%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: colt_final.dvi %%Pages: 9 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips -o colt_final.ps colt_final %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1998.05.03:1824 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{ statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 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/Text { ishow } def /idef { dup where { pop pop pop } { exch def } ifelse } def /ifill { 0 begin gsave patternGrayLevel -1 ne { fgred bgred fgred sub patternGrayLevel mul add fggreen bggreen fggreen sub patternGrayLevel mul add fgblue bgblue fgblue sub patternGrayLevel mul add setrgbcolor eofill } { eoclip originalCTM setmatrix pathbbox /t exch def /r exch def /b exch def /l exch def /w r l sub ceiling cvi def /h t b sub ceiling cvi def /imageByteWidth w 8 div ceiling cvi def /imageHeight h def bgred bggreen bgblue setrgbcolor eofill fgred fggreen fgblue setrgbcolor w 0 gt h 0 gt and { l b translate w h scale w h true [w 0 0 h neg 0 h] { patternproc } imagemask } if } ifelse grestore end } dup 0 8 dict put def /istroke { gsave brushDashOffset -1 eq { [] 0 setdash 1 setgray } { brushDashArray brushDashOffset setdash fgred fggreen fgblue setrgbcolor } ifelse brushWidth setlinewidth originalCTM setmatrix stroke grestore } def /ishow { 0 begin gsave fgred fggreen fgblue setrgbcolor /fontDict printFont printSize scalefont dup setfont def /descender fontDict begin 0 [FontBBox] 1 get FontMatrix end transform exch pop def /vertoffset 1 printSize sub descender sub def { 0 vertoffset moveto show /vertoffset vertoffset printSize sub def } forall grestore end } dup 0 3 dict put def /patternproc { 0 begin /patternByteLength patternString length def /patternHeight patternByteLength 8 mul sqrt cvi def /patternWidth patternHeight def /patternByteWidth patternWidth 8 idiv def /imageByteMaxLength imageByteWidth imageHeight mul stringLimit patternByteWidth sub min def /imageMaxHeight imageByteMaxLength imageByteWidth idiv patternHeight idiv patternHeight mul patternHeight max def /imageHeight imageHeight imageMaxHeight sub store /imageString imageByteWidth imageMaxHeight mul patternByteWidth add string def 0 1 imageMaxHeight 1 sub { /y exch def /patternRow y patternByteWidth mul patternByteLength mod def /patternRowString patternString patternRow patternByteWidth getinterval def /imageRow y imageByteWidth mul def 0 patternByteWidth imageByteWidth 1 sub { /x exch def imageString imageRow x add patternRowString putinterval } for } for imageString end } dup 