From 2039ca958765cf4c841b4b574ccae9e957c40378 Mon Sep 17 00:00:00 2001 From: Simon Cruanes Date: Thu, 30 Apr 2020 22:21:03 -0400 Subject: [PATCH] add some documents --- doc/IntMap98.pdf | Bin 0 -> 385183 bytes doc/balacing_wbt.pdf | Bin 0 -> 1545918 bytes doc/bbt_full.pdf | Bin 0 -> 1107369 bytes doc/build_deps.ml | 32 + doc/containers.md | 81 ++ doc/dune | 8 + doc/finger_tree.pdf | Bin 0 -> 172801 bytes doc/intro.txt | 166 +++ doc/rrb_vector.pdf | Bin 0 -> 1756736 bytes doc/scheme_wbt.ps | 3127 ++++++++++++++++++++++++++++++++++++++++++ doc/wbt_original.pdf | Bin 0 -> 591317 bytes 11 files changed, 3414 insertions(+) create mode 100644 doc/IntMap98.pdf create mode 100644 doc/balacing_wbt.pdf create mode 100644 doc/bbt_full.pdf create mode 100755 doc/build_deps.ml create mode 100644 doc/containers.md create mode 100644 doc/dune create mode 100644 doc/finger_tree.pdf create mode 100644 doc/intro.txt create mode 100644 doc/rrb_vector.pdf create mode 100644 doc/scheme_wbt.ps create mode 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y(matter)m(.)428 1153 y FA(\017)23 +b Fu(split_lt)14 b FB(and)e Fu(split_gt)i FB(can)f(be)f(implemented)h(to)e +(take)h(time)g Fx(O)q Fw(\(log)7 b Fx(n)p Fw(\))474 1209 y +FB(in)12 b(the)f(size)g(of)h(the)f(tr)o(ee.)428 1312 y FA(\017)23 +b Fu(concat\(v,l,r\))18 b FB(can)d(be)f(implemented)i(to)e(take)g(time)h +Fx(O)q Fw(\(log)7 b Fx(n)12 b FA(\000)g Fw(log)7 b Fx(m)p Fw(\))474 +1368 y FB(wher)o(e)k(the)g(lar)o(ger)h(of)f Fu(l)h FB(and)f +Fu(r)h FB(has)g(size)f Fx(n)h FB(and)f(the)g(smaller)h(has)g(size)f +Fx(m)p FB(.)428 1471 y FA(\017)23 b FB(The)12 b(entir)o(e)h +Fu(union)h FB(operation)e(takes)g(worst)h(case)f(time)g Fx(O)q +Fw(\()p Fx(n)f Fw(+)g Fx(m)p Fw(\))h FB(wher)o(e)h Fx(n)474 +1527 y FB(and)e Fx(m)g FB(ar)o(e)g(the)h(sizes)f(of)h(the)f(tr)o(ees)g(being) +h(combined.)428 1629 y FA(\017)23 b FB(In)15 b(cases)f(wher)o(e)h(one)f(tr)o +(ee)g(is)g(small)h(or)f(the)h(tr)o(ees)f(have)g(dense)g(r)o(egions)h(that)474 +1686 y(do)c(not)g(overlap)g(the)g(operation)h(is)f(considerably)i(faster)m(.) +360 1799 y(The)18 b(function)i Fu(concat3)g FB(is)f(used)f(to)g(join)g(two)g +(tr)o(ees)g(with)g(an)g(element)h(that)e(is)360 1855 y(between)10 +b(the)h(values)g(in)h(the)e(left)h(tr)o(ee)g(and)f(the)h(values)g(in)g(the)g +(right)h(tr)o(ee.)k(If)11 b(the)g(left)360 1911 y(and)j(right)h(ar)o(guments) +h(would)e(make)h(a)f(balanced)g(tr)o(ee)g(then)h(they)f(can)h(be)f(joined)360 +1968 y(immediately)-5 b(.)19 b(If)12 b(one)h(tr)o(ee)e(is)i(signi\256cantly)h +(lar)o(ger)f(then)f(it)g(is)h(scanned)f(to)g(\256nd)h(the)360 +2024 y(lar)o(gest)h(subtr)o(ee)g(on)f(the)h(side)f(`facing')h(the)g(smaller)f +(tr)o(ee)g(that)h(is)g(small)g(enough)g(to)360 2081 y(balance)j(with)g(the)g +(smaller)g(tr)o(ee.)32 b(The)17 b(tr)o(ee)f(is)h(joined)g(at)f(this)i +(position)f(and)g(the)360 2137 y(higher)12 b(levels)g(ar)o(e)f(r)o(ebalanced) +g(if)h(necessary)-5 b(.)360 2265 y Fu(fun)28 b(concat3)h(\(v,E,r\))g(=)f +(add\(r,v\))415 2321 y(|)f(concat3)i(\(v,l,E\))g(=)f(add\(l,v\))415 +2378 y(|)f(concat3)i(\(v,)f(l)g(as)g(T\(v1,n1,l1,r1\))q(,)i(r)e(as)f +(T\(v2,n2,l2,r)q(2\)\))k(=)469 2434 y(if)d(weight*n1)i(<)d(n2)h(then)g +(T'\(v2,concat)q(3\(v,)q(l,l)q(2\),r)q(2\))469 2490 y(else)g(if)g(weight*n2)i +(<)e(n1)f(then)i(T'\(v1,l1,conca)q(t3\(v)q(,r1,)q(r\)\))605 +2547 y(else)g(N\(v,l,r\))p 360 2591 600 2 v 410 2618 a Fr(9)427 +2634 y Fq(de\256ned)8 b(as)773 2673 y Fl(\()p Fi(f)k Fj(\010)c +Fi(g)q Fl(\)\()p Fi(x)p Fl(\))42 b(=)f Fi(g)q Fl(\()p Fi(x)p +Fl(\))k Fq(if)10 b Fi(g)q Fl(\()p Fi(x)p Fl(\))f Fq(is)h(de\256ned)985 +2719 y Fl(=)41 b Fi(f)t Fl(\()p Fi(x)p Fl(\))g Fq(otherwise)60 +2914 y FB(CSTR)12 b(92-10)1516 b(17)p eop +%%Page: 18 20 +bop -30 37 a FB(9.)1242 b Fz(SET)p FA(\002)p Fz(SET)p FA(!)p +Fz(SET)14 b(OPERA)m(TIONS)p -30 62 1795 3 v 270 281 a FB(When)9 +b(the)f(tr)o(ees)g(ar)o(e)g(nearly)g(the)g(same)g(size,)h(then)f +Fu(concat3)i FB(does)e(very)g(little)g(work.)270 350 y(A)15 +b(tr)o(ee)f(is)h(split)h(by)f(discar)o(ding)h(all)e(the)h(unwanted)g +(elements)g(and)g(subtr)o(ees,)h(and)270 406 y(joining)c(together)f(all)h +(the)f(wanted)g(parts)h(using)h Fu(concat3)p FB(.)270 515 y +Fu(fun)28 b(split_lt)i(\(E,x\))e(=)g(E)325 572 y(|)f(split_lt)j +(\(T\(v,_,l,r\),x\))h(=)379 628 y(if)d(lt\(x,v\))h(then)f(split_lt\(l,x)q(\)) +379 685 y(else)g(if)g(lt\(v,x\))h(then)g(concat3\(v,l,sp)q(lit_)q(lt\()q +(r,x\))q(\))515 741 y(else)g(l)270 850 y(split_lt)17 b FB(takes)e(time)g +Fx(O)q Fw(\(log)7 b Fx(n)p Fw(\))p FB(.)28 b(At)16 b(\256rst)f(it)g(might)i +(be)e(expected)g(to)f(take)h(time)270 907 y Fx(O)q Fw(\(log)382 +885 y Fs(2)409 907 y Fx(n)p Fw(\))d FB(as)h(each)f(of)h(the)f +Fx(O)q Fw(\(log)7 b Fx(n)p Fw(\))13 b FB(r)o(ecursive)g(calls)g(might)g(call) +g Fu(concat3)i FB(which)270 963 y(might)e(take)e(logarithmic)i(time)f +(itself.)18 b(This)13 b(does)e(not)h(happen)h(because)f(the)g(or)o(der)270 +1019 y(of)19 b(the)g(calls)g(to)f Fu(concat3)j 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+y(of)e(the)g(algorithm)h(is)g Fu(uni_bd)p FB(,)h(which)f(cur)o(es)g(the)f +(ills)i(of)e Fu(union')p FB(.)23 b(The)13 b(\256rst)h(base)270 +394 y(case)c(tests)g(whether)g(the)g(second)g(parameter)g(is)g(empty)-5 +b(.)17 b(This)11 b(is)f(worthwhile)h(again)270 450 y(because)h(the)g +(invariant)g(ensur)o(es)h(that)e(this)i(parameter)e(r)o(educes)h(in)h(size)e +(in)i(at)e(least)270 507 y(one)16 b(of)f(the)g(r)o(ecursive)h(calls.)30 +b(In)16 b(the)g(second)g(base)f(case)g(of)h Fu(uni_bd)h FB(,)g(when)f(the)270 +563 y(\256rst)10 b(tr)o(ee)f(is)i(empty)-5 b(,)10 b(it)g(is)g(no)g(longer)g +(necessary)h(to)e(call)h Fu(split_gt)i FB(on)e(the)g(r)o(esult)g(of)270 +620 y Fu(split_lt)k FB(as)f(we)f(alr)o(eady)g(know)g(that)h +Fu(v)f FB(is)h(between)f Fu(lo)h FB(and)f Fu(hi)p FB(.)20 b(The)12 +b(subtr)o(ees)270 676 y(can)g(be)f(trimmed)g(as)h(appr)o(opriate.)270 +745 y(The)d(algorithm)g(starts)g(with)g(the)g(range)g Fw(\()p +FA(\0001)p Fx(;)f Fw(+)p FA(1)p Fw(\))p FB(.)16 b(It)9 b(is)h(undesirable)f +(to)g(pr)o(ogram)270 801 y(this)k(explicity)g(for)f(a)f(number)i(of)f(r)o +(easons.)19 b(If)12 b(values)g(ar)o(e)f(chosen)i(to)f(r)o(epr)o(esent)g(the) +270 858 y(in\256nities,)17 b(the)c(algorithm)i(will)f(fail)g(when)g(those)g +(values)g(appear)g(in)g(a)f(set.)24 b(Ther)o(e)270 914 y(may)15 +b(be)f(no)h(such)h(value,)g(for)f(example,)h(it)f(is)g(always)g(possible)g +(to)g(\256nd)g(a)g(gr)o(eater)270 971 y(alphabetic)d(string)g(by)g(appending) +h(a)e(`z':)16 b(`z')10 b Fx(<)i FB(`zz')e Fx(<)i FB(`zzz')d(etc.)270 +1040 y(The)g(solution)h(used)f(in)h(\256gur)o(e)g(5)f(is)h(specialise)g(to)e +(the)h(functions)i Fu(trim)f FB(and)f Fu(uni_bd)270 1096 y +FB(for)g(the)g(cases)g(when)h(one)e(or)h(both)g(of)g Fu(hi)h +Fw(=)j(+)p FA(1)c FB(and)g Fu(lo)h Fw(=)j FA(\0001)p FB(.)k(This)9 +b(accounts)h(for)270 1153 y(all)i(of)f(the)g(additional)g(functions.)19 +b(Specialising)13 b(a)e(function)i(is)f(simple,)h(if)e(tedious:)338 +1245 y FA(\017)23 b FB(Copy)12 b(the)f(function)i(and)e(give)h(it)f(a)g(new)h +(name.)338 1333 y FA(\017)23 b FB(Remove)11 b(the)g(parameter)m(,)f(say)i +Fu(lo)p FB(.)338 1421 y FA(\017)23 b FB(Subsititute)12 b FA(\0001)g +FB(for)f Fu(lo)h FB(in)g(the)g(body)f(of)g(the)h(function.)338 +1509 y FA(\017)23 b FB(Use)14 b(the)h(pr)o(operties)f(of)h +FA(\0001)p FB(,)h(like)f FA(\0001)g FB(|<)e(lo|)h(is)h(alwats)f(true,)h(to)f +(r)o(emove)384 1566 y(all)d(r)o(efer)o(ences)h(to)f(it.)270 +1658 y(The)16 b(case)f(wher)o(e)h(both)g Fu(hi)g FB(and)g Fu(lo)g +FB(ar)o(e)g(in\256nities)h(has)f(the)g(corr)o(ect)g(signatur)o(e)g(to)270 +1715 y(r)o(eplace)f Fu(union)p FB(.)29 b(Hedge)15 b(versions)g(of)g(set)g +(dif)o(fer)o(ence)h(and)f(intersection)h(may)f(be)270 1771 +y(derived)10 b(in)i(the)g(same)f(way)-5 b(.)270 1840 y(The)17 +b(run)h(time)g(of)f Fu(hedge_union)j FB(is)e Fx(O)q Fw(\()p +Fx(n)p Fw(\))p FB(,)h(like)f Fu(union)p FB(,)i(but)e(it)g(is)f(faster)h(than) +270 1896 y Fu(union)7 b FB(because)g(it)g(does)g(fewer)g(operations:)17 +b(the)7 b Fu(split)g FB(function)q(s)g(ar)o(e)g(called)g(onl)q(y)270 +1953 y(to)h(build)i(subtr)o(ees)f(that)g(ar)o(e)g(necessary)-5 +b(.)16 b(T)l(est)8 b(runs)i(indicate)f(that)g Fu(hedge_union)i +FB(is)270 2009 y(usually)f(20\045)e(faster)g(than)h Fu(union)p +FB(.)17 b(In)9 b(the)g(worst)f(case,)h Fx(S)s(mal)q(l)q(S)s(et)s +FA([)s Fx(B)r(ig)r(S)s(et)p FB(,)g(it)f(runs)270 2066 y(in)16 +b(about)f(the)g(same)g(time.)29 b(Whether)16 b(this)g(is)f(worth)h(the)f +(additional)g(complexity)270 2122 y(depends)k(on)h(the)f(application.)41 +b(Small)19 b(constant)h(factors)f(ar)o(e)f(only)i(worth)f(this)270 +2179 y(degr)o(ee)11 b(of)g(ef)o(fort)g(when)h(the)g(pr)o(ogram)f(is)h(a)f +(commonly)i(used)e(part)h(of)f(a)g(library)-5 b(.)-30 2327 +y Fy(11)66 b(Additional)21 b(op)r(erations:)29 b(rank)22 b(and)f(indexing)270 +2438 y FB(Each)15 b(tr)o(ee)g(node)g(contains)i(a)e(count)h(of)f(the)g +(elements)h(in)g(the)f(tr)o(ee)g(r)o(ooted)f(at)h(that)270 +2495 y(node.)h(This)11 b(was)f(justi\256ed)g(in)h(section)g(3\320the)f +(information)h(is)f(used)h(to)e(ensur)o(e)i(that)270 2551 y(only)j(balanced)g +(tr)o(ees)f(wer)o(e)g(built,)i(and)f(is)g(useful)g(in)h(its)e(own)h(right.)24 +b(W)m(ithout)15 b(the)270 2608 y(count)j(data)e(it)h(would)g(take)g +Fx(O)q Fw(\()p Fx(n)p Fw(\))f FB(time)h(to)g(count)g(the)h(number)f(of)g +(elements)h(by)270 2664 y(traversing)11 b(the)h(tr)o(ee)f(instead)g(of)g +(constant)h(time.)270 2733 y(The)f(count)h(data)d(can)j(also)f(be)f(used)i +(to)e(determine)h(the)g(rank)g(of)g(an)g(element)g(and)f(to)270 +2789 y(r)o(etrieve)j(an)h(element)g(by)g(its)g(rank.)25 b(The)14 +b(elements)g(ar)o(e)f(ranked)h(accor)o(ding)g(to)g(the)-30 +2914 y(22)1517 b(CSTR)11 b(92-10)p eop +%%Page: 23 25 +bop 60 40 a Fz(ADDITIONAL)13 b(OPERA)m(TIONS)p FB(:)18 b Fz(RANK)12 +b(AND)g(INDEXING)819 b FB(11.)p 60 62 1795 3 v 360 315 a Fb(fun)25 +b(trim)f(\(lo,hi,E\))g(=)h(E)410 365 y(|)g(trim)f(\(lo,hi,s)g(as)h +(T\(v,_,l,r\)\))f(=)460 415 y(if)49 b(lt\(lo,v\))g(then)534 +464 y(if)h(lt\(v,hi\))f(then)g(s)534 514 y(else)h(trim\(lo,hi,l\))460 +564 y(else)24 b(trim\(lo,hi,r\))360 664 y(fun)h(uni_bd)f(\(s,E,lo,hi\))g(=)h +(s)410 713 y(|)g(uni_bd)f(\(E,T\(v,_,l,r\),lo,hi\))f(=)509 +763 y(concat3\(v,split_gt\(l,lo\),split_lt\(r,hi\)\))410 813 +y(|)i(uni_bd)f(\(T\(v,_,l1,r1\),)g(s2)g(as)h(T\(v2,_,l2,r2\),lo,hi\))e(=)509 +863 y(concat3\(v,)709 913 y(uni_bd\(l1,trim\(lo,v,s2\),lo,v\),)709 +962 y(uni_bd\(r1,trim\(v,hi,s2\),v,hi\)\))485 1012 y(\(*)h(inv:)50 +b(lo)24 b(<)h(v)g(<)g(hi)g(*\))435 1112 y(\(*all)f(the)h(other)f(versions)g +(of)h(uni)g(and)f(trim)h(are)435 1162 y(specializations)e(of)i(the)g(above)f +(two)h(functions)f(with)435 1212 y(lo=-in\256nity)g(and/or)g(hi=+in\256nity)g +(*\))360 1311 y(fun)h(trim_lo)f(\(_)h(,E\))f(=)h(E)410 1361 +y(|)g(trim_lo)f(\(lo,s)g(as)h(T\(v,_,_,r\)\))f(=)509 1411 y(if)h(lt\(lo,v\))f +(then)h(s)g(else)f(trim_lo\(lo,r\))360 1461 y(fun)h(trim_hi)f(\(_)h(,E\))f(=) +h(E)410 1510 y(|)g(trim_hi)f(\(hi,s)g(as)h(T\(v,_,l,_\)\))f(=)509 +1560 y(if)h(lt\(v,hi\))f(then)h(s)g(else)f(trim_hi\(hi,l\))360 +1660 y(fun)h(uni_hi)f(\(s,E,hi\))g(=)h(s)410 1710 y(|)g(uni_hi)f +(\(E,T\(v,_,l,r\),hi\))f(=)509 1759 y(concat3\(v,l,split_lt\(r,hi\)\))410 +1809 y(|)i(uni_hi)f(\(T\(v,_,l1,r1\),)g(s2)g(as)h(T\(v2,_,l2,r2\),hi\))e(=) +509 1859 y(concat3\(v,)709 1909 y(uni_hi\(l1,trim_hi\(v,s2\),v\),)709 +1959 y(uni_bd\(r1,trim\(v,hi,s2\),v,hi\)\))360 2058 y(fun)i(uni_lo)f +(\(s,E,lo\))g(=)h(s)410 2108 y(|)g(uni_lo)f(\(E,T\(v,_,l,r\),lo\))f(=)509 +2158 y(concat3\(v,split_gt\(l,lo\),r\))410 2208 y(|)i(uni_lo)f +(\(T\(v,_,l1,r1\),)g(s2)g(as)h(T\(v2,_,l2,r2\),lo\))e(=)509 +2258 y(concat3\(v,)709 2307 y(uni_bd\(l1,trim\(lo,v,s2\),lo,v\),)709 +2357 y(uni_lo\(r1,trim_lo\(v,s2\),v\)\))360 2457 y(fun)i(hedge_union)f +(\(s,E\))g(=)h(s)410 2507 y(|)g(hedge_union)f(\(E,s2)g(as)h(T\(v,_,l,r\)\))f +(=)g(s2)410 2556 y(|)h(hedge_union)f(\(T\(v,_,l1,r1\),)f(s2)i(as)g +(T\(v2,_,l2,r2\)\))e(=)509 2606 y(concat3\(v,)709 2656 y +(uni_hi\(l1,trim_hi\(v,s2\),v\),)709 2706 y(uni_lo\(r1,trim_lo\(v,s2\),v\)\)) +907 2854 y FB(Fig.)17 b(5:)g(Hedge)11 b(union)60 2914 y(CSTR)h(92-10)1516 +b(23)p eop +%%Page: 24 26 +bop -30 33 a FB(12.)1546 b Fz(SUMMAR)o(Y)p -30 62 1795 3 v +270 281 a FB(or)o(dering)16 b(r)o(elation)g Fu(<)p FB(.)30 +b(The)16 b(minimum)h(element)f(has)g(rank)g(0,)h(the)e(next)i(smallest)270 +337 y(has)h(rank)g(1)g(and)f(so)h(on.)36 b(By)18 b(basing)h(the)e(rank)h(on)g +(0,)i(the)d(rank)i(of)e(an)h(element)270 394 y(is)e(simply)h(the)e(number)i +(of)e(elements)h(to)f(the)g(left)h(of)f(that)g(element.)29 +b(The)16 b(rank)f(is)270 450 y(computed)9 b(by)g(summing)i(the)d(sizes)h(of)g +(all)g(the)f(left-subtr)o(ees)i(not)e(taken)h(on)g(the)g(path)270 +507 y(to)i(the)g(element.)16 b(If)c(the)f(element)g(doesn't)g(appear)g(in)g +(the)h(tr)o(ee)e(then)i(a)f Fu(Subscript)270 563 y FB(exception)h(is)g +(signalled.)270 670 y Fu(exception)30 b(Subscript)270 783 y(fun)e(rank)g +(\(E,x\))h(=)f(raise)g(Subscript)325 839 y(|)f(rank)h(\(T\(v,n,l,r\),)j(x\))d +(=)379 896 y(if)g(xv)g(then)h +(rank\(r,x\))g(+)f(size)g(l)g(+)f(1)515 1009 y(else)i(size)f(l)270 +1115 y FB(Indexing)12 b(uses)e(the)g(size)g(information)h(to)e(navigate)h(to) +f(the)h(element.)16 b(At)10 b(each)g(node)270 1172 y(the)19 +b(index)g(is)g(checked)g(against)g(the)f(size)h(of)f(the)h(left)f(subtr)o +(ee.)39 b(If)19 b(the)f(index)h(is)270 1228 y(smaller)c(than)h(the)f(size)f +(of)h(the)g(subtr)o(ee)g(then)h(the)f(element)g(must)g(be)g(somewher)o(e)270 +1284 y(in)g(that)e(subtr)o(ee.)25 b(If)15 b(it)f(is)g(gr)o(eater)g(then)g(it) +g(must)h(be)f(in)h(the)f(right)g(subtr)o(ee,)i(but)e(the)270 +1341 y(number)e(of)g(elements)g(in)g(the)f(left)h(subtr)o(ee)g(and)f(the)h +(one)f(at)g(curr)o(ent)i(node)e(must)h(be)270 1397 y(discounted)f(\256rst.)17 +b(As)12 b(always,)f(if)g(the)g(element)g(is)g(in)h(neither)f(subtr)o(ee)g +(then)h(it)e(must)270 1454 y(be)h(at)g(the)h(curr)o(ent)g(node.)270 +1560 y Fu(fun)28 b(index)h(\(E,_\))f(=)g(raise)h(Subscript)325 +1617 y(|)e(index)i(\(T\(v,_,l,r\),)h(i\))e(=)379 1673 y(let)g(val)g(nl)g(=)g +(size)g(l)379 1730 y(in)488 1786 y(if)g(inl)i(then)f(index\(r,i-nl-)q(1\))625 1899 +y(else)g(v)379 1956 y(end)-30 2107 y Fy(12)66 b(Summa)n(ry)270 +2219 y FB(V)-5 b(ery)17 b(little)h(in)g(this)g(r)o(eport)f(is)h(new)l(.)35 +b(Balanced)17 b(binary)h(sear)o(ch)g(tr)o(ees)f(have)g(been)270 +2275 y(ar)o(ound)d(for)f(a)h(long)g(time.)24 b(In)14 b(particular)m(,)h(the)e +(methods)h(of)f(analysis)i(and)f(r)o(esults,)270 2331 y(\()p +Fx(O)q Fw(\(log)7 b Fx(n)p Fw(\))k FB(insertion,)i Fx(O)q Fw(\()p +Fx(n)p Fw(\))e FB(union)i(etc.,)e(ar)o(e)g(long)h(established)g(facts.)17 +b(This)12 b(pr)o(esen-)270 2388 y(tation,)g(however)m(,)e(has)i(several)f +(novel)g(featur)o(es:)338 2494 y FA(\017)23 b FB(the)11 b(functional)i(pr)o +(ogramming)g(style)338 2589 y FA(\017)23 b FB(the)11 b(abstraction)h(away)f +(fr)o(om)g(the)h(balancing)g(algorithm)338 2683 y FA(\017)23 +b FB(the)11 b Fu(hedge_union)j FB(algorithm)270 2789 y(Fr)o(om)e(this)g(exer) +o(cise)g(we)f(conclude)h(that)-30 2914 y(24)1517 b(CSTR)11 +b(92-10)p eop +%%Page: 25 27 +bop 60 40 a Fz(SUMMAR)o(Y)1548 b FB(12.)p 60 62 1795 3 v 428 +281 a FA(\017)23 b FB(It)11 b(is)g(nearly)h(as)f(easy)g(to)g(implement)h +(balanced)f(tr)o(ees)g(as)g(unbalanced)h(tr)o(ees)f(if)474 +337 y(the)g(concept)h(of)f(`r)o(ebalancing)i(constructors')e(is)h(taken)f(on) +h(boar)o(d.)428 431 y FA(\017)23 b FB(Rebalancing)e(constructors)g(need)f(an) +h(absolute)f(measur)o(e)h(of)f(the)g(size)g(or)474 488 y(height)11 +b(of)g(the)f(tr)o(ee.)16 b(A)-5 b(VL)11 b(tr)o(ees)f(which)i(encode)e(a)g +(height)i Fv(differ)o(ence)c FB(between)474 544 y(left)15 b(and)h(right)h +(subtr)o(ees)f(ar)o(e)f(not)h(easy)g(to)f(code)h(because)g(the)g(r)o +(ebalancing)474 601 y(depends)11 b(on)h(the)f Fv(change)h FB(in)g(height)g +(rather)f(than)h(an)f(absolute)h(measur)o(e.)428 694 y FA(\017)23 +b FB(Bounded)7 b(Balance)g(binary)g(tr)o(ees)g(ar)o(e)g(mor)o(e)g(useful)g +(than)g(other)g(type)g(of)g(binary)474 751 y(tr)o(ee.)428 845 +y FA(\017)23 b FB(All)15 b(the)f(balanced)g(tr)o(ee)f(schemes)i(have)f +(logarithmic)h(insert)g(and)f(delete,)g(so)474 901 y(when)19 +b(comparing)g(them)f(the)h(constant)g(factor)e(is)i(very)f(important.)38 +b(One)474 958 y(thing)18 b(that)f(af)o(fects)h(the)f(constant)h(factor)f(is)h +(the)f(size)h(of)f(the)g(node\320every)474 1014 y(node)11 b(has)h(to)f(be)h +(initialized)g(and)g(competes)f(for)h(space,)g(putting)h(pr)o(esur)o(e)g(on) +474 1070 y(the)e(memory)g(allocation)h(system.)17 b(Small)12 +b(is)g(beautifull.)360 1177 y(The)i(pr)o(ogram)f(described)h(her)o(e)g(was)f +(written)g(in)i(r)o(esponse)f(to)f(a)g(competition)h(put)360 +1233 y(forwar)o(d)9 b(by)h(Andr)o(ew)g(Appel.)18 b(W)m(ithout)11 +b(this)f(spur)h(I)f(would)g(never)f(have)g(taken)h(the)360 +1290 y(time)h(to)g(\256nish)i(the)e(pr)o(ogram,)h(or)e(write)h(this)h(r)o +(eport.)17 b(I)11 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np [ 2 12.16 ] 0 setdash 594 1818 a 609 1812 li st [] 0 setdash +[] 0 setdash np 609 1812 a closepath st np [ 2 12.16 ] 0 setdash +609 1812 a 624 1806 li st [] 0 setdash [] 0 setdash np 624 +1806 a closepath st np [ 2 11.81 ] 0 setdash 624 1806 a 639 +1801 li st [] 0 setdash [] 0 setdash np 639 1801 a closepath +st np [ 2 12.16 ] 0 setdash 639 1801 a 654 1795 li st [] 0 setdash +[] 0 setdash np 654 1795 a closepath st np [ 2 11.81 ] 0 setdash +654 1795 a 669 1790 li st [] 0 setdash [] 0 setdash np 669 +1790 a closepath st np [ 2 11.81 ] 0 setdash 669 1790 a 684 +1785 li st [] 0 setdash [] 0 setdash np 684 1785 a closepath +st np [ 2 11.81 ] 0 setdash 684 1785 a 699 1780 li st [] 0 setdash +[] 0 setdash np 699 1780 a closepath st np [ 2 11.52 ] 0 setdash +699 1780 a 714 1776 li st [] 0 setdash [] 0 setdash np 714 +1776 a closepath st np [ 2 11.81 ] 0 setdash 714 1776 a 729 +1771 li st [] 0 setdash [] 0 setdash np 729 1771 a closepath +st np [ 2 11.52 ] 0 setdash 729 1771 a 744 1767 li st [] 0 setdash +[] 0 setdash np 744 1767 a closepath st np [ 2 11.52 ] 0 setdash +744 1767 a 759 1763 li st [] 0 setdash [] 0 setdash np 759 +1763 a closepath st np [ 2 11.52 ] 0 setdash 759 1763 a 774 +1759 li st [] 0 setdash [] 0 setdash np 774 1759 a closepath +st np [ 2 11.52 ] 0 setdash 774 1759 a 789 1755 li st [] 0 setdash +[] 0 setdash np 789 1755 a closepath st np [ 2 11.52 ] 0 setdash +789 1755 a 804 1751 li st [] 0 setdash [] 0 setdash np 804 +1751 a closepath st np [ 2 11.52 ] 0 setdash 804 1751 a 819 +1747 li st [] 0 setdash [] 0 setdash np 819 1747 a closepath +st np [ 2 11.30 ] 0 setdash 819 1747 a 834 1744 li st [] 0 setdash +[] 0 setdash np 834 1744 a closepath st np [ 2 11.30 ] 0 setdash +834 1744 a 849 1741 li st [] 0 setdash [] 0 setdash np 849 +1741 a closepath st np [ 2 11.52 ] 0 setdash 849 1741 a 864 +1737 li st [] 0 setdash [] 0 setdash np 864 1737 a closepath +st np [ 2 11.30 ] 0 setdash 864 1737 a 879 1734 li st [] 0 setdash +[] 0 setdash np 879 1734 a closepath st np [ 2 11.30 ] 0 setdash +879 1734 a 894 1731 li st [] 0 setdash [] 0 setdash np 894 +1731 a closepath st np [ 2 11.52 ] 0 setdash 894 1731 a 909 +1727 li st [] 0 setdash [] 0 setdash np 909 1727 a closepath +st np [ 2 11.30 ] 0 setdash 909 1727 a 924 1724 li st [] 0 setdash +[] 0 setdash np 924 1724 a closepath st np [ 2 11.30 ] 0 setdash +924 1724 a 939 1721 li st [] 0 setdash [] 0 setdash np 939 +1721 a closepath st np [ 2 11.13 ] 0 setdash 939 1721 a 954 +1719 li st [] 0 setdash [] 0 setdash np 954 1719 a closepath +st np [ 2 11.30 ] 0 setdash 954 1719 a 969 1716 li st [] 0 setdash +[] 0 setdash np 969 1716 a closepath st np [ 2 11.30 ] 0 setdash +969 1716 a 984 1713 li st [] 0 setdash [] 0 setdash np 984 +1713 a closepath st np [ 2 11.30 ] 0 setdash 984 1713 a 999 +1710 li st [] 0 setdash [] 0 setdash np 999 1710 a closepath +st np [ 2 11.13 ] 0 setdash 999 1710 a 1014 1708 li st [] 0 setdash +[] 0 setdash np 1014 1708 a closepath st np [ 2 11.30 ] 0 setdash +1014 1708 a 1029 1705 li st [] 0 setdash [] 0 setdash np 1029 +1705 a closepath st np [ 2 11.13 ] 0 setdash 1029 1705 a 1044 +1703 li st [] 0 setdash [] 0 setdash np 1044 1703 a closepath +st np [ 2 11.30 ] 0 setdash 1044 1703 a 1059 1700 li st [] 0 setdash +[] 0 setdash np 1059 1700 a closepath st np [ 2 11.13 ] 0 setdash +1059 1700 a 1074 1698 li st [] 0 setdash [] 0 setdash np 1074 +1698 a closepath st np [ 2 11.13 ] 0 setdash 1074 1698 a 1089 +1696 li st [] 0 setdash [] 0 setdash np 1089 1696 a closepath +st np [ 2 11.30 ] 0 setdash 1089 1696 a 1104 1693 li st [] 0 setdash +[] 0 setdash np 1104 1693 a closepath st np [ 2 11.13 ] 0 setdash +1104 1693 a 1119 1691 li st [] 0 setdash [] 0 setdash np 1119 +1691 a closepath st np [ 2 11.13 ] 0 setdash 1119 1691 a 1134 +1689 li st [] 0 setdash [] 0 setdash np 1134 1689 a closepath +st np [ 2 11.13 ] 0 setdash 1134 1689 a 1149 1687 li st [] 0 setdash +[] 0 setdash np 1149 1687 a closepath st np [ 2 11.13 ] 0 setdash +1149 1687 a 1164 1685 li st [] 0 setdash [] 0 setdash np 1164 +1685 a closepath st np [ 2 11.13 ] 0 setdash 1164 1685 a 1179 +1683 li st [] 0 setdash [] 0 setdash np 1179 1683 a closepath +st np [ 2 11.13 ] 0 setdash 1179 1683 a 1194 1681 li st [] 0 setdash +[] 0 setdash np 1194 1681 a closepath st np [ 2 11.13 ] 0 setdash +1194 1681 a 1209 1679 li st [] 0 setdash [] 0 setdash np 1209 +1679 a closepath st np [ 2 11.13 ] 0 setdash 1209 1679 a 1224 +1677 li st [] 0 setdash [] 0 setdash np 1224 1677 a closepath +st np [ 2 11.13 ] 0 setdash 1224 1677 a 1239 1675 li st [] 0 setdash +[] 0 setdash np 1239 1675 a closepath st np [ 2 11.13 ] 0 setdash +1239 1675 a 1254 1673 li st [] 0 setdash [] 0 setdash np 1254 +1673 a closepath st np [ 2 11.03 ] 0 setdash 1254 1673 a 1269 +1672 li st [] 0 setdash [] 0 setdash np 1269 1672 a closepath +st np [ 2 11.13 ] 0 setdash 1269 1672 a 1284 1670 li st [] 0 setdash +[] 0 setdash np 1284 1670 a closepath st np [ 2 11.13 ] 0 setdash +1284 1670 a 1299 1668 li st [] 0 setdash [] 0 setdash np 1299 +1668 a closepath st np [ 2 11.13 ] 0 setdash 1299 1668 a 1314 +1666 li st [] 0 setdash [] 0 setdash np 1314 1666 a closepath +st np [ 2 11.03 ] 0 setdash 1314 1666 a 1329 1665 li st [] 0 setdash +[] 0 setdash np 1329 1665 a closepath st np [ 2 11.13 ] 0 setdash +1329 1665 a 1344 1663 li st [] 0 setdash [] 0 setdash np 1344 +1663 a closepath st np [ 2 11.13 ] 0 setdash 1344 1663 a 1359 +1661 li st [] 0 setdash [] 0 setdash np 1359 1661 a closepath +st np [ 2 11.03 ] 0 setdash 1359 1661 a 1374 1660 li st [] 0 setdash +[] 0 setdash np 1374 1660 a closepath st np [ 2 11.13 ] 0 setdash +1374 1660 a 1389 1658 li st [] 0 setdash [] 0 setdash np 1389 +1658 a closepath st np [ 2 11.03 ] 0 setdash 1389 1658 a 1404 +1657 li st [] 0 setdash [] 0 setdash np 1404 1657 a closepath +st np [ 2 11.13 ] 0 setdash 1404 1657 a 1419 1655 li st [] 0 setdash +[] 0 setdash np 1419 1655 a closepath st np [ 2 11.03 ] 0 setdash +1419 1655 a 1434 1654 li st [] 0 setdash [] 0 setdash np 1434 +1654 a closepath st np [ 2 11.03 ] 0 setdash 1434 1654 a 1449 +1653 li st [] 0 setdash [] 0 setdash np 1449 1653 a closepath +st np [ 2 11.13 ] 0 setdash 1449 1653 a 1464 1651 li st [] 0 setdash +[] 0 setdash np 1464 1651 a closepath st np [ 2 11.03 ] 0 setdash +1464 1651 a 1479 1650 li st [] 0 setdash [] 0 setdash np 1479 +1650 a closepath st np [ 2 11.13 ] 0 setdash 1479 1650 a 1494 +1648 li st [] 0 setdash [] 0 setdash np 1494 1648 a closepath +st np [ 2 11.03 ] 0 setdash 1494 1648 a 1509 1647 li st [] 0 setdash +[] 0 setdash np 1509 1647 a closepath st np [ 2 11.03 ] 0 setdash +1509 1647 a 1524 1646 li st [] 0 setdash [] 0 setdash np 1524 +1646 a closepath st np [ 2 11.03 ] 0 setdash 1524 1646 a 1539 +1645 li st [] 0 setdash [] 0 setdash np 1539 1645 a closepath +st np [ 2 11.13 ] 0 setdash 1539 1645 a 1554 1643 li st [] 0 setdash +[] 0 setdash np 1554 1643 a closepath st np [ 2 11.03 ] 0 setdash +1554 1643 a 1569 1642 li st [] 0 setdash [] 0 setdash np 1569 +1642 a closepath st np [ 2 11.03 ] 0 setdash 1569 1642 a 1584 +1641 li st [] 0 setdash [] 0 setdash np 1584 1641 a closepath +st np [ 2 11.03 ] 0 setdash 1584 1641 a 1599 1640 li st [] 0 setdash +[] 0 setdash np 1599 1640 a closepath st np [ 2 11.03 ] 0 setdash +1599 1640 a 1614 1639 li st [] 0 setdash [] 0 setdash np 1614 +1639 a closepath st np [ 2 11.13 ] 0 setdash 1614 1639 a 1629 +1637 li st [] 0 setdash [] 0 setdash np 1629 1637 a closepath +st np [ 2 11.03 ] 0 setdash 1629 1637 a 1644 1636 li st [] 0 setdash +[] 0 setdash np 1644 1636 a closepath st np [ 2 11.03 ] 0 setdash +1644 1636 a 1659 1635 li st [] 0 setdash [] 0 setdash np 1659 +1635 a closepath st np [ 2 11.03 ] 0 setdash 1659 1635 a 1674 +1634 li st [] 0 setdash [] 0 setdash np 1674 1634 a closepath +st np [ 2 11.03 ] 0 setdash 1674 1634 a 1689 1633 li st [] 0 setdash +[] 0 setdash np 1689 1633 a closepath st np [ 2 11.03 ] 0 setdash +1689 1633 a 1704 1632 li st [] 0 setdash [] 0 setdash np 1704 +1632 a closepath st np [ 2 11.03 ] 0 setdash 1704 1632 a 1719 +1631 li st [] 0 setdash [] 0 setdash np 1719 1631 a closepath +st 734 728 a Fx(\013)f FA(\024)g Fx(w)357 936 y(\013)g FA(\025)457 +906 y Fw(1)p 452 926 34 2 v 452 968 a Fx(w)1475 1591 y(\013)g +FA(\024)1609 1560 y Fx(w)p 1570 1580 113 2 v 1570 1622 a(w)d +Fw(+)h(2)778 1414 y Fx(\013)h FA(\024)872 1384 y Fw(2)p Fx(w)f +Fw(+)f(1)p 872 1404 135 2 v 874 1446 a Fx(w)908 1433 y Fs(2)937 +1446 y FA(\000)g Fw(1)1022 1669 y(\()p Fx(w)1073 1676 y Fm(c)1090 +1669 y Fx(;)e(\013)1140 1676 y Fm(c)1156 1669 y Fw(\))1216 +1772 y FB(\(4.646,)1350 1754 y Fs(1)p 1349 1761 17 2 v 1349 +1788 a(2)1371 1772 y FB(\))p np 219 1798 a closepath st np +[ 2 14.00 ] 0 setdash 219 1798 a 237 1798 li st [] 0 setdash +[] 0 setdash np 237 1798 a closepath st np [ 2 13.00 ] 0 setdash +237 1798 a 254 1798 li st [] 0 setdash [] 0 setdash np 254 +1798 a closepath st np [ 2 13.00 ] 0 setdash 254 1798 a 271 +1798 li st [] 0 setdash [] 0 setdash np 271 1798 a closepath +st np [ 2 13.00 ] 0 setdash 271 1798 a 288 1798 li st [] 0 setdash +[] 0 setdash np 288 1798 a closepath st np [ 2 13.00 ] 0 setdash +288 1798 a 305 1798 li st [] 0 setdash [] 0 setdash np 305 +1798 a closepath st np [ 2 13.00 ] 0 setdash 305 1798 a 322 +1798 li st [] 0 setdash [] 0 setdash np 322 1798 a closepath +st np [ 2 13.00 ] 0 setdash 322 1798 a 339 1798 li st [] 0 setdash +[] 0 setdash np 339 1798 a closepath st np [ 2 14.00 ] 0 setdash +339 1798 a 357 1798 li st [] 0 setdash [] 0 setdash np 357 +1798 a closepath st np [ 2 13.00 ] 0 setdash 357 1798 a 374 +1798 li st [] 0 setdash [] 0 setdash np 374 1798 a closepath +st np [ 2 13.00 ] 0 setdash 374 1798 a 391 1798 li st [] 0 setdash +[] 0 setdash np 391 1798 a closepath st np [ 2 13.00 ] 0 setdash +391 1798 a 408 1798 li st [] 0 setdash [] 0 setdash np 408 +1798 a closepath st np [ 2 13.00 ] 0 setdash 408 1798 a 425 +1798 li st [] 0 setdash [] 0 setdash np 425 1798 a closepath +st np [ 2 13.00 ] 0 setdash 425 1798 a 442 1798 li st [] 0 setdash +[] 0 setdash np 442 1798 a closepath st np [ 2 13.00 ] 0 setdash +442 1798 a 459 1798 li st [] 0 setdash [] 0 setdash np 459 +1798 a closepath st np [ 2 14.00 ] 0 setdash 459 1798 a 477 +1798 li st [] 0 setdash [] 0 setdash np 477 1798 a closepath +st np [ 2 13.00 ] 0 setdash 477 1798 a 494 1798 li st [] 0 setdash +[] 0 setdash np 494 1798 a closepath st np [ 2 13.00 ] 0 setdash +494 1798 a 511 1798 li st [] 0 setdash [] 0 setdash np 511 +1798 a closepath st np [ 2 13.00 ] 0 setdash 511 1798 a 528 +1798 li st [] 0 setdash [] 0 setdash np 528 1798 a closepath +st np [ 2 13.00 ] 0 setdash 528 1798 a 545 1798 li st [] 0 setdash +[] 0 setdash np 545 1798 a closepath st np [ 2 13.00 ] 0 setdash +545 1798 a 562 1798 li st [] 0 setdash [] 0 setdash np 562 +1798 a closepath st np [ 2 13.00 ] 0 setdash 562 1798 a 579 +1798 li st [] 0 setdash [] 0 setdash np 579 1798 a closepath +st np [ 2 14.00 ] 0 setdash 579 1798 a 597 1798 li st [] 0 setdash +[] 0 setdash np 597 1798 a closepath st np [ 2 13.00 ] 0 setdash +597 1798 a 614 1798 li st [] 0 setdash [] 0 setdash np 614 +1798 a closepath st np [ 2 13.00 ] 0 setdash 614 1798 a 631 +1798 li st [] 0 setdash [] 0 setdash np 631 1798 a closepath +st np [ 2 13.00 ] 0 setdash 631 1798 a 648 1798 li st [] 0 setdash +[] 0 setdash np 648 1798 a closepath st np [ 2 13.00 ] 0 setdash +648 1798 a 665 1798 li st [] 0 setdash [] 0 setdash np 665 +1798 a closepath st np [ 2 13.00 ] 0 setdash 665 1798 a 682 +1798 li st [] 0 setdash [] 0 setdash np 682 1798 a closepath +st np [ 2 13.00 ] 0 setdash 682 1798 a 699 1798 li st [] 0 setdash +[] 0 setdash np 699 1798 a closepath st np [ 2 14.00 ] 0 setdash +699 1798 a 717 1798 li st [] 0 setdash [] 0 setdash np 717 +1798 a closepath st np [ 2 13.00 ] 0 setdash 717 1798 a 734 +1798 li st [] 0 setdash [] 0 setdash np 734 1798 a closepath +st np [ 2 13.00 ] 0 setdash 734 1798 a 751 1798 li st [] 0 setdash +[] 0 setdash np 751 1798 a closepath st np [ 2 13.00 ] 0 setdash +751 1798 a 768 1798 li st [] 0 setdash [] 0 setdash np 768 +1798 a closepath st np [ 2 13.00 ] 0 setdash 768 1798 a 785 +1798 li st [] 0 setdash [] 0 setdash np 785 1798 a closepath +st np [ 2 13.00 ] 0 setdash 785 1798 a 802 1798 li st [] 0 setdash +[] 0 setdash np 802 1798 a closepath st np [ 2 13.00 ] 0 setdash +802 1798 a 819 1798 li st [] 