6th ACM International Conference on Supercomputing, Washington D.C., July 1992.
Status: R

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Fm(\000)p Fq(3)p Fp(y)371 235 y Fo(5)430 231 y Fm(\000)p Fq(2)p
Fp(y)498 235 y Fo(4)557 231 y Fm(\024)71 b Fq(27)154 272 y
Fm(\000)p Fq(42)p Fp(x)244 276 y Fo(1)303 272 y Fq(+8)p Fp(y)371
276 y Fo(5)430 272 y Fq(+5)p Fp(y)498 276 y Fo(4)557 272 y
Fm(\024)41 b(\000)p Fq(72)-75 348 y(Suc)o(h)16 b(an)f(op)q(eration)j(is)d
(not)h(p)q(ossible)h(with)f(the)f(v)n(ariable)j Fp(x)834 352
y Fo(1)851 348 y Fq(.)23 b(So)-75 389 y(w)o(e)13 b(de\014ne)g(t)o(w)o(o)g
(constrained)i(v)n(ariables:)320 463 y Fp(x)342 467 y Fo(1)370
463 y Fq(=)c Fp(y)430 467 y Fo(6)455 463 y Fm(\000)e Fp(y)513
467 y Fo(7)-75 539 y Fq(and)14 b(the)f(\014nal)h(system)f(of)g(constrained)i
(v)n(ariables)h(is:)113 612 y(18)p Fp(y)170 616 y Fo(6)229
612 y Fm(\000)p Fq(18)p Fp(y)316 616 y Fo(7)375 612 y Fm(\000)p
Fq(3)p Fp(y)443 616 y Fo(5)501 612 y Fm(\000)p Fq(2)p Fp(y)569
616 y Fo(4)628 612 y Fm(\024)71 b Fq(27)83 654 y Fm(\000)p
Fq(42)p Fp(y)170 658 y Fo(6)229 654 y Fq(+42)p Fp(y)316 658
y Fo(7)375 654 y Fq(+8)p Fp(y)443 658 y Fo(5)501 654 y Fq(+5)p
Fp(y)569 658 y Fo(4)628 654 y Fm(\024)42 b(\000)p Fq(72)-75
729 y(More)14 b(generally)m(,)h(if)e(more)h(than)g(one)f(free)g(v)n(ariable)j
(remains,)e(it)g(is)-75 771 y(p)q(ossible)f(to)d(transform)h(a)g(system)f
(with)h Fp(n)g Fq(free)f(v)n(ariables)j Fp(x)808 775 y Fn(i)831
771 y Fq(in)o(to)e(a)-75 812 y(system)h(with)h Fp(n)7 b Fq(+)g(1)12
b(constrained)i(v)n(ariables)g(b)o(y)f(setting)g(\(classical)-75
854 y(\\single)i(added)f(v)n(ariable")h(tec)o(hnique\):)325
928 y Fp(x)347 932 y Fn(i)370 928 y Fq(=)c Fp(y)430 932 y Fn(i)451
928 y Fm(\000)e Fp(y)509 932 y Fo(0)-75 1026 y Fr(4)50 b(SOL)-6
b(VING)17 b(THE)f(FINAL)h(PROBLEM)-75 1105 y Fq(The)h(metho)q(ds)h(whic)o(h)f
(will)i(b)q(e)e(examined)i(here)e(use)g(essen)o(tially)-75
1146 y(tec)o(hniques)13 b(and)e(algorithms)i(deriving)g(from)d(the)h(famous)g
(simplex)-75 1188 y(metho)q(d,)j(of)e(G.B.)h(Dan)o(tzig.)19
b(This)14 b(w)o(ell-kno)o(wn)g(metho)q(d)g(will)h(not)-75 1229
y(b)q(e)d(describ)q(ed)h(in)g(detail)g(\(for)e(details)i(see)f([7,)f(22])g
(or)h(other)g(b)q(o)q(oks\).)-75 1271 y(W)m(e)d(just)g(recall)i(some)e
(terminology)i(that)e(will)i(b)q(e)e(used)h(throughout)-75
1312 y(this)k(section.)-17 1354 y(The)f(aim)h(of)f(the)h(simplex)h(metho)q(d)
g(is)f(to)f(solv)o(e)i(the)e(follo)o(wing)-75 1395 y(optimization)j(problem:)