0 12 dict put def /min { dup 3 2 roll dup 4 3 roll lt { exch } if pop } def /max { dup 3 2 roll dup 4 3 roll gt { exch } if pop } def /midpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 x1 add 2 div y0 y1 add 2 div end } dup 0 4 dict put def /thirdpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 2 mul x1 add 3 div y0 2 mul y1 add 3 div end } dup 0 4 dict put def /subspline { 0 begin /movetoNeeded exch def y exch get /y3 exch def x exch get /x3 exch def y exch get /y2 exch def x exch get /x2 exch def y exch get /y1 exch def x exch get /x1 exch def y exch get /y0 exch def x exch get /x0 exch def x1 y1 x2 y2 thirdpoint /p1y exch def /p1x exch def x2 y2 x1 y1 thirdpoint /p2y exch def /p2x exch def x1 y1 x0 y0 thirdpoint p1x p1y midpoint /p0y exch def /p0x exch def x2 y2 x3 y3 thirdpoint p2x p2y midpoint /p3y exch def /p3x exch def movetoNeeded { p0x p0y moveto } if p1x p1y p2x p2y p3x p3y curveto end } dup 0 17 dict put def /storexyn { /n exch def /y n array def /x n array def n 1 sub -1 0 { /i exch def y i 3 2 roll put x i 3 2 roll put } for } def /SSten { fgred fggreen fgblue setrgbcolor dup true exch 1 0 0 -1 0 6 -1 roll matrix astore } def /FSten { dup 3 -1 roll dup 4 1 roll exch newpath 0 0 moveto dup 0 exch lineto exch dup 3 1 roll exch lineto 0 lineto closepath bgred bggreen bgblue setrgbcolor eofill SSten } def /Rast { exch dup 3 1 roll 1 0 0 -1 0 6 -1 roll matrix astore } def /arrowhead { 0 begin transform originalCTM itransform /taily exch def /tailx exch def transform originalCTM itransform /tipy exch def /tipx exch def /dy tipy taily sub def /dx tipx tailx sub def /angle dx 0 ne dy 0 ne or { dy dx atan } { 90 } ifelse def gsave originalCTM setmatrix tipx tipy translate angle rotate newpath arrowHeight neg arrowWidth 2 div moveto 0 0 lineto arrowHeight neg arrowWidth 2 div neg lineto patternNone not { originalCTM setmatrix /padtip arrowHeight 2 exp 0.25 arrowWidth 2 exp mul add sqrt brushWidth mul arrowWidth div def /padtail brushWidth 2 div def tipx tipy translate angle rotate padtip 0 translate arrowHeight padtip add padtail add arrowHeight div dup scale arrowheadpath ifill } if brushNone not { originalCTM setmatrix tipx tipy translate angle rotate arrowheadpath istroke } if grestore end } dup 0 9 dict put def /arrowheadpath { newpath arrowHeight neg arrowWidth 2 div moveto 0 0 lineto arrowHeight neg arrowWidth 2 div neg lineto } def /leftarrow { 0 begin y exch get /taily exch def x exch get /tailx exch def y exch get /tipy exch def x exch get /tipx exch def brushLeftArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def /rightarrow { 0 begin y exch get /tipy exch def x exch get /tipx exch def y exch get /taily exch def x exch get /tailx exch def brushRightArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def %I Idraw 10 Grid 8 8 Begin %I b u %I cfg u %I cbg u %I f u %I p u %I t [ 0.799705 0 0 0.799705 0 0 ] concat /originalCTM matrix currentmatrix def Begin %I Line %I b 65535 2 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 0.5 -0 -0 0.5 214 433.5 ] concat %I 84 429 292 429 Line %I 2 End Begin %I Line %I b 65520 1 0 0 [12 4] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 0.5 -0 -0 0.5 214 433.5 ] concat %I 68 365 292 429 Line %I 2 End Begin %I Line %I b 65520 1 0 0 [12 4] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 0.5 -0 -0 0.5 214 433.5 ] concat %I 84 365 292 365 Line %I 2 End Begin %I Line %I b 65520 1 0 0 [12 4] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 0.5 -0 -0 0.5 214 433.5 ] concat %I 68 237 292 301 Line %I 2 End Begin %I Line %I b 65520 1 0 0 [12 4] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 0.5 -0 -0 0.5 214 433.5 ] concat %I 292 109 68 109 Line %I 2 End Begin %I Line %I b 65520 1 0 0 [12 4] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 0.5 -0 -0 0.5 214 433.5 ] concat %I 68 173 292 109 Line %I 2 End Begin %I Line %I b 65520 1 0 0 [12 4] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 0.5 -0 -0 0.5 214 433.5 ] concat %I 292 237 68 173 Line %I 2 End Begin %I Line %I b 65535 2 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 0.5 -0 -0 0.5 214 433.5 ] concat %I 68 109 292 173 Line %I 2 End Begin %I Line %I b 65535 2 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 0.544811 -0 -0 0.625 200.825 412.125 ] concat %I 84 173 292 173 Line %I 2 End Begin %I Line %I b 65535 2 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 0.5 -0 -0 0.5 214 433.5 ] concat %I 84 301 292 365 Line %I 2 End Begin %I Line %I b 65535 2 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 0.5 -0 -0 0.5 214 433.5 ] concat %I 68 429 292 365 Line %I 2 End Begin %I Pict %I b u %I cfg u %I cbg u %I f u %I p u %I t u Begin %I Elli %I b 65535 0 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 -0 -0 1 174 230 ] concat %I 74 418 8 8 Elli End Begin %I Elli %I b 65535 0 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 -0 -0 1 173 69.5 ] concat %I 74 418 8 8 Elli End Begin %I Elli %I b 65535 0 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 -0 -0 1 172.5 100.5 ] concat %I 74 418 8 8 Elli End Begin %I Elli %I b 65535 0 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 -0 -0 1 173.5 133 ] concat %I 74 418 8 8 Elli End Begin %I Elli %I b 65535 0 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 -0 -0 1 173.5 164.5 ] concat %I 74 418 8 8 Elli End Begin %I Elli %I b 65535 0 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 -0 -0 1 173 197.5 ] concat %I 74 418 8 8 Elli End End %I eop Begin %I Pict %I b u %I cfg u %I cbg u %I f u %I p u %I t [ 1 0 0 1 115 0.5 ] concat Begin %I Elli %I b 65535 0 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 -0 -0 1 174 230 ] concat %I 74 418 8 8 Elli End Begin %I Elli %I b 65535 0 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 -0 -0 1 173 69.5 ] concat %I 74 418 8 8 Elli End Begin %I Elli %I b 65535 0 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 -0 -0 1 172.5 100.5 ] concat %I 74 418 8 8 Elli End Begin %I Elli %I b 65535 0 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 -0 -0 1 173.5 133 ] concat %I 74 418 8 8 Elli End Begin %I Elli %I b 65535 0 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 -0 -0 1 173.5 164.5 ] concat %I 74 418 8 8 Elli End Begin %I Elli %I b 65535 0 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 1 SetP %I t [ 1 -0 -0 1 173 197.5 ] concat %I 74 418 8 8 Elli End End %I eop End %I eop showpage end %%EndDocument endTexFig 1025 621 a Ft(Figure)f(1:)k Fj(Graphs)d Fm(G)1375 627 y Fi(D)1418 621 y Fj(and)g Fm(G)1528 627 y Fk(S)1552 621 y Fj(.)k(Edges)14 b(rep)o(resent)j(examples)1025 666 y(with)f(non-zero)g(p)o(robabilit)o(y)h(under)h Fv(D)q Fj(.)23 b(Solid)16 b(edges)h(rep)o(resent)1025 712 y(examples)g (observed)h(in)g(some)e(\014nite)i(sample)e Fm(S)r Fj(.)29 b(Notice)18 b(that)1025 757 y(given)13 b(our)h(assumptions,)f(even)h (without)g(seeing)g(any)g(lab)q(els)g(the)1025 803 y(lea)o(rning)h (algo)o(rithm)f(can)h(deduce)i(that)e(any)h(t)o(w)o(o)e(examples)h(b)q (e-)1025 849 y(longing)k(to)h(the)h(same)e(connected)j(comp)q(onent)f (in)f Fm(G)1896 855 y Fk(S)1940 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Ft(\))558 1154 y Fk(m)590 1171 y Fm(:)270 b Ft(\(1\))-75 1295 y(F)m(or)21 b(instance,)i(if)e(the)h(graph)f Fm(G)472 1301 y Fi(D)522 1295 y Ft(has)g(only)f Fm(k)j Ft(connected)-75 1341 y(comp)q(onen)o(ts,)16 b(then)g(w)o(e)g(can)h(ac) o(hiev)o(e)f(error)h Fm(\017)e Ft(with)h(at)g(most)-75 1387 y Fm(O)q Ft(\()p Fm(k)q(=\017)p Ft(\))d(examples.)