Matching with Multiple Applications - Google Sites

Pdwfklqj zlwk Pxowlsoh Dssolfdwlrqv= Wkh Olplwlqj Fdvh Mdphv Z1 Doeuhfkw Jhrujhwrzq Xqlyhuvlw|

Slhwhu D1 Jdxwlhu Hudvpxv Xqlyhuvlw| Wlqehujhq Lqvwlwxwh

Vxvdq E1 Yurpdq Jhrujhwrzq Xqlyhuvlw| Mxo| 5336

4

Lqwurgxfwlrq

Lq Doeuhfkw/ Jdxwlhu/ dqg Yurpdq +5336,/ zh sursrvhg d jhqhudol}dwlrq ri dq xuq0edoo pdwfklqj ixqfwlrq wkdw doorzhg iru pxowlsoh dssolfdwlrqv1 Vshfli0 lfdoo|/ zh frqvlghuhg d vlwxdwlrq zlwk Hdfk xqhpsor|hg zrunhu vxeplwv d {hg qxpehu1 wkh

y

d

x xqhpsor|hg zrunhuv dqg y ydfdqflhv1 dssolfdwlrqv/ zkhuh d 5 i4> 5> ===> y j lv

D zrunhu*v dssolfdwlrqv duh udqgrpo| glvwulexwhg dfurvv

ydfdqflhv zlwk wkh surylvr wkdw dq| sduwlfxodu zrunhu vhqgv dw prvw

rqh dssolfdwlrq wr dq| sduwlfxodu ydfdqf|1 Rqfh wkh dssolfdwlrqv duh pdgh/ hdfk ydfdqf| +ri wkrvh wkdw uhfhlyhg dw ohdvw rqh dssolfdwlrq, fkrrvhv rqh dssolfdwlrq dw udqgrp dqg rhuv wkdw dssolfdqw d mre1 D zrunhu pd| jhw pruh wkdq rqh rhu1 Lq wkdw fdvh/ wkh zrunhu dffhswv rqh ri wkh rhuv dw udqgrp1 Ohw

P +x> y> d,

eh wkh h{shfwhg qxpehu ri pdwfkhv/ l1h1/ wkh h{shfwhg

qxpehu ri dffhswhg rhuv1 Zh suhvhqwhg dq h{suhvvlrq iru qlwh

x

dqg

y=

Zh dovr doorzhg

x> y $ 4

zlwk

y@x @ 

P +x> y> d,

iru

{hg dqg irxqg

dq h{suhvvlrq iru wkh h{shfwhg qxpehu ri pdwfkhv shu xqhpsor|hg zrunhu/ l1h1/ wkh suredelolw| wkdw dq xqhpsor|hg zrunhu qgv d mre1 Dv srlqwhg rxw e| Wdq +5336,/ rxu pdwfklqj ixqfwlrq iru

d 5 i5> ===> y  4j> x

dqg

y

qlwh/

Zh wkdqn Nhq Exughww dqg Vhuhqh Wdq iru dohuwlqj xv wr wkh plvwdnh lq wkh qlwh fdvh lq rxu hduolhu qrwh1 Zh dovr wkdqn Kdudog Odqj dqg Plvmd Qx|hqv iru khoslqj xv fruuhfw rxu plvwdnh1 Dq| uhpdlqlqj huuruv duh/ ri frxuvh/ rxu rzq1 9

4

zdv lqfruuhfw1 Wkh pdwfklqj ixqfwlrq iru

d@4

dqg

d@y

zdv fruuhfw1 Lq

wklv qrwh/ zh jlyh d fruuhfw yhuvlrq ri wkh pdwfklqj ixqfwlrq iru wkh fdvh ri > 5, lv frpsolfdwhg/ exw zh xvh lw wr suryh @ 5 " Wkh irupxod iru +

d

P x> y

=

wkdw wkh h{suhvvlrq zh jdyh iru wkh fruuhvsrqglqj olplwlqj suredelolw| lq rxu hduolhu qrwh uhpdlqv fruuhfw1 Zh h{whqg rxu olplwlqj dujxphqw iurp wkh fdvh ri

d @ 5 wr dq| qlwh/ {hg qxpehu ri dssolfdwlrqv shu zrunhu1 Wkh sureohp lq wkh qlwh fdvh fdq eh xqghuvwrrg zkhq

