subtasks recursively.
83:
1.1 noro 84: %A typical example is {\it quicksort}, where an array to be sorted is
85: %partitioned into two sub-arrays and the algorithm is applied to each
86: %sub-array. In each level of recursion, two subtasks are generated
87: %and one can ask other OpenXM servers to execute them.
88: %Though it makes little contribution to the efficiency in the case of
89: %quicksort, we present an Asir program of this distributed quicksort
90: %to demonstrate that OpenXM gives an easy way to test this algorithm.
91: %In the program, a predefined constant {\tt LevelMax} determines
92: %whether new servers are launched or whole subtasks are done on the server.
93: %
94: %\begin{verbatim}
95: %#define LevelMax 2
96: %extern Proc1, Proc2;
97: %Proc1 = -1$ Proc2 = -1$
98: %
99: %/* sort [A[P],...,A[Q]] by quicksort */
100: %def quickSort(A,P,Q,Level) {
101: % if (Q-P < 1) return A;
102: % Mp = idiv(P+Q,2); M = A[Mp]; B = P; E = Q;
103: % while (1) {
104: % while (A[B] < M) B++;
105: % while (A[E] > M && B <= E) E--;
106: % if (B >= E) break;
107: % else { T = A[B]; A[B] = A[E]; A[E] = T; E--; }
108: % }
109: % if (E < P) E = P;
110: % if (Level < LevelMax) {
111: % /* launch new servers if necessary */
112: % if (Proc1 == -1) Proc1 = ox_launch(0);
113: % if (Proc2 == -1) Proc2 = ox_launch(0);
114: % /* send the requests to the servers */
115: % ox_rpc(Proc1,"quickSort",A,P,E,Level+1);
116: % ox_rpc(Proc2,"quickSort",A,E+1,Q,Level+1);
117: % if (E-P < Q-E) {
118: % A1 = ox_pop_local(Proc1);
119: % A2 = ox_pop_local(Proc2);
120: % }else{
121: % A2 = ox_pop_local(Proc2);
122: % A1 = ox_pop_local(Proc1);
123: % }
124: % for (I=P; I<=E; I++) A[I] = A1[I];
125: % for (I=E+1; I<=Q; I++) A[I] = A2[I];
126: % return(A);
127: % }else{
128: % /* everything is done on this server */
129: % quickSort(A,P,E,Level+1);
130: % quickSort(A,E+1,Q,Level+1);
131: % return(A);
132: % }
133: %}
134: %\end{verbatim}
1.3 noro 135: %
1.4 noro 136: A typical example is a parallelization of the Cantor-Zassenhaus
137: algorithm for polynomial factorization over finite fields.
138: which is a recursive algorithm.
139: At each level of the recursion, a given polynomial can be
140: divided into two non-trivial factors with some probability by using
141: a randomly generated polynomial as a {\it separator}.
142: We can apply the following simple parallelization:
143: When two non-trivial factors are generated on a server,
144: one is sent to another server and the other factor is factorized on the server
145: itself.
1.1 noro 146: %\begin{verbatim}
147: %/* factorization of F */
148: %/* E = degree of irreducible factors in F */
149: %def c_z(F,E,Level)
150: %{
151: % V = var(F); N = deg(F,V);
152: % if ( N == E ) return [F];
153: % M = field_order_ff(); K = idiv(N,E); L = [F];
154: % while ( 1 ) {
155: % /* gererate a random polynomial */
156: % W = monic_randpoly_ff(2*E,V);
157: % /* compute a power of the random polynomial */
158: % T = generic_pwrmod_ff(W,F,idiv(M^E-1,2));
159: % if ( !(W = T-1) ) continue;
160: % /* G = GCD(F,W^((M^E-1)/2)) mod F) */
161: % G = ugcd(F,W);
162: % if ( deg(G,V) && deg(G,V) < N ) {
163: % /* G is a non-trivial factor of F */
164: % if ( Level >= LevelMax ) {
165: % /* everything is done on this server */
166: % L1 = c_z(G,E,Level+1);
167: % L2 = c_z(sdiv(F,G),E,Level+1);
168: % } else {
169: % /* launch a server if necessary */
170: % if ( Proc1 < 0 ) Proc1 = ox_launch();
171: % /* send a request with Level = Level+1 */
172: % /* ox_c_z is a wrapper of c_z on the server */
173: % ox_cmo_rpc(Proc1,"ox_c_z",lmptop(G),E,
174: % setmod_ff(),Level+1);
175: % /* the rest is done on this server */
176: % L2 = c_z(sdiv(F,G),E,Level+1);
177: % L1 = map(simp_ff,ox_pop_cmo(Proc1));
178: % }
179: % return append(L1,L2);
180: % }
181: % }
182: %}
183: %\end{verbatim}
184: %
185: %
186: %
187: %
188: %
189: %
190: %
1.2 noro 191:
192: \subsubsection{Product of univariate polynomials}
193:
194: Shoup \cite{Shoup} showed that the product of univariate polynomials
195: with large degrees and large coefficients can be computed efficiently
196: by FFT over small finite fields and Chinese remainder theorem.