-13 1432 y(More)h(generally)m(,)e(w)o(e)i(can)f(use)h(the)g(t)o(w)o(o)f (views)g(to)g(ac)o(hiev)o(e)-75 1478 y(a)f(tradeo\013)g(b)q(et)o(w)o (een)i(the)f(n)o(um)o(b)q(er)e(of)g(lab)q(eled)h(and)g(unlab)q(eled)-75 1524 y(examples)20 b(needed.)40 b(If)21 b(w)o(e)g(consider)g(the)h (graph)e Fm(G)807 1530 y Fk(S)852 1524 y Ft(\(the)-75 1569 y(graph)g(with)g(one)g(edge)h(for)f(eac)o(h)h(observ)o(ed)g (example\),)f(w)o(e)-75 1615 y(can)f(see)h(that)f(as)g(w)o(e)g(observ)o (e)g(more)f(unlab)q(eled)h(examples,)-75 1661 y(the)14 b(n)o(um)o(b)q(er)e(of)h(connected)i(comp)q(onen)o(ts)e(will)e(drop)j (as)f(com-)-75 1706 y(p)q(onen)o(ts)i(merge)e(together,)h(un)o(til)f (\014nally)g(they)i(are)f(the)g(same)-75 1752 y(as)g(the)h(comp)q(onen) o(ts)e(of)h Fm(G)357 1758 y Fi(D)385 1752 y Ft(.)k(F)m(urthermore,)c (for)f(a)h(giv)o(en)g(set)-75 1798 y Fm(S)r Ft(,)f(if)f(w)o(e)h(no)o(w) g(select)h(a)f(random)e(subset)j(of)e Fm(m)h Ft(of)g(them)f(to)g(la-) -75 1843 y(b)q(el,)g(the)g(probabilit)o(y)d(that)j(the)g(lab)q(el)f(of) g(a)g(random)f(\()p Fm(m)t Ft(+)t(1\)st)-75 1889 y(example)h(c)o(hosen) h(from)f(the)h(remaining)e(p)q(ortion)h(of)h Fm(S)i Ft(cannot)-75 1935 y(b)q(e)h(deduced)g(b)o(y)f(the)g(algorithm)d(is)294 2007 y Fg(X)274 2096 y Fk(c)289 2100 y Ff(j)305 2096 y Fi(2)p Fk(G)353 2100 y Ff(S)386 2016 y Fm(s)405 2022 y Fk(j)423 1982 y Fg(\000)442 1998 y Fi(j)p Fk(S)r Fi(j\000)p Fk(s)526 2002 y Ff(j)476 2030 y Fk(m)541 1982 y Fg(\001)p 386 2037 174 2 v 418 2049 a(\000)452 2064 y Fi(j)p Fk(S)r Fi(j)437 2097 y Fk(m)p Fl(+1)508 2049 y Fg(\001)565 2047 y Fm(;)-75 2171 y Ft(where)16 b Fm(s)65 2177 y Fk(j)98 2171 y Ft(is)e(the)i(n)o(um)o(b)q(er)e(of)g(edges)i(in)f(comp)q(onen)o (t)f Fm(c)806 2177 y Fk(j)838 2171 y Ft(of)g Fm(S)r Ft(.)-75 2216 y(If)g Fm(m)d Fv(\034)h(j)p Fm(S)r Fv(j)p Ft(,)h(the)h(ab)q(o)o(v) o(e)g(form)o(ula)d(is)j(appro)o(ximately)236 2283 y Fg(X)216 2372 y Fk(c)231 2376 y Ff(j)247 2372 y Fi(2)p Fk(G)295 2376 y Ff(S)335 2294 y Fm(s)354 2300 y Fk(j)p 328 2313 51 2 v 328 2351 a Fv(j)p Fm(S)r Fv(j)391 2264 y Fg(\022)421 2322 y Ft(1)9 b Fv(\000)505 2294 y Fm(s)524 2300 y Fk(j)p 498 2313 V 498 2351 a Fv(j)p Fm(S)r Fv(j)554 2264 y Fg(\023)584 2272 y Fk(m)623 2322 y Fm(;)-75 2446 y Ft(in)k(analogy)g(to)h(Equation) f(1.)-13 2492 y(In)f(fact,)f(w)o(e)h(can)f(use)h(recen)o(t)h(results)g (in)e(the)h(study)g(of)f(ran-)-75 2537 y(dom)i(graph)h(pro)q(cesses)j ([9)o(])d(to)g(describ)q(e)i(quan)o(titativ)o(ely)d(ho)o(w)p -75 2574 250 2 v -23 2601 a Fd(1)-6 2617 y Fo(T)m(o)d(mak)o(e)g(this)h (more)g(plausible)i(in)e(the)f(con)o(text)h(of)f(w)o(eb)g(pages,)-75 2658 y(think)j(of)e Fc(x)89 2662 y Fd(1)117 2658 y Fo(as)h(not)f(the)h (do)q(cumen)o(t)g(itself)h(but)e(rather)h(some)g(small)-75 2700 y(set)h(of)g(attributes)h(of)f(the)g(do)q(cumen)o(t.)1025 -33 y Ft(w)o(e)e(exp)q(ect)i(the)f(comp)q(onen)o(ts)f(in)g Fm(G)1586 -27 y Fk(S)1622 -33 y Ft(to)g(con)o(v)o(erge)h(to)f(those)i (of)1025 12 y Fm(G)1058 18 y Fi(D)1099 12 y Ft(as)g(w)o(e)h(see)g(more) e(unlab)q(eled)h(examples,)f(based)i(on)f(prop-)1025 58 y(erties)f(of)g(the)g(distribution)f Fv(D)q Ft(.)18 b(F)m(or)11 b(a)g(giv)o(en)h(connected)h(com-)1025 103 y(p)q(onen)o(t)j Fm(H)i Ft(of)d Fm(G)1300 109 y Fi(D)1328 103 y Ft(,)h(let)f Fm(\013)1444 109 y Fk(H)1491 103 y Ft(b)q(e)h(the)g(v)n(alue)f(of)g(the)h(minim)n(um)1025 149 y(cut)f(of)e Fm(H)18 b Ft(\(the)d(minim)n(um)o(,)c(o)o(v)o(er)j (all)g(cuts)h(of)f Fm(H)s Ft(,)g(of)g(the)h(sum)1025 195 y(of)d(the)i(w)o(eigh)o(ts)f(on)g(the)h(edges)g(in)f(the)h(cut\).)k (In)13 b(other)h(w)o(ords,)1025 240 y Fm(\013)1052 246 y Fk(H)1096 240 y Ft(is)f(the)i(probabilit)o(y)d(that)h(a)g(random)f (example)h(will)f(cross)1025 286 y(this)h(sp)q(eci\014c)i(minim)n(um)9 b(cut.)18 b(Clearly)m(,)12 b(for)h(our)g(sample)f Fm(S)k Ft(to)1025 332 y(con)o(tain)f(a)g(spanning)g(tree)i(of)e Fm(H)s Ft(,)g(and)g(therefore)i(to)f(include)1025 377 y(all)g(of)h Fm(H)k Ft(as)c(one)h(comp)q(onen)o(t,)g(it)f(m)o(ust)g(ha) o(v)o(e)g(at)h(least)g(one)1025 423 y(edge)13 b(in)g(that)h(minim)n(um) 9 b(cut.)18 b(Th)o(us,)13 b(the)h(exp)q(ected)i(n)o(um)o(b)q(er)1025 469 y(of)c(unlab)q(eled)h(samples)e(needed)k(for)d(this)h(to)g(o)q (ccur)g(is)g(at)g(least)1025 514 y(1)p Fm(=\013)1094 520 y Fk(H)1124 514 y Ft(.)k(Of)c(course,)g(there)h(are)e(man)o(y)f (cuts)i(in)f Fm(H)j Ft(and)d(to)g(ha)o(v)o(e)1025 560 y(a)f(spanning)h(tree)h(one)g(m)o(ust)e(include)h(at)g(least)h(one)f (edge)h(from)1025 606 y(ev)o(ery)19 b(cut.)33 b(Nonetheless,)21 b(Karger)f([9)o(])e(sho)o(ws)h(that)g(this)g(is)1025 651 y(nearly)12 b(su\016cien)o(t)h(as)g(w)o(ell.)k(Sp)q(eci\014cally)m (,)11 b(Theorem)i(2.1)e(of)h([9)o(])1025 697 y(sho)o(ws)i(that)h Fm(O)q Ft(\(\(log)6 b Fm(N)f Ft(\))p Fm(=\013)1462 703 y Fk(H)1493 697 y Ft(\))15 b(unlab)q(eled)f(samples)f(are)i(su\016-) 1025 743 y(cien)o(t)f(to)h(ensure)h(that)e(a)g(spanning)g(tree)i(is)e (found)g(with)h(high)1025 788 y(probabilit)o(y)m(.)1234 773 y Fl(2)1269 788 y Ft(So,)f(if)f Fm(\013)f Ft(=)h(min)1530 794 y Fk(H)1561 788 y Fv(f)p Fm(\013)1609 794 y Fk(H)1640 788 y Fv(g)p Ft(,)h(then)h Fm(O)q Ft(\(\(log)6 b Fm(N)f Ft(\))p Fm(=\013)p Ft(\))1025 834 y(unlab)q(eled)11 b(samples)f(are)i (su\016cien)o(t)g(to)f(ensure)h(that)g(the)g(n)o(um-)1025 880 y(b)q(er)j(of)g(connected)h(comp)q(onen)o(ts)f(in)f(our)h(sample)f (is)h(equal)f(to)1025 925 y(the)f(n)o(um)o(b)q(er)f(in)g Fv(D)q Ft(,)h(minim)o(izing)c(the)14 b(n)o(um)o(b)q(er)e(of)g(lab)q (eled)h(ex-)1025 