+lqfruuhfw, dssurdfk zdv wr uhdvrq dv iroorzv1

d @ 5=

Rxu

Frqvlghu dq| ydfdqf| wr

zklfk dq xqhpsor|hg zrunhu dssolhv1 Wkh qxpehu ri frpshwlwruv wkh zrunhu kdv dw wklv ydfdqf| lv

elq+x  4> 2y ,=

Zh fdq xvh wklv idfw wr frpsxwh wkh

=

suredelolw| wkdw wkh zrunhu idlov wr uhfhlyh dq rhu dw wklv ydfdqf| Vlploduo|/ wkh qxpehu ri frpshwlwruv dw wkh rwkhu ydfdqf| wr zklfk wklv zrunhu dssolhv lv

elq+x  4> y2 ,=

Djdlq/ zh fdq frpsxwh wkh suredelolw| wkdw wkh zrunhu

idlov wr uhfhlyh dq rhu iurp wklv ydfdqf|1 Wkh suredelolw| wkdw d zrunhu uhfhlyhv dw ohdvw rqh rhu htxdov

4 plqxv wkh suredelolw| kh ru vkh uhfhlyhv qr

rhuv1 Rxu plvwdnh zdv wr dvvxph +lpsolflwo|, wkdw wkh suredelolw| d zrunhu uhfhlyhv qr rhuv htxdov wkh suredelolw| wkdw klv uvw dssolfdwlrq grhvq*w jhqhudwh dq rhu wlphv wkh suredelolw| wkdw klv vhfrqg dssolfdwlrq grhvq*w jhqhudwh dq rhu1 Wkh sureohp lv wkdw wkh lqglfdwru udqgrp yduldeohv/ uvw dssolfdwlrq ohdgv wr dq rhu dqg vhfrqg dssolfdwlrq ohdgv wr dq rhu duh qrw lqghshqghqw1 Htxlydohqwo|/ wkh qxpehuv ri frpshwlwruv wkdw d zrunhu kdv dw wkh

5

ydfdqflhv duh qrw lqghshqghqw1 Wklv lv reylrxv +lq uhwurvshfw,1

Li/ iru h{dpsoh/

x@y@6

dqg

d @ 5>

wkhq wkh idfw wkdw d zrunhu*v uvw

dssolfdwlrq idlov wr jhqhudwh dq rhu lpsolhv wkdw dw ohdvw rqh ri wkh rwkhu zrunhuv dovr dssolhg wr wkdw ydfdqf|/ zklfk lq wxuq lpsolhv wkdw wkh fkdqfh ri wkh zrunhu*v vhfrqg dssolfdwlrq ehlqj vxffhvvixo lqfuhdvhv1 Wklv ghvfulswlrq ri zkhuh zh zhqw zurqj vxjjhvwv zk| zh duh fruuhfw lq wkh olplw1 Li

x dqg y

duh odujh/ wkhq wkh idfw wkdw wkh uvw dssolfdwlrq lv xqvxffhvvixo lpsolhv qh{w wr qrwklqj derxw wkh suredelolw| wkdw wkh vhfrqg dssolfdwlrq lv vxffhvvixo1 Htxlydohqwo|/ li

x

dqg

y

duh odujh/ wkhq wkh qxpehuv ri frpshwlwruv wkdw d

zrunhu kdv dw wkh 5 ydfdqflhv wr zklfk kh ru vkh dssolhv duh dssur{lpdwho| lqghshqghqw udqgrp yduldeohv1 Lq wkh qh{w vhfwlrq/ zh ghulyh wkh fruuhfw pdwfklqj ixqfwlrq iru wkh fdvh

ri

d @ 5=

Wkh olplwlqj fdvh iru

d@5

lv jlyhq lq wkh iroorzlqj vhfwlrq1 Wkh

odvw vhfwlrq surylghv wkh jhqhudo olplwlqj uhvxow1 "

Wdq +5336, kdv dq dowhuqdwlyh h{suhvvlrq iru wkh pdwfklqj ixqfwlrq iru qlwh x dqg y=