0 setdash [] 0 setdash np 819 +1798 a closepath st np [ 2 14.00 ] 0 setdash 819 1798 a 837 +1798 li st [] 0 setdash [] 0 setdash np 837 1798 a closepath +st np [ 2 13.00 ] 0 setdash 837 1798 a 854 1798 li st [] 0 setdash +[] 0 setdash np 854 1798 a closepath st np [ 2 13.00 ] 0 setdash +854 1798 a 871 1798 li st [] 0 setdash [] 0 setdash np 871 +1798 a closepath st np [ 2 13.00 ] 0 setdash 871 1798 a 888 +1798 li st [] 0 setdash [] 0 setdash np 888 1798 a closepath +st np [ 2 13.00 ] 0 setdash 888 1798 a 905 1798 li st [] 0 setdash +[] 0 setdash np 905 1798 a closepath st np [ 2 13.00 ] 0 setdash +905 1798 a 922 1798 li st [] 0 setdash [] 0 setdash np 922 +1798 a closepath st np [ 2 13.00 ] 0 setdash 922 1798 a 939 +1798 li st [] 0 setdash [] 0 setdash np 939 1798 a closepath +st np [ 2 14.00 ] 0 setdash 939 1798 a 957 1798 li st [] 0 setdash +[] 0 setdash np 957 1798 a closepath st np [ 2 13.00 ] 0 setdash +957 1798 a 974 1798 li st [] 0 setdash [] 0 setdash np 974 +1798 a closepath st np [ 2 13.00 ] 0 setdash 974 1798 a 991 +1798 li st [] 0 setdash [] 0 setdash np 991 1798 a closepath +st np [ 2 13.00 ] 0 setdash 991 1798 a 1008 1798 li st [] 0 setdash +[] 0 setdash np 1008 1798 a closepath st np [ 2 13.00 ] 0 setdash +1008 1798 a 1025 1798 li st [] 0 setdash [] 0 setdash np 1025 +1798 a closepath st np [ 2 13.00 ] 0 setdash 1025 1798 a 1042 +1798 li st [] 0 setdash [] 0 setdash np 1042 1798 a closepath +st np [ 2 13.00 ] 0 setdash 1042 1798 a 1059 1798 li st [] 0 setdash +[] 0 setdash np 1059 1798 a closepath st np [ 2 14.00 ] 0 setdash +1059 1798 a 1077 1798 li st [] 0 setdash [] 0 setdash np 1077 +1798 a closepath st np [ 2 13.00 ] 0 setdash 1077 1798 a 1094 +1798 li st [] 0 setdash [] 0 setdash np 1094 1798 a closepath +st np [ 2 13.00 ] 0 setdash 1094 1798 a 1111 1798 li st [] 0 setdash +[] 0 setdash np 1111 1798 a closepath st np [ 2 13.00 ] 0 setdash +1111 1798 a 1128 1798 li st [] 0 setdash [] 0 setdash np 1128 +1798 a closepath st np [ 2 13.00 ] 0 setdash 1128 1798 a 1145 +1798 li st [] 0 setdash [] 0 setdash np 1145 1798 a closepath +st np [ 2 13.00 ] 0 setdash 1145 1798 a 1162 1798 li st [] 0 setdash +[] 0 setdash np 1162 1798 a closepath st np [ 2 13.00 ] 0 setdash +1162 1798 a 1179 1798 li st [] 0 setdash [] 0 setdash np 1179 +1798 a closepath st np [ 2 14.00 ] 0 setdash 1179 1798 a 1197 +1798 li st [] 0 setdash [] 0 setdash np 1197 1798 a closepath +st np [ 2 13.00 ] 0 setdash 1197 1798 a 1214 1798 li st [] 0 setdash +[] 0 setdash np 1214 1798 a closepath st np [ 2 13.00 ] 0 setdash +1214 1798 a 1231 1798 li st [] 0 setdash [] 0 setdash np 1231 +1798 a closepath st np [ 2 13.00 ] 0 setdash 1231 1798 a 1248 +1798 li st [] 0 setdash [] 0 setdash np 1248 1798 a closepath +st np [ 2 13.00 ] 0 setdash 1248 1798 a 1265 1798 li st [] 0 setdash +[] 0 setdash np 1265 1798 a closepath st np [ 2 13.00 ] 0 setdash +1265 1798 a 1282 1798 li st [] 0 setdash [] 0 setdash np 1282 +1798 a closepath st np [ 2 13.00 ] 0 setdash 1282 1798 a 1299 +1798 li st [] 0 setdash [] 0 setdash np 1299 1798 a closepath +st np [ 2 14.00 ] 0 setdash 1299 1798 a 1317 1798 li st [] 0 setdash +[] 0 setdash np 1317 1798 a closepath st np [ 2 13.00 ] 0 setdash +1317 1798 a 1334 1798 li st [] 0 setdash [] 0 setdash np 1334 +1798 a closepath st np [ 2 13.00 ] 0 setdash 1334 1798 a 1351 +1798 li st [] 0 setdash [] 0 setdash np 1351 1798 a closepath +st np [ 2 13.00 ] 0 setdash 1351 1798 a 1368 1798 li st [] 0 setdash +[] 0 setdash np 1368 1798 a closepath st np [ 2 13.00 ] 0 setdash +1368 1798 a 1385 1798 li st [] 0 setdash [] 0 setdash np 1385 +1798 a closepath st np [ 2 13.00 ] 0 setdash 1385 1798 a 1402 +1798 li st [] 0 setdash [] 0 setdash np 1402 1798 a closepath +st np [ 2 13.00 ] 0 setdash 1402 1798 a 1419 1798 li st [] 0 setdash +[] 0 setdash np 1419 1798 a closepath st np [ 2 14.00 ] 0 setdash +1419 1798 a 1437 1798 li st [] 0 setdash [] 0 setdash np 1437 +1798 a closepath st np [ 2 13.00 ] 0 setdash 1437 1798 a 1454 +1798 li st [] 0 setdash [] 0 setdash np 1454 1798 a closepath +st np [ 2 13.00 ] 0 setdash 1454 1798 a 1471 1798 li st [] 0 setdash +[] 0 setdash np 1471 1798 a closepath st np [ 2 13.00 ] 0 setdash +1471 1798 a 1488 1798 li st [] 0 setdash [] 0 setdash np 1488 +1798 a closepath st np [ 2 13.00 ] 0 setdash 1488 1798 a 1505 +1798 li st [] 0 setdash [] 0 setdash np 1505 1798 a closepath +st np [ 2 13.00 ] 0 setdash 1505 1798 a 1522 1798 li st [] 0 setdash +[] 0 setdash np 1522 1798 a closepath st np [ 2 13.00 ] 0 setdash +1522 1798 a 1539 1798 li st [] 0 setdash [] 0 setdash np 1539 +1798 a closepath st np [ 2 14.00 ] 0 setdash 1539 1798 a 1557 +1798 li st [] 0 setdash [] 0 setdash np 1557 1798 a closepath +st np [ 2 13.00 ] 0 setdash 1557 1798 a 1574 1798 li st [] 0 setdash +[] 0 setdash np 1574 1798 a closepath st np [ 2 13.00 ] 0 setdash +1574 1798 a 1591 1798 li st [] 0 setdash [] 0 setdash np 1591 +1798 a closepath st np [ 2 13.00 ] 0 setdash 1591 1798 a 1608 +1798 li st [] 0 setdash [] 0 setdash np 1608 1798 a closepath +st np [ 2 13.00 ] 0 setdash 1608 1798 a 1625 1798 li st [] 0 setdash +[] 0 setdash np 1625 1798 a closepath st np [ 2 13.00 ] 0 setdash +1625 1798 a 1642 1798 li st [] 0 setdash [] 0 setdash np 1642 +1798 a closepath st np [ 2 13.00 ] 0 setdash 1642 1798 a 1659 +1798 li st [] 0 setdash [] 0 setdash np 1659 1798 a closepath +st np [ 2 14.00 ] 0 setdash 1659 1798 a 1677 1798 li st [] 0 setdash +[] 0 setdash np 1677 1798 a closepath st np [ 2 13.00 ] 0 setdash +1677 1798 a 1694 1798 li st [] 0 setdash [] 0 setdash np 1694 +1798 a closepath st np [ 2 13.00 ] 0 setdash 1694 1798 a 1711 +1798 li st [] 0 setdash [] 0 setdash np 1711 1798 a closepath +st np 243 2098 a 237 2092 li st 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zj%0WKo<)MK=`;0A^3fX}*i!+#VDZK6V=Ye|m%nVDXG0eqD4Gc>>0c*~n6N$DqSW8B5FcB9*pwrEWMBp?C=JXH? zi(pKM1;dw(%(;LQ?a$nI+EJ)k?7do=G!-Q z3Mpp_;@WmfH>i-<-_7X5@}G8llcZ-hlqqlazxLeM;5k{BDAkITN9RH6`Qt^I-@Ky5 zS>iE{PPuPfN7iMN)Ksv0O{{$AcToxQrOBo;#-;x Gen.filter_map + (function + | `File, f when CCString.suffix ~suf:".odoc" f -> Some f + | _ -> None + ) + |> Gen.flat_map + (fun f -> Gen.of_list ["-load"; f]) + |> Gen.to_list +;; + +let out = "deps.dot";; + +let cmd = + "ocamldoc -dot -o " ^ out ^ " " ^ String.concat " " odoc_files +;; + +print_endline ("run: " ^ cmd);; +Unix.system cmd;; +print_endline ("output in " ^ out);; diff --git a/doc/containers.md b/doc/containers.md new file mode 100644 index 00000000..d96f7683 --- /dev/null +++ b/doc/containers.md @@ -0,0 +1,81 @@ +# More about OCaml-containers + +This document contains more information on some modules of Containers. + +```ocaml +# #require "containers";; +``` + +## Hash combinators: `CCHash` + +Although OCaml provides polymorphic hash tables (`('a,'b) Hashtbl.t`) +using the polymorphic equality `(=)` and hash `Hashtbl.hash`, it is often +safer and more efficient to use `Hashtbl.Make` (or the extended `CCHashtbl.Make`) +with custom equality and hash functions. + +`CCHash` provides combinators for writing hash functions: + +```ocaml +# module H = CCHash;; +module H = CCHash + +# let hash1 : (int * bool) list H.t = H.(list (pair int bool));; +val hash1 : (int * bool) list H.t = +``` + +```ocaml non-deterministic=output +# hash1 [1, true; 2, false; 3, true];; +- : int = 636041136 +# hash1 CCList.(1 -- 1000 |> map (fun i->i, i mod 2 = 0));; +- : int = 845685523 +# hash1 CCList.(1 -- 1001 |> map (fun i->i, i mod 2 = 0));; +- : int = 381026697 +``` + +The polymorphic hash function is still present, as `CCHash.poly`. +The functions `CCHash.list_comm` and `CCHash.array_comm` allow to hash +lists and arrays while ignoring the order of elements: all permutations +of the input will have the same hash. + +## Parser Combinator: `CCParse` + +The module `CCParse` defines basic parser combinators on strings. +Adapting [angstrom's tutorial example](https://github.com/inhabitedtype/angstrom#usage) +gives the following snippet. +Note that backtracking is explicit in `CCParse`, hence +the use of `try_` to allow it in some places. +Explicit memoization with `memo` and `fix_memo` is also possible. + +```ocaml +open CCParse.Infix +module P = CCParse + +let parens p = P.try_ (P.char '(') *> p <* P.char ')' +let add = P.char '+' *> P.return (+) +let sub = P.char '-' *> P.return (-) +let mul = P.char '*' *> P.return ( * ) +let div = P.char '/' *> P.return ( / ) +let integer = + P.chars1_if (function '0'..'9'->true|_->false) >|= int_of_string + +let chainl1 e op = + P.fix (fun r -> + e >>= fun x -> P.try_ (op <*> P.return x <*> r) <|> P.return x) + +let expr : int P.t = + P.fix (fun expr -> + let factor = parens expr <|> integer in + let term = chainl1 factor (mul <|> div) in + chainl1 term (add <|> sub)) +``` + +Now we can parse strings using `expr`: + +```ocaml +# P.parse_string expr "4*1+2";; (* Ok 6 *) +- : int P.or_error = Result.Ok 6 + +# P.parse_string expr "4*(1+2)";; (* Ok 12 *) +- : int P.or_error = Result.Ok 12 +``` + diff --git a/doc/dune b/doc/dune new file mode 100644 index 00000000..2dbb14d7 --- /dev/null +++ b/doc/dune @@ -0,0 +1,8 @@ + +;(alias +; (name runtest) +; (deps containers.md) +; (action (progn +; (run mdx test %{deps}) +; (diff? %{deps} %{deps}.corrected)))) +; diff --git a/doc/finger_tree.pdf b/doc/finger_tree.pdf new file mode 100644 index 0000000000000000000000000000000000000000..051d4218aeaeae53915a138e14b76a47605b0c37 GIT binary patch literal 172801 zcma&MQ?Mw(vaUO9+qP}nwr$(CZQHgz=CEzs<~(z)*eCA8emM6fJ3FH?qB|?QqrSh2 zR6#_HmXVGLigf%sz6gqqfPui?$O?*whhE0i&fLXLc$fL_eT(8W~5 z)Y#s{l#dU}*~Q7!&=$%gr&UWfd20-D#&hb#+pb5nN=xYJ=OCF@JMMTrjkx}(Duqr1 zsXID?gaRO?rSrYM{H)j;AW4f7JiWRI3Jqi0rQh$)HP{}#<`5d~cj|9l8@fGxUXN1r 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