302 1440 y Fd(P)346 1451 y Fn(n)346 1484 y(j)q Fo(=1)407 1471
y Fp(c)424 1475 y Fn(j)440 1471 y Fp(x)462 1475 y Fn(j)489
1471 y Fq(=)10 b Fp(z)336 1521 y Fq(min)e Fp(z)k Fq(=)e Fp(z)497
1505 y Fo(0)130 1534 y Fd(P)174 1544 y Fn(n)174 1577 y(j)q
Fo(=1)235 1564 y Fp(a)255 1568 y Fn(ij)283 1564 y Fp(x)305
1568 y Fn(j)331 1564 y Fm(\024)h Fp(d)392 1568 y Fn(i)405 1564
y Fp(;)49 b(i)10 b Fm(2)h(f)p Fq(1)p Fp(;)6 b Fq(2)p Fp(;)g(:)g(:)h(:)f(m)p
Fm(g)216 1611 y Fp(x)238 1615 y Fn(j)265 1611 y Fm(\025)k Fq(0)p
Fp(;)49 b(j)13 b Fm(2)d(f)p Fq(1)p Fp(;)d Fq(2)p Fp(;)f(:)g(:)g(:)g(n)p
Fm(g)-75 1687 y Fq(for)13 b(real)h(v)n(ariables)h Fp(x)240
1691 y Fn(i)253 1687 y Fq(.)-17 1728 y(The)g(primal)h(simplex)h(algorithm)g
(can)f(b)q(e)f(applied)j(when)d(the)-75 1770 y Fp(d)-55 1774
y Fn(i)-23 1770 y Fq(are)k(non-negativ)o(e.)37 b(The)19 b(dual)i(simplex)g
(algorithm)g(can)e(b)q(e)-75 1811 y(applied)h(when)f(the)f
Fp(c)258 1815 y Fn(j)292 1811 y Fq(are)g(non-negativ)o(e.)34
b(Dumm)o(y)18 b(v)n(ariables,)-75 1853 y(one)c(p)q(er)g(inequalit)o(y)m(,)j
(are)c(in)o(tro)q(duced)j(and)f(inequalities)i(are)d(then)-75
1894 y(referred)h(to)g(b)o(y)g(the)g(asso)q(ciated)i(v)n(ariable.)24
b(F)m(or)15 b(instance,)h(at)f(the)-75 1936 y(\014rst)e(step,)g(w)o(e)g(ha)o
(v)o(e)133 2010 y Fp(z)f Fq(=)56 b Fp(c)266 2014 y Fo(1)283
2010 y Fp(x)305 2014 y Fo(1)379 2010 y Fq(+)p Fp(c)426 2014
y Fo(2)442 2010 y Fp(x)464 2014 y Fo(2)523 2010 y Fq(+)6 b
Fp(:)g(:)g(:)48 b Fm(\000)55 b Fq(0)94 2051 y Fp(x)116 2055
y Fn(n)p Fo(+)p Fn(i)182 2051 y Fq(:)41 b Fp(a)254 2055 y Fn(i)p
Fo(1)283 2051 y Fp(x)305 2055 y Fo(1)364 2051 y Fq(+)p Fp(a)414
2055 y Fn(i)p Fo(2)442 2051 y Fp(x)464 2055 y Fo(2)523 2051
y Fq(+)6 b Fp(:)g(:)g(:)48 b Fm(\024)41 b Fp(d)743 2055 y Fn(i)-75
2127 y Fq(V)m(ariables)15 b(app)q(earing)g(in)g(the)e(economic)i(function)f
Fp(z)h Fq(are)e(referred)-75 2168 y(to)g(as)h Fh(out)f(b)n(asis)f(variables)p
Fq(.)j(V)m(ariables)g(whic)o(h)g(reference)e(inequal-)-75 2210
y(ities)k(are)f(called)i Fh(b)n(asis)d(variables)p Fq(.The)e(simplex)18
b(is)f(based)g(on)f(the)-75 2251 y(observ)n(ation)h(that)e(if)g(the)g(co)q
(e\016cien)o(ts)i Fp(c)535 2255 y Fn(j)565 2251 y Fq(in)f Fp(z)g
Fq(are)f(non-negativ)o(e)-75 2293 y(and)e(if)f(the)h(co)q(e\016cien)o(ts)g
Fp(d)312 2297 y Fn(i)337 2293 y Fq(are)g(non-negativ)o(e,)h(then)f(the)f
(problem)-75 2334 y(is)f(solv)o(ed)g(b)o(y)g(giving)h(the)e(v)n(alue)h(0)f