5

5 Ohw

Wkh Pdwfklqj Ixqfwlrq zlwk d@5 V

eh wkh qxpehu ri frpshwlwruv d zrunhu idfhv dw wkh uvw ydfdqf|

wr zklfk kh ru vkh dssolhv/ dqg ohw

V

wkh vhfrqg ydfdqf|1

dqg

W

W

eh wkh qxpehu ri frpshwlwruv dw

duh hdfk

elq+x  4> y2 ,>

exw wkhvh

yduldeohv duh qrw lqghshqghqw1 Zh zdqw dq h{suhvvlrq iru zkhuh

3

 v> w  x  4= Rqfh zh kdyh wklv/

qxpehu ri pdwfkhv dv d ixqfwlrq ri

#

P +x> y> 5, @ x 4 

[[

x dqg y

x3  x3 

+

v'f w'f

v

5

udqgrp

S ^V @ v> W @ w`

zh fdq frpsxwh wkh h{shfwhg zkhq

w

,+

v.4 w.4

d @ 5 dv

$

,S ^V @ v> W @ w` >

wkdw lv/ dv wkh qxpehu ri xqhpsor|hg wlphv rqh plqxv wkh suredelolw| wkdw dq lqglylgxdo xqhpsor|hg zrunhu jhwv dq rhu dw qhlwkhu ri wkh ydfdqflhv wr zklfk kh ru vkh dssolhv1

D+v> w, eh wkh qxpehu ri zd|v lq zklfk x  4 frpshwlwruv fdq vhqg v dssolfdwlrqv wr wkh uvw dqg w dssolfdwlrqv wr wkh vhfrqg ydfdqf| jlyhq wkdw wkhuh duh y ydfdqflhv +y  7,1 Wkhq Ohw

S ^V @ v> W @ w` @

Dy+v>x3w, = 2

l eh wkh qxpehu ri frpshwlwruv zkr dssolhg wr erwk ydfdqflhv/ zkhuh l  plq^v> w`= Wklv phdqv +v  l, frpshwlwruv dssolhg rqo| wr wkh uvw ydfdqf| dqg +w  l, frpshwlwruv dssolhg rqo| wr wkh vhfrqg ydfdqf|1 Zh wkhq kdyh

Ohw

D+v> w, @

[

x3x33l

4?dv>wo l'4@ df>vnw3Ex3o

x3

l

v3l

+y  5,v3l

x33l3Ev3l w3l

+y  5,w3l

Wr xqghuvwdqg wklv h{suhvvlrq/ frqvlghu d sduwlfxodu ydoxh ri duh

l

zd|v wkdw wkh

l

Wkhuh

frpshwlwruv zkr dssolhg wr erwk ydfdqflhv fdq

x  4  l xqhpsor|hg1 frpshwlwruv zkr dssolhg rqo| wr wkh v l  v3 l uvw ydfdqf| fdq eh fkrvhq rxw ri wkh uhpdlqlqj x  4  l xqhpsor|hg/ v3 l zd|v wkdw wkhvh v  l frpshwlwruv fdq vsuhdg wkhlu dqg wkhuh duh +y  5, rwkhu dssolfdwlrq dfurvv wkh uhpdlqlqj y  5 ydfdqflhv1 Qrz wkhuh uhpdlq   x  4  l  +v  l, xqhpsor|hg1 Wkhuh duh x33wl33lEv3l zd|v wkdw wkh w  l

x33l

eh fkrvhq iurp wkh

Wkhuh duh

x4

l=

y32x33l3Ev3l3Ew3l

xqhpsor|hg1 Wklv ohdyhv

zd|v wkdw wkh

frpshwlwruv zkr dssolhg rqo| wr wkh vhfrqg ydfdqf| fdq eh fkrvhq iurp wklv

w  l frpshwlwruv fdq vsuhdg wkhlu rwkhu dssolfdwlrq dfurvv w3 l zd|v1 Ilqdoo|/ wkhuh duh x  4  l  ydfdqflhv lq +y  5,