197: It can be easily parallelized:
198:
199: \begin{tabbing}
200: Input :\= $f_1, f_2 \in {\bf Z}[x]$ such that $deg(f_1), deg(f_2) < 2^M$\\
201: Output : $f = f_1f_2$ \\
202: $P \leftarrow$ \= $\{m_1,\cdots,m_N\}$ where $m_i$ is an odd prime, \\
203: \> $2^{M+1}|m_i-1$ and $m=\prod m_i $ is sufficiently large. \\
204: Separate $P$ into disjoint subsets $P_1, \cdots, P_L$.\\
205: for \= $j=1$ to $L$ $M_j \leftarrow \prod_{m_i\in P_j} m_i$\\
206: Compute $F_j$ such that $F_j \equiv f_1f_2 \bmod M_j$\\
207: \> and $F_j \equiv 0 \bmod m/M_j$ in parallel.\\
208: \> (The product is computed by FFT.)\\
209: return $\phi_m(\sum F_j)$\\
210: (For $a \in {\bf Z}$, $\phi_m(a) \in (-m/2,m/2)$ and $\phi_m(a)\equiv a \bmod m$)
211: \end{tabbing}
212:
213: Figure \ref{speedup}
214: shows the speedup factor under the above distributed computation
215: on Risa/Asir. For each $n$, two polynomials of degree $n$
216: with 3000bit coefficients are generated and the product is computed.
217: The machine is FUJITSU AP3000,
218: a cluster of Sun workstations connected with a high speed network
219: and MPI over the network is used to implement OpenXM.
220: \begin{figure}[htbp]
221: \epsfxsize=8.5cm
222: \epsffile{speedup.ps}
223: \caption{Speedup factor}
224: \label{speedup}
225: \end{figure}
226:
227: If the number of servers is $L$ and the inputs are fixed, then the cost to
228: compute $F_j$ in parallel is $O(1/L)$, whereas the cost
229: to send and receive polynomials is $O(L)$ if {\tt ox\_push\_cmo()} and
230: {\tt ox\_pop\_cmo()} are repeatedly applied on the client.
231: Therefore the speedup is limited and the upper bound of
232: the speedup factor depends on the ratio of
233: the computational cost and the communication cost for each unit operation.
234: Figure \ref{speedup} shows that
235: the speedup is satisfactory if the degree is large and $L$
236: is not large, say, up to 10 under the above environment.
237: If OpenXM provides collective operations for broadcast and reduction
238: such as {\tt MPI\_Bcast} and {\tt MPI\_Reduce} respectively, the cost of
239: sending $f_1$, $f_2$ and gathering $F_j$ may be reduced to $O(\log_2L)$
240: and we can expect better results in such a case. In order to implement
241: such operations we need new specifications for inter-sever communication
242: and the session management, which will be proposed as OpenXM-RFC 102.
243: We note that preliminary experiments show the collective operations
244: work well on OpenXM.
245:
246: %\subsubsection{Competitive distributed computation by various strategies}
247: %
248: %SINGULAR \cite{Singular} implements {\it MP} interface for distributed
249: %computation and a competitive Gr\"obner basis computation is
250: %illustrated as an example of distributed computation.
251: %Such a distributed computation is also possible on OpenXM as follows:
252: %
253: %The client creates two servers and it requests
254: %Gr\"obner basis comutations from the homogenized input and the input itself
255: %to the servers.
256: %The client watches the streams by {\tt ox\_select()}
257: %and the result which is returned first is taken. Then the remaining
258: %server is reset.
259: %
260: %\begin{verbatim}
261: %/* G:set of polys; V:list of variables */
262: %/* O:type of order; P0,P1: id's of servers */
263: %def dgr(G,V,O,P0,P1)
264: %{
265: % P = [P0,P1]; /* server list */
266: % map(ox_reset,P); /* reset servers */
267: % /* P0 executes non-homogenized computation */
268: % ox_cmo_rpc(P0,"dp_gr_main",G,V,0,1,O);
269: % /* P1 executes homogenized computation */
270: % ox_cmo_rpc(P1,"dp_gr_main",G,V,1,1,O);
271: % map(ox_push_cmd,P,262); /* 262 = OX_popCMO */
272: % F = ox_select(P); /* wait for data */
273: % /* F[0] is a server's id which is ready */
274: % R = ox_get(F[0]);
275: % if ( F[0] == P0 ) {
276: % Win = "nonhomo"; Lose = P1;
277: % } else {
278: % Win = "homo"; Lose = P0;
279: % }
280: % ox_reset(Lose); /* reset the loser */
281: % return [Win,R];
282: %}
283: %\end{verbatim}
284:
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