(to)g(ev)o(ery)h(out)f(basis)h(v)n(ariable)i(and)-75 2376 y(the)f(v)n(alue)h
Fp(d)108 2380 y Fn(i)132 2376 y Fq(to)f(the)f(basis)i(v)n(ariable)h
Fp(x)500 2380 y Fn(n)p Fo(+)p Fn(i)555 2376 y Fq(.)j(The)11
b(goal)i(is)f(therefore)-75 2417 y(to)g(obtain)h(suc)o(h)g(a)f(situation)i(b)
o(y)e(transformations)i(of)d(the)h(system.)-17 2459 y(The)19
b(basic)j(transformation)f(used)g(is)f(a)g(Gaussian)i(elimina-)-75
2500 y(tion)17 b(algorithm)i(that)e(tak)o(es)f(a)h(basis)h(v)n(ariable)g(out)
f(of)f(the)h(basis)-75 2542 y(and)i(inserts)g(an)f(out-basis)h(v)n(ariable)h
(in)o(to)f(the)f(basis.)33 b(Piv)o(oting)-75 2583 y(on)18 b(\()p
Fp(x)20 2587 y Fn(k)38 2583 y Fp(;)6 b(x)77 2587 y Fn(n)p Fo(+)p
Fn(r)137 2583 y Fq(\))17 b(\()p Fp(x)206 2587 y Fn(k)242 2583
y Fq(out)h(of)f(the)g(basis)i(and)f Fp(x)633 2587 y Fn(n)p
Fo(+)p Fn(r)709 2583 y Fq(in)h(the)e(basis\))-75 2625 y(means)f(replacing)i
Fp(x)234 2629 y Fn(k)268 2625 y Fq(b)o(y)e(1)p Fp(=a)382 2629
y Fn(r)q(k)417 2625 y Fq(\()p Fp(d)452 2629 y Fn(r)480 2625
y Fm(\000)10 b Fp(x)542 2629 y Fn(n)p Fo(+)p Fn(r)612 2625
y Fm(\000)652 2595 y Fd(P)696 2605 y Fn(n)696 2638 y(j)q Fo(=1)p
Fn(;j)q Fk(6)p Fo(=)p Fn(k)821 2625 y Fp(a)841 2629 y Fn(r)q(j)873
2625 y Fp(x)895 2629 y Fn(j)911 2625 y Fq(\))-75 2673 y(in)16
b(the)g(whole)h(system.)25 b(Inequalit)o(y)17 b Fp(x)508 2677
y Fn(n)p Fo(+)p Fn(r)583 2673 y Fq(is)g(renamed)f Fp(x)805
2677 y Fn(k)824 2673 y Fq(.)24 b(The)-75 2715 y(co)q(e\016cien)o(t)14
b Fp(a)124 2719 y Fn(r)q(k)171 2715 y Fq(is)f(called)i(the)d
Fh(pivot)p Fq(.)j(The)e(piv)o(ot)h(m)o(ust)e(b)q(e)h(c)o(hosen)-75
2756 y(suc)o(h)g(that,)f(in)h(the)f(primal)h(algorithm,)h(the)e(righ)o
(t-hand)i(sides)g(\()p Fp(d)898 2760 y Fn(i)911 2756 y Fq(\))1025
-33 y(remain)k(non-negativ)o(e)h(and)g(in)f(the)f(dual)i(algorithm)g(the)f
(co)q(e\016-)1025 8 y(cien)o(ts)10 b Fp(c)1146 12 y Fn(j)1172
8 y Fq(remain)h(non-negativ)o(e.)18 b(V)m(arian)o(ts)11 b(of)f(the)g
(simplex-based)1025 50 y(metho)q(ds)15 b(di\013er)h(essen)o(tially)i(in)d
(the)g(strategy)g(used)g(for)g(c)o(ho)q(osing)1025 91 y(the)e(piv)o(ot.)1082
133 y(No)o(w)f(when)g(the)h(system)f(to)g(solv)o(e)i(constrains)g(v)n
(ariables)g(to)f(b)q(e)1025 174 y(in)o(teger,)h(simplex)i(metho)q(d)e(m)o
(ust)g(b)q(e)g(adapted,)g(this)h(is)f(explained)1025 216 y(in)j(the)g(next)g
(sections.)29 b(It)16 b(should)j(b)q(e)e(noted)g(that)g(in)o(teger)h(pro-)
1025 257 y(gramming)c(sp)q(eci\014c)h(tec)o(hniques)g(are)f(not)f(recen)o(t.)
18 b(They)c(w)o(ere)f(es-)1025 299 y(sen)o(tially)20 b(dev)o(elop)q(ed)g(b)o
(y)e(R.E.)g(Gomory)h(and)f(are)g(describ)q(ed)i(in)1025 340
y(classic)14 b(b)q(o)q(oks)g(suc)o(h)g(that)f(\([8],[20)o(])p
Fp(:)6 b(:)g(:)g Fq(\).)1025 432 y Fr(4.1)49 b(THE)17 b(RUDIMENT)l(ARY)e
(PRIMAL)1138 474 y(ALL-INTEGER)i(ALGORITHM)g(\(R.P)l(.A.I.\))1025
539 y Fq(This)h(algorithm)h(is)f(a)f(to)q(ol)h(used)g(extensiv)o(ely)i(in)f
(metho)q(ds)f(pre-)1025 581 y(sen)o(ted)c(in)h(sections)g(4.2)f(and)h(4.3.)k
(Its)14 b(aim)h(is)f(to)g(solv)o(e)h(the)f(prob-)1025 622 y(lem:)1401
627 y Fd(P)1445 637 y Fn(n)1445 671 y(j)q Fo(=1)1506 657 y
Fp(c)1523 661 y Fn(j)1539 657 y Fp(x)1561 661 y Fn(j)1588 657
y Fq(=)d Fp(z)1436 707 y Fq(min)c Fp(z)12 b Fq(=)f Fp(z)1597
691 y Fo(0)1229 720 y Fd(P)1273 730 y Fn(n)1273 764 y(j)q Fo(=1)1334
750 y Fp(a)1354 754 y Fn(ij)1382 750 y Fp(x)1404 754 y Fn(j)1431
750 y Fm(\024)f Fp(d)1491 754 y Fn(i)1504 750 y Fp(;)49 b(i)11
b Fm(2)f(f)p Fq(1)p Fp(;)d Fq(2)p Fp(;)f(:)g(:)g(:)g(m)p Fm(g)1251
798 y Fp(x)1273 802 y Fn(j)1300 798 y Fm(2)k Fp(Z)q(;)c(x)1402
802 y Fn(j)1429 798 y Fm(\025)k Fq(0)p Fp(;)49 b(j)13 b Fm(2)d(f)p
Fq(1)p Fp(;)c Fq(2)p Fp(;)h(:)f(:)g(:)g(n)p Fm(g)1025 852 y
Fq(W)m(e)k(assume)g(that)h(all)g(input)g(data)g(are)f(in)o(teger)h(and)g
(that)f(the)g(righ)o(t-)1025 894 y(hand)j(sides)i Fp(d)1233
898 y Fn(i)1259 894 y Fq(are)e(non-negativ)o(e.)1082 935 y(The)e(principle)j
(of)d(the)g(algorithm)i(is)f(v)o(ery)f(close)i(to)e(that)g(of)g(the)1025
977 y(algorithm)h(for)e(real)i(v)n(ariables)g(describ)q(ed)h(ab)q(o)o(v)o(e.)
k(The)10 b(criteria)i(for)1025 1018 y(piv)o(ot)g(selection,)h(optimalit)o(y)m
(,)h(and)e(in\014nitude)i(are)d(the)g(same.)17 b(The)1025 1060
y(in)o(teger)d(algorithm)h(di\013ers)f(on)f(these)h(p)q(oin)o(ts:)1082
1111 y Fm(\017)19 b Fq(it)14 b(is)f(clear)h(that)f(when)g(the)g(c)o(hosen)h
(piv)o(ot)g Fp(a)1779 1115 y Fn(r)q(k)1826 1111 y Fq(is)g Fm(\006)p
Fq(1)f(\(more)1120 1153 y(exactly)i(1)f(since)g(the)g(piv)o(ot)g(is)g(p)q
(ositiv)o(e\),)i(then)d(a)h(Gaussian)1120 1194 y(eliminati)q(on)21
b(can)c(b)q(e)h(p)q(erformed)g(and)h(the)e(co)q(e\016cien)o(ts)i(of)1120
1236 y(the)14 b(resulting)i(system)e(are)f(still)j(in)o(tegers.)j(No)o(w,)13
b(when)h(the)1120 1277 y(piv)o(ot)19 b(c)o(hoice)f(criterion)h(generates)f(a)
f(piv)o(ot)i(with)e(a)h(v)n(alue)1120 1319 y(other)c(than)f(1,)g(from)g(the)g
(inequalit)o(y)j(includin)q(g)g(it:)1431 1351 y Fd(P)1475 1362
y Fn(n)1475 1395 y(j)q Fo(=1)1536 1382 y Fp(a)1556 1386 y Fn(r)q(j)1588
1382 y Fp(x)1610 1386 y Fn(j)1637 1382 y Fm(\024)10 b Fp(d)1697
1386 y Fn(r)1120 1456 y Fq(a)17 b(new)g(\(in)o(teger\))g(inequalit)o(y)j(is)d
(constructed,)h(referred)f(to)1120 1498 y(as)d(a)f(\\cut")1304
1482 y Fo(2)1321 1498 y Fq(:)1305 1530 y Fd(P)1349 1540 y Fn(n)1349
1574 y(j)q Fo(=1)1404 1560 y Fm(b)p Fp(a)1441 1564 y Fn(r)q(j)1473
1560 y Fp(=)p Fm(j)p Fp(a)1523 1564 y Fn(r)q(k)1557 1560 y
Fm(jc)p Fp(x)1607 1564 y Fn(j)1633 1560 y Fm(\024)e(b)p Fp(d)1711
1564 y Fn(r)1728 1560 y Fp(=)p Fm(j)p Fp(a)1778 1564 y Fn(r)q(k)1813
1560 y Fm(jc)1120 1629 y Fq(This)17 b(new)f(inequali)q(t)o(y)j(is)d(added)i
(to)e(the)g(system,)h(and)g(the)1120 1671 y(elemen)o(t)d(lo)q(cated)g(on)g
(column)g Fp(k)q Fq(,)e(whose)h(co)q(e\016cien)o(t)h(is)g(no)o(w)1120
1712 y(1)19 b(\(and)f(is)h(also)g(a)f(p)q(ossible)j(piv)o(ot)e(according)h
(to)e(simplex)1120 1754 y(criteria\))d(is)e(c)o(hosen)h(as)g(the)f(piv)o(ot.)