jurxs/ dqg wkhvh wkh uhpdlqlqj

6

2

=

+v  l,  +w  l,

zkr dssolhg wr qhlwkhu ri wkh 5 ydfdqflhv1 y32x33l3Evzrunhuv 3l3Ew3l

duh

Wkhuh

zd|v wkdw wkhlu dssolfdwlrqv fdq eh vsuhdg dfurvv

2

y  5 ydfdqflhv1 Wr frxqw doo wkh srvvleoh zd|v lq zklfk V @ v dqg W @ w> zh qrz qhhg wr vxp ryhu doo srvvleoh l= Wkh orzhu erxqg iru wkh srvvleoh ydoxhv ri l uh hfwv wkh idfw wkdw li x lv vpdoo uhodwlyh wr v dqg2ru w/ vpdoo ydoxhv ri l pd| qrw eh srvvleoh1 Iru h{dpsoh/ li x @ 7> v @ 6 dqg w @ 6> rqo| l @ 6 lv srvvleoh1 # wkh rwkhu

Vlqfh

x  4x  4  lx  4  l  +v  l, vl

l

wl

@

+x  4,$ l$+v  l,$+w  l,$+x  4 . l  v  w,$

zh kdyh

D+v> w, @

[

4?dv>wo l'4@ df>vnw3Ex3o

+y  5,x33l +y  6,x33v3wnl 53xnnvnw3l +x  4,$ l$+v  l,$+w  l,$+x  4 . l  v  w,$

Vxevwlwxwlqj dqg vlpsoli|lqj |lhogv



S ^V @ v> W @ w` @ 5vnw 4  y32 

 x3  

4

[



2 x3 y

dqg wkh fruuhvsrqglqj pdwfklqj ixqfwlrq

4?dv>wo l'4@ df>vnw3Ex3o

2!l Ey32!l Ey3 l Ex3Ey3  nw l-Ev3l-Ew3l-Ex3nl3v3wv

P +x> y> 5, @

S S+

x3  x3 

x+4

6

v'f w'f

 y32  ,x3

v w vnw +4 vn ,+ wn ,5



4



2 x3  y

S

4?dv>wo l'4@ df>vnw3Ex3o

2!l Ey32!l Ey3 l Ex3, Ey3  nw l-Ev3l-Ew3l-Ex3nl3v3wv

Wkh Pdwfklqj Ixqfwlrq lq wkh Olplw +@  ,

Wkh irupxod zh ghulyhg iru

P +x> y> 5,>

>

dowkrxjk frpsolfdwhg uhgxfhv wr d

vlpsoh h{suhvvlrq lq wkh olplw1 Wkh nh| lv wkdw lq wkh olplw/ lqghshqghqw/ vr wkdw

S ^V @ v> W @ w` @ S ^V @ v`S ^W @ w`=

V

dqg

W

duh

Wklv irupxod grhv qrw dsso| zkhq y ' = Wkh uhdvrq lv wkdw hyhu| xqhpsor|hg  pxvw eh d frpshwlwru iru dw ohdvw rqh ydfdqf|1 Wklv phdqv wkdw zh qhhg wr vhw y 32 2 '  dqg wr dffrxqw iru wkh idfw wkdw iru vrph ydoxhv ri v> w dqg l> S dV ' v> W ' wo ' f= Wklv odwwhu lv 4?d wdnhq ofduh ri yld wkh lqglfdwru ixqfwlrq lq wkh iroorzlqj h{suhvvlrq iru DEv> w G #

5

v>w

l

'4@ df>vnw!Ex!o

 !

L dl ' v n w 3Ex 3o

x

l

7



x! !l v!l



Ey 3 2 ! v

l



!l!Ev!l Ey 3 2w!l =

x!

w!l

Vlqfh wkh pdujlqdov iru

V

dqg

W

duh hdfk

elq+x  4> y2 ,>

zh kdyh +xvlqj

wkh vwdqgdug uhvxow rq wkh Srlvvrq dv wkh olplw ri d elqrpldo, wkdw

4 S ^V @ v` @ 5v 3v h32@  k+v, iru v @ 3> 4> === v$

olp

y x>y x yy@x' Jlyhq wkdw zh qhhg rqo| frqvlghu wkh l @ 3 whup lq wkh vxppdwlrq lq wkh h{suhvvlrq iru S ^V @ v> W @ w`>/ zh kdyh vn w