971 y(amples)f(needed.)1087 1018 y(F)m(or)19 b(instance,)j(supp)q(ose)f Fm(N)q(=)p Ft(2)e(p)q(oin)o(ts)h(in)f Fm(X)1824 1024 y Fl(1)1863 1018 y Ft(are)h(p)q(osi-)1025 1063 y(tiv)o(e)e(and)g Fm(N)q(=)p Ft(2)h(are)g(negativ)o(e,)g(and)g (similarly)c(for)k Fm(X)1908 1069 y Fl(2)1927 1063 y Ft(,)g(and)1025 1109 y(the)h(distribution)f Fv(D)i Ft(is)e(uniform)f (sub)r(ject)j(to)e(placing)g(zero)1025 1155 y(probabilit)o(y)12 b(on)j(illegal)d(examples.)19 b(In)c(this)g(case,)g(eac)o(h)g(legal) 1025 1200 y(example)f(has)j(probabilit)o(y)d Fm(p)h Ft(=)h(2)p Fm(=)n(N)1642 1185 y Fl(2)1660 1200 y Ft(.)24 b(T)m(o)16 b(reduce)i(the)e(ob-)1025 1246 y(serv)o(ed)i(graph)f(to)g(t)o(w)o(o)f (connected)j(comp)q(onen)o(ts)e(w)o(e)g(do)g(not)1025 1292 y(need)e(to)g(see)h(all)e Fm(O)q Ft(\()p Fm(N)1388 1277 y Fl(2)1407 1292 y Ft(\))h(edges,)h(ho)o(w)o(ev)o(er.)21 b(All)14 b(w)o(e)i(need)g(are)1025 1337 y(t)o(w)o(o)d(spanning)g (trees.)20 b(The)14 b(minim)n(um)9 b(cut)15 b(for)e(eac)o(h)h(comp)q (o-)1025 1383 y(nen)o(t)f(has)g(v)n(alue)f Fm(pN)q(=)p Ft(2,)h(so)g(b)o(y)g(Karger's)g(result,)h Fm(O)q Ft(\()p Fm(N)d Ft(log)c Fm(N)e Ft(\))1025 1429 y(unlab)q(eled)19 b(examples)f(su\016ce.)34 b(\(This)19 b(simple)e(case)j(can)f(b)q(e) 1025 1474 y(analyzed)13 b(easily)h(from)e(\014rst)i(principles)h(as)f (w)o(ell.\))1087 1521 y(More)k(generally)m(,)g(w)o(e)h(can)f(b)q(ound)g (the)h(n)o(um)o(b)q(er)e(of)h(con-)1025 1566 y(nected)12 b(comp)q(onen)o(ts)f(w)o(e)h(exp)q(ect)h(to)e(see)i(\(and)e(th)o(us)h (the)g(n)o(um-)1025 1612 y(b)q(er)g(of)f(lab)q(eled)h(examples)f (needed)j(to)d(pro)q(duce)i(a)f(p)q(erfect)h(h)o(y-)1025 1658 y(p)q(othesis)g(if)e(w)o(e)i(imagine)d(the)j(algorithm)c(is)j (allo)o(w)o(ed)f(to)i(select)1025 1703 y(whic)o(h)j(unlab)q(eled)h (examples)f(will)g(b)q(e)h(lab)q(eled\))g(in)g(terms)g(of)1025 1749 y(the)c(n)o(um)o(b)q(er)f(of)h(unlab)q(eled)g(examples)f Fm(m)1693 1755 y Fk(u)1728 1749 y Ft(as)h(follo)o(ws.)k(F)m(or)12 b(a)1025 1795 y(giv)o(en)18 b Fm(\013)i(<)g Ft(1,)f(consider)h(a)f (greedy)g(pro)q(cess)i(in)e(whic)o(h)f(an)o(y)1025 1840 y(cut)k(of)f(v)n(alue)g(less)i(that)e Fm(\013)h Ft(in)f Fm(G)1594 1846 y Fi(D)1644 1840 y Ft(has)h(all)f(its)h(edges)g(re-)1025 1886 y(mo)o(v)o(ed,)13 b(and)j(this)f(pro)q(cess)i(is)f(then)g(rep)q (eated)h(un)o(til)e(no)g(con-)1025 1932 y(nected)i(comp)q(onen)o(t)d (has)i(suc)o(h)h(a)e(cut.)23 b(Let)17 b Fm(N)1781 1938 y Fk(C)r(C)1835 1932 y Ft(\()p Fm(\013)p Ft(\))e(b)q(e)h(the)1025 1977 y(n)o(um)o(b)q(er)f(of)h(connected)i(comp)q(onen)o(ts)e (remaining.)23 b(If)15 b(w)o(e)i(let)1025 2023 y Fm(\013)11 b Ft(=)h Fm(c)7 b Ft(log)o(\()p Fm(N)e Ft(\))p Fm(=m)1312 2029 y Fk(u)1334 2023 y Ft(,)11 b(where)i Fm(c)e Ft(is)g(the)h(constan) o(t)g(from)d(Karger's)1025 2069 y(theorem,)18 b(and)h(if)e Fm(m)1367 2075 y Fk(u)1408 2069 y Ft(is)i(large)f(enough)g(so)h(that)g (there)h(are)1025 2114 y(no)11 b(singleton)h(comp)q(onen)o(ts)g(\(comp) q(onen)o(ts)g(ha)o(ving)f(no)h(edges\))1025 2160 y(remaining)i(after)j (the)g(ab)q(o)o(v)o(e)g(pro)q(cess,)i(then)e Fm(N)1807 2166 y Fk(C)r(C)1861 2160 y Ft(\()p Fm(\013)p Ft(\))g(is)f(an)p 1025 2201 V 1076 2228 a Fd(2)1094 2243 y Fo(This)j(theorem)f(is)h(in)g (a)g(mo)q(del)g(in)g(whic)o(h)g(eac)o(h)g(edge)g Fc(e)f Fn(in-)1025 2285 y(dep)n(enden)o(tly)10 b Fo(app)q(ears)k(in)g(the)g (observ)o(ed)g(graph)h(with)e(probabili)q(t)o(y)1025 2326 y Fc(mp)1078 2330 y Fb(e)1094 2326 y Fo(,)21 b(where)e Fc(p)1262 2330 y Fb(e)1298 2326 y Fo(is)h(the)f(w)o(eigh)o(t)h(of)f (edge)h Fc(e)f Fo(and)h Fc(m)f Fo(is)g(the)h(ex-)1025 2368 y(p)q(ected)12 b(n)o(um)o(b)q(er)h(of)f(edges)h(c)o(hosen.)k(\(Sp) q(eci\014call)q(y)m(,)e(Karger)d(is)h(con-)1025 2409 y(cerned)k(with)g(the)f(net)o(w)o(ork)h(reliabil)q(it)o(y)j(problem)e (in)f(whic)o(h)g(eac)o(h)1025 2451 y(edge)11 b(go)q(es)g(\\do)o(wn")g (indep)q(enden)o(tl)q(y)i(with)e(some)g(kno)o(wn)g(probabil-)1025 2492 y(it)o(y)j(and)g(y)o(ou)g(w)o(an)o(t)f(to)h(kno)o(w)g(the)g (probabilit)o(y)i(that)e(connectivit)o(y)1025 2534 y(is)d(main)o (tained.\))18 b(Ho)o(w)o(ev)o(er,)11 b(it)f(is)h(not)g(hard)g(to)g(con) o(v)o(ert)g(this)g(to)g(the)1025 2575 y(setting)16 b(w)o(e)e(are)h (concerned)h(with,)g(in)f(whic)o(h)h(a)f(\014xed)h Fc(m)e Fo(samples)1025 2617 y(are)d(dra)o(wn,)h(eac)o(h)f(indep)q(enden)o(tl)q (y)j(from)d(the)h(distribution)j(de\014ned)1025 2658 y(b)o(y)g(the)g Fc(p)1167 2662 y Fb(e)1184 2658 y Fo('s.)22 b(In)15 b(fact,)g(Karger)g(in)h([10])e(handles)j(this)f(con)o(v)o (ersion)1025 2700 y(formally)m(.)p eop %%Page: 5 5 5 4 bop -75 -33 a Ft(upp)q(er)15 b(b)q(ound)f(on)g(the)h(exp)q(ected)h (n)o(um)o(b)q(er)d(of)h(lab)q(eled)g(exam-)-75 12 y(ples)19 b(needed)h(to)f(co)o(v)o(er)g(all)f(of)g Fv(D)q Ft(.)32 b(On)19 b(the)h(other)f(hand,)g(if)-75 58 y(w)o(e)f(let)f Fm(\013)g Ft(=)g(1)p Fm(=)p Ft(\(2)p Fm(m)261 64 y Fk(u)283 58 y Ft(\),)h(then)431 41 y Fl(1)p 431 48 17 2 v 431 72 a(2)453 58 y Fm(N)486 64 y Fk(C)r(C)540 58 y Ft(\()p Fm(\013)p Ft(\))f(is)g(a)g(lo)o(w)o(er)g(b)q(ound)-75 103 y(since)c(the)g(ab)q(o)o(v)o(e)f(greedy)h(pro)q(cess)h(m)o(ust)d (ha)o(v)o(e)h(made)f(at)h(most)-75 149 y Fm(N)-42 155 y Fk(C)r(C)19 149 y 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