1082 1807 y Fm(\017)19 b Fq(Ev)o(ery)c(time)f(a)g(\\cut")h(v)n(ariable)g
(returns)g(to)f(the)g(basis)h(after)1120 1849 y(a)d(piv)o(oting)h(op)q
(eration,)g(the)e(inequalit)o(y)j(relativ)o(e)f(to)e(this)h(cut)1120
1890 y(v)n(ariable)k(is)d(remo)o(v)o(ed)h(from)f(the)g(system)h(\(this)f(p)q
(oin)o(t)i(is)f(not)1120 1932 y(strictly)h(obligatory\).)1025
1983 y(Let)d(us)i(consider)g(the)f(problem)1234 2048 y Fp(z)g
Fq(=)71 b(2)p Fp(x)1407 2052 y Fo(1)1466 2048 y Fq(+2)p Fp(x)1537
2052 y Fo(2)1595 2048 y Fm(\000)p Fq(5)p Fp(x)1666 2052 y Fo(3)1725
2048 y Fm(\000)41 b Fq(0)1234 2090 y Fp(x)1256 2094 y Fo(4)1284
2090 y Fq(:)g Fm(\000)p Fq(3)p Fp(x)1407 2094 y Fo(1)1466 2090
y Fm(\000)p Fq(2)p Fp(x)1537 2094 y Fo(2)1614 2090 y Fq(+)p
Fp(x)1666 2094 y Fo(3)1725 2090 y Fm(\024)g Fq(3)1234 2131
y Fp(x)1256 2135 y Fo(5)1284 2131 y Fq(:)g Fm(\000)p Fq(2)p
Fp(x)1407 2135 y Fo(1)1466 2131 y Fq(+3)p Fp(x)1537 2135 y
Fo(2)1595 2131 y Fm(\000)p Fq(3)p Fp(x)1666 2135 y Fo(3)1725
2131 y Fm(\024)g Fq(4)1234 2173 y Fp(x)1256 2177 y Fo(6)1284
2173 y Fq(:)71 b(4)p Fp(x)1407 2177 y Fo(1)1485 2173 y Fm(\000)p
Fp(x)1537 2177 y Fo(2)1595 2173 y Fq(+2)p Fp(x)1666 2177 y
Fo(3)1725 2173 y Fm(\024)41 b Fq(3)1025 2239 y(The)11 b(criterion)i
(designates)g(the)e(v)n(ariable)j Fp(x)1665 2243 y Fo(3)1693
2239 y Fq(as)d(the)g(piv)o(ot)i(and)f(the)1025 2281 y(inequalit)o(y)k
Fp(x)1225 2285 y Fo(6)1242 2281 y Fq(.)i(The)c(v)n(alue)h(of)e(the)g(co)q
(e\016cien)o(t)i Fp(a)1761 2285 y Fn(r)q(k)1809 2281 y Fq(is)f(2,)g(so)f(a)h
(cut)1025 2322 y(is)f(added:)1268 2387 y Fp(x)1290 2391 y Fo(7)1318
2387 y Fq(:)41 b(2)p Fp(x)1411 2391 y Fo(1)1470 2387 y Fm(\000)p
Fp(x)1522 2391 y Fo(2)1580 2387 y Fq(+)p Fp(x)1632 2391 y Fo(3)1691
2387 y Fm(\024)g Fq(1)1025 2454 y(and)13 b(w)o(e)g(can)g(execute)h(the)f(piv)
o(oting)i(op)q(eration)g(\()p Fp(x)1788 2458 y Fo(3)1805 2454
y Fp(;)6 b(x)1844 2458 y Fo(7)1861 2454 y Fq(\):)1234 2520
y Fp(z)13 b Fq(=)52 b(12)p Fp(x)1407 2524 y Fo(1)1466 2520
y Fm(\000)p Fq(3)p Fp(x)1537 2524 y Fo(2)1595 2520 y Fq(+5)p
Fp(x)1666 2524 y Fo(7)1725 2520 y Fm(\000)41 b Fq(5)1234 2562
y Fp(x)1256 2566 y Fo(4)1284 2562 y Fq(:)g Fm(\000)p Fq(5)p
Fp(x)1407 2566 y Fo(1)1485 2562 y Fm(\000)p Fp(x)1537 2566
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1023 y(nique,)f(D)o(\023)-18 b(epartemen)o(t)12 b(de)f(math)o(\023)-18
b(ematiques)12 b(et)e(informatique,)4 1064 y(Juin)k(1989.)-56
1123 y([4])19 b Fa(Berr)n(y,)13 b(M.,)h(Chen,)f(D.,)h(K)o(oss,)g(P.,)f(Kuck,)
g(D.,)h(and)f(Lo,)4 1165 y(S.)35 b Fq(The)19 b(PERFECT)g(Club)g(b)q(enc)o
(hmarks:)31 b(e\013ectiv)o(e)20 b(p)q(er-)4 1206 y(formance)13
b(ev)n(aluation)j(of)c(sup)q(ercomputers.)19 b(CSRD)14 b(Rep)q(ort)4
1248 y(827,)24 b(Univ)o(ersit)o(y)f(of)f(Illinois,)k(Urbana-Champaign,)g(Ma)o
(y)4 1289 y(1989.)-56 1348 y([5])19 b Fa(Cousot,)12 b(P.,)f(and)g(Cousot,)h
(R.)g Fq(Abstract)e(in)o(terpretation:)4 1390 y(A)k(uni\014ed)j(lattice)f(mo)
q(del)g(for)f(static)h(analysis)h(of)d(programs)4 1431 y(b)o(y)f
(construction)h(or)e(appro)o(ximation)j(of)d(\014xp)q(oin)o(ts.)18
b(In)12 b Fh(Pr)n(o)n(c.)4 1473 y(of)d(the)h(F)m(orth)f(A)o(CM)i(Confer)n
(enc)n(e)c(on)j(Principles)e(of)h(Pr)n(o)n(gr)n(am-)4 1514
y(ming)k(L)n(anguages)g Fq(\(1977\).)-56 1573 y([6])19 b Fa(Cousot,)c(P.,)g
(and)h(Halbw)m(a)o(chs,)f(N.)21 b Fq(Automatic)15 b(disco)o(v-)4
1614 y(ery)h(of)g(linear)i(restrain)o(ts)f(among)g(v)n(ariables)i(of)d(a)g
(program.)4 1656 y(In)d Fh(Pr)n(o)n(c.)g(of)g(the)g(Fifth)f(A)o(CM)i(Confer)n
(enc)n(e)e(on)h(Principles)e(of)4 1697 y(Pr)n(o)n(gr)n(amming)i(L)n(anguages)
g Fq(\(1978\).)-56 1756 y([7])19 b Fa(D)o(antzig,)24 b(G.)42
b Fh(Line)n(ar)20 b(Pr)n(o)n(gr)n(amming)h(and)f(Extensions)p
Fq(.)4 1798 y(Princeton)29 b(Univ)o(ersit)o(y)g(Press,)i(Princeton,)h
(New-Jersey)m(,)4 1839 y(1963.)-56 1898 y([8])19 b Fa(Garfinkel,)f(R.)g(S.,)g
(and)g(Nemha)o(user,)h(G.)e(N.)27 b Fh(Inte)n(ger)4 1940 y(Pr)n(o)n(gr)n
(amming)p Fq(.)16 b(Wiley-In)o(terscience,)g(1972.)-56 1999
y([9])j Fa(Giboulot,)32 b(M.)d(C.,)34 b(Lebon,)d(E.)e(R.,)k(Lo)o(yer,)g(M.,)4
2040 y(Popo)o(vitch,)10 b(G.,)h(Shafie,)f(H.,)h(and)g(Thomasset,)f(F.)g
Fq(P)o(ar-)4 2082 y(allel)15 b(execution)h(of)d(F)m(ortran)h(programs)g(on)g
(the)f(EWS)i(w)o(ork-)4 2123 y(station.)g(In)10 b Fh(Computing)g(Metho)n(ds)g
(in)i(Applie)n(d)d(Scienc)n(es)g(and)4 2165 y(Engine)n(ering)21
b Fq(\(Philadelph)q(ia,)26 b(Jan.)20 b(1990\),)j(R.)e(Glo)o(winski)4
2206 y(and)14 b(A.)e(Lic)o(hnewsky)m(,)i(Eds.,)f(SIAM,)f(pp.)h(413{423.)-75
2265 y([10])19 b Fa(Goff,)14 b(G.,)h(Kenned)o(y,)g(K.,)g(and)g(Tseng,)g
(C.-W.)k Fq(Practi-)4 2306 y(cal)12 b(dep)q(endence)h(testing.)i(In)c
Fh(Pr)n(o)n(c.)g(of)g(the)g(1991)f(A)o(CM)j(SIG-)4 2348 y(PLAN)18
b(Confer)n(enc)n(e)f(on)h(Pr)n(o)n(gr)n(amming)h(L)n(anguage)d(Design)4
2389 y(and)f(Implementation)g Fq(\(T)m(oron)o(to,)h(On)o(tario,)h(Canada,)g
(June)4 2431 y(26-28)c(1991\),)h(S.)e(Notices,)i(Ed.,)e(v)o(ol.)i(26,)f(A)o
(CM,)f(pp.)h(15{29.)-75 2490 y([11])19 b Fa(Granger,)11 b(P.)g
Fq(Static)f(analysis)h(of)e(arithmetic)i(congruences.)4 2531
y Fh(Int)i(J)h(Computer)e(Math)i(30)j Fq(\(1989\),)c(165{190.)-75
2590 y([12])19 b Fa(Klappholz,)14 b(D.,)g(Psarris,)i(K.,)f(and)f(K)o(ong,)h
(X.)j Fq(On)13 b(the)4 2632 y(p)q(erfect)j(accuracy)h(of)f(an)h(appro)o
(ximate)h(subscript)g(analysis)4 2673 y(test.)25 b(In)16 b
Fh(Pr)n(o)n(c)n(e)n(e)n(dings)d(of)j(A)o(CM)g(1990)f(Internationa)o(l)e(Con-)
4 2715 y(fer)n(enc)n(e)d(on)h(Sup)n(er)n(c)n(omputin)o(g)g
Fq(\(Amsterdam,)g(Holland,)i(June)4 2756 y(1990\).)1025 -33
y([13])18 b Fa(Knuth,)k(D.)e(E.)34 b Fh(The)19 b(A)o(rt)g(of)f(Computer)g(Pr)
n(o)n(gr)n(amming,)1103 8 y(V)m(ol)23 b(2,)i(Seminumeric)n(al)c(A)o
(lgorithms)p Fq(.)45 b(Second)25 b(Edition,)1103 50 y(Addison-W)m(esley)n(,)
15 b(Reading,)f(Massac)o(h)o(usetts,)g(1981.)1025 108 y([14])k
Fa(K)o(ong,)g(X.,)f(Klappholz,)g(D.,)g(and)g(Psarris,)h(K.)25
b Fq(The)15 b(I)1103 149 y(test:)23 b(an)16 b(impro)o(v)o(ed)h(dep)q(endence)
h(test)d(for)h(automatic)h(par-)1103 191 y(alleliza)q(tion)22
b(and)c(v)o(ectorization.)34 b Fh(IEEE)18 b(T)m(r)n(ansactions)e(on)1103
232 y(Par)n(al)r(lel)c(and)g(Distribute)n(d)e(Systems)h(2)p
Fq(,)h(3)g(\(July)h(1991\),)f(342{)1103 274 y(349.)1025 332
y([15])18 b Fa(Lenstra,)j(H.)f(W.)33 b Fq(In)o(teger)18 b(programming)i(with)
f(a)f(\014xed)1103 373 y(n)o(um)o(b)q(er)i(of)e(v)n(ariables.)35
b Fh(Mathematics)16 b(of)i(Op)n(er)n(ations)f(R)n(e-)1103 415
y(se)n(ar)n(ch)c(8)p Fq(,)f(4)h(\(1983\),)g(538{548.)1025 473
y([16])18 b Fa(Lichnewsky,)j(A.,)g(and)f(Thomasset,)h(F.)33
b Fq(In)o(tro)q(ducing)1103 514 y(sym)o(b)q(olic)14 b(problem)f(solving)h
(tec)o(hniques)g(in)e(the)g(dep)q(endence)1103 556 y(testing)f(phases)g(of)e
(a)g(v)o(ectorizer.)k(In)c Fh(Pr)n(o)n(c)n(e)n(e)n(dings)f(of)i(1988)f(In-)
1103 597 y(ternational)e(Confer)n(enc)n(e)h(on)h(Sup)n(er)n(c)n(omputin)o(g)g
Fq(\(Sain)o(t-Malo,)1103 639 y(F)m(rance,)k(July)h(1988\).)1025
697 y([17])k Fa(Ma)n(yd)o(an,)g(D.)e(E.,)i(Hennessy,)f(J.)g(L.,)g(and)g(Lam,)
g(M.)g(S.)1103 739 y Fq(E\016cien)o(t)e(and)g(exact)f(data)g(dep)q(endence)h
(analysis.)22 b(In)14 b Fh(Pr)n(o)n(c.)1103 780 y(of)j(the)f(1991)f(A)o(CM)j
(SIGPLAN)d(Confer)n(enc)n(e)f(on)j(Pr)n(o)n(gr)n(am-)1103 822
y(ming)g(L)n(anguage)e(Design)g(and)i(Implementation)f Fq(\(T)m(oron)o(to,)
1103 863 y(On)o(tario,)21 b(Canada,)g(June)f(26-28)f(1991\),)i(S.)d(Notices,)
j(Ed.,)1103 905 y(v)o(ol.)14 b(26,)f(A)o(CM,)f(pp.)h(1{14.)1025
963 y([18])18 b Fa(Minoux,)c(M.)k Fh(Pr)n(o)n(gr)n(ammation)12
b(math)o(\023)-19 b(ematique,)11 b(th)o(\023)-19 b(eorie)12
b(et)1103 1004 y(algorithmes,)g(tomes)g(1)i(et)f(2)p Fq(.)j(Duno)q(d,)e
(1983.)1025 1062 y([19])k Fa(Nemha)o(user,)e(G.)e(N.,)h(and)g(W)o(olsey,)g
(L.)f(A.)20 b Fh(Inte)n(ger)12 b(and)1103 1104 y(Combinatorial)g
(Optimization)p Fq(.)i(Wiley-In)o(terscienc)q(e,)h(1988.)1025
1162 y([20])j Fa(Salkin,)e(H.)f(M.,)g(and)g(Ma)n(thur,)h(K.)k
Fh(F)m(oundations)11 b(of)j(In-)1103 1204 y(te)n(ger)f(Pr)n(o)n(gr)n(amming)p
Fq(.)j(North-Holland,)e(1989.)1025 1262 y([21])k Fa(Schrijver,)d(A.)h
Fh(The)n(ory)c(of)h(Line)n(ar)e(and)h(Inte)n(ger)g(Pr)n(o)n(gr)n(am-)1103
1303 y(ming)p Fq(.)k(Wiley-In)o(terscien)q(ce,)f(1986.)1025
1361 y([22])j Fa(Simonnard,)12 b(B.)g Fh(Pr)n(o)n(gr)n(ammation)d(lin)o(\023)
-19 b(eair)n(e)p Fq(.)10 b(Duno)q(d,)h(P)o(aris,)1103 1403
y(1962.)1025 1461 y([23])18 b Fa(Sogno,)12 b(J.-C.)i Fq(Analysis)f(of)d
(standard)i(and)g(new)f(algorithms)1103 1502 y(for)g(the)f(in)o(teger)h
(constrain)o(t)h(satisfaction)h(problem.)h(Rapp)q(ort)1103
1544 y(de)g(rec)o(herc)o(he,)f(INRIA,)f(1992.)18 b(to)12 b(app)q(ear.)1025
1602 y([24])18 b Fa(W)l(alla)o(ce,)d(D.)e(R.)k Fq(Dep)q(endence)e(of)d(m)o
(ulti-dimensi)q(ona)q(l)k(ar-)1103 1644 y(ra)o(y)21 b(references.)38
b(In)20 b Fh(Pr)n(o)n(c)n(e)n(e)n(dings)e(of)h(1988)g(Internationa)o(l)1103
1685 y(Confer)n(enc)n(e)j(on)h(Sup)n(er)n(c)n(omputing)h Fq(\(Sain)o(t-Malo,)
29 b(F)m(rance,)1103 1727 y(July)14 b(1988\).)1025 1785 y([25])k
Fa(Wegman,)13 b(M.)g(N.,)g(and)g(Zadeck,)g(F.)f(K.)j Fq(Constan)o(t)d(prop-)
1103 1826 y(agation)i(with)f(conditional)j(branc)o(hes.)i(In)12
b Fh(Pr)n(o)n(c.)h(of)f(the)g(12th)1103 1868 y(A)o(CM)19 b(Confer)n(enc)n(e)c
(on)i(Principles)f(of)h(Pr)n(o)n(gr)n(amming)g(L)n(an-)1103
1909 y(guages)d Fq(\(1985\),)f(pp.)g(291{299.)1025 1967 y([26])18
b Fa(W)o(olfe,)29 b(M.)49 b Fh(Optimizing)22 b(Sup)n(er)n(c)n(ompilers)f(for)
j(Sup)n(er-)1103 2009 y(c)n(omputers)p Fq(.)34 b(PhD)20 b(thesis,)i(Univ.)e
(of)f(Illinoi)q(s)j(at)d(Urbana-)1103 2050 y(Champaign,)c(Urbana,)e
(Illinois,)j(1982.)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF

%%%%%%%%%%%%% fin papier