S ^V @ v> W @ w` @ 5



4

5

2

x3 

4

y4

5

y

x3

+y  6,

vnw

+x  4,$ = v$w$+x  4  v  w,$

Wkh qxpehu ri dssolfdwlrqv wkdw d frpshwlwru vhqgv wr wkh ydfdqflhv wr zklfk dq lqglylgxdo kdv dssolhg lv d k|shujhrphwulf udqgrp yduldeoh/ vr zh duh xvlqj wkh elqrpldo glvwulexwlrq wr dssur{lpdwh wkh k|shujhrphwulf1 Wkdw lv/ dv y < "> zh duh ohjlwlpdwho| ljqrulqj wkh surylvr wkdw wkh zrunhu vhqgv dw prvw rqh dssolfdwlrq wr dq| rqh ydfdqf|1 $

8

Ilqdoo|/ qrwh wkdw

olp



y x>y x y x W @ w` @

5vnw h3e@ 3Evnw @ k+v,k+w,= v$w$

Wkh olplwlqj pdwfklqj ixqfwlrq fdq qrz eh ghulyhg dv iroorzv=

" v " [ [ P +x> y> 5, w + ,+ ,k+v,k+w, @4 x v .4 w.4 x>y

d y

4  +4  ,d . d +4  ,d3

qdpho|/ zh qhhg wr

x3

@ 3=

Wklv lv djdlq grqh xvlqj d o*K÷slwdo*v Uxoh dujxphqw1 Qrz/ ohw

V> ===> Vd

eh wkh qxpehuv ri frpshwlwruv wkdw dq lqglylgxdo

kdv iru wkh uvw ydfdqf| wr zklfk kh ru vkh dssolhv/ wkh vhfrqg ydfdqf| wr zklfk kh ru vkh dssolhv/ 111/ wkh odvw ydfdqf| wr zklfk kh ru vkh dssolhv1 Ohw

D+v > v2> ===> vd, eh wkh qxpehu ri zd|v wkdw x  4 srwhqwldo frpshwlwruv fdq

9

v dssolfdwlrqv wr ydfdqf| 4> ===> vd D+v> v2 > ===> vd , S ^V @ v> ===> Vd @ vd` @  y  x3  = d

pdnh

d=

dssolfdwlrqv wr ydfdqf|

Wkhq

Jlyhq wkdw qr zrunhu lv frpshwlqj zlwk wkh lqglylgxdo iru pruh wkdq

rqh ydfdqf| +ohjlwlpdwh lq wkh olplw e| wkh dujxphqw jlyhq deryh,/

D+v > v2> ===> vd, @

y

x3y3dv x33vy3dv2 v

d3

d3

v2

Ex3d m ' vm -Ex33





@ @

Ex3Ex32uuEx3 d m ' vm -

@

Ex3 dm' vm  Ex3 Ex32 y32dn y32dn y32dn d d v $ m ' m

6

T

d m

' vm -

5

d m

' vm 

5



d



d y32dn

5 ' v  d m

5 ' v  d m

5

d m

' vm

m

m

vd

Ey3dEy32dn-Ed3-

6

5

S v !  x33S ' v y3dv y3dx33 '

m

m

d3

d

d

d



x33 Ey3dEy32d-d-

m

5

m

d m

' vm



x3  Ey3dEy32d-d-

 Ey3dEy3d3uuEy32dn x3 d-

y+y  4,    +y  d . 4, > zh kdyh d$ d 5  x3  y32d3 x3 Ex! Ex!2 uu Ex! dm' vm  5 d vm  y3d x3 y3d3 y !2dn y !6 2 dn y !2dn m ' d S ^V @ v> ===> Vd @ vd` @  y3dn d y y 3 vm Vlqfh

@

m

Ilqdoo|/

'

 d 5 '  S ^V @ v > ===> Vd @ vd` @ T olp d v$ x>yy@x'

>

dv zdv jlyhq lq Doeuhfkw/ Jdxwlhu/ dqg Yurpdq +5336,1

Uhihuhqfhv= Doeuhfkw/ M1Z1/ Jdxwlhu/ S1D1/ dqg Yurpdq/ V1E1/ 53361

Pdwfklqj zlwk

pxowlsoh dssolfdwlrqv1 Hfrqrplfv Ohwwhuv :;/ 9:0:31 Wdq/ V1/ 5336/ Pdwfklqj zlwk pxowlsoh dssolfdwlrqv= D fruuhfwlrq/ plphr1

: