8QLYHUVLWtD GHJOL 6WXGL GL 5RPD 7UH A linear algorithm

ROMA
TRE
DIA
t GHJOL 6WXGL GL 5RPD 7UH
8QLYHUVLWD
'LSDUWLPHQWR GL ,QIRUPDWLFD H $XWRPD]LRQH
9LD GHOOD 9DVFD 1DYDOH ^ 5RPD ,WDO\
$ OLQHDU DOJRULWKP IRU WKH HGJH
KDPLOWRQLDQ FRPSOHWLRQ QXPEHU
RI D WUHH
$ $JQHWLV 3 'HWWL & 0HORQL ' 3DFFLDUHOOL
57',$
0DU]R 'LSDUWLPHQWR GL ,QJHJQHULD GHOO
,QIRUPD]LRQH 8QLYHUVLWtD GL 6LHQD ,WDO\
DJQHWLV#GLLXQLVLLW
'LSDUWLPHQWR GL ,QIRUPDWLFD H $XWRPD]LRQH 8QLYHUVLWtD GL 5RPD 7UH ,WDO\
GHWWL#GLDXQLURPDLW
PHORQL#GLDXQLURPDLW
SDFFLDUHOOL#GLDXQLURPDLW
$%675$&7
7KH (GJH +DPLOWRQLDQ &RPSOHWLRQ QXPEHU RI D JUDSK * LV WKH PLQLPXP QXPEHU RI
HGJHV WR EH DGGHG WR LWV OLQH JUDSK /* WR PDNH /* KDPLOWRQLDQ 7KLV SUREOHP LV
NQRZQ WR EH 13 ^FRPSOHWH IRU JHQHUDO JUDSKV ZKHUHDV D 2Q DOJRULWKP H[LVWV ZKHQ *
LV D WUHH ,Q WKLV SDSHU D OLQHDU DOJRULWKP IRU nQGLQJ WKH (GJH +DPLOWRQLDQ &RPSOHWLRQ
QXPEHU RI D WUHH LV SURSRVHG
NH\ZRUG DOJRULWKPV KDPLOWRQLDQ SDWK KDPLOWRQLDQ FRPSOHWLRQ QXPEHU WUHH OLQH
JUDSK
,QWURGXFWLRQ
7KH KDPLOWRQLDQ FRPSOHWLRQ QXPEHU RI D JUDSK *
9 ( LV WKH PLQLPXP QXPEHU
RI HGJHV ZKLFK QHHG WR EH DGGHG WR * WR PDNH LW KDPLOWRQLDQ DQG ZLOO EH GHQRWHG DV
+&1 * ,Q WKLV SDSHU ZH VD\ WKDW D JUDSK * LV KDPLOWRQLDQ LI LW KDV D KDPLOWRQLDQ
SDWK DV RSSRVHG WR WKH PRUH XVXDO GHnQLWLRQ ZKLFK UHIHUV WR KDPLOWRQLDQ F\FOHV ,Q
SDUWLFXODU ZH LQYHVWLJDWH WKH SUREOHP UHVWULFWHG WR D SDUWLFXODU FODVV RI JUDSKV FDOOHG
OLQH JUDSKV 7KH OLQH JUDSK /* RI *
9 ( LV D JUDSK KDYLQJ M(M QRGHV HDFK
QRGH RI /* EHLQJ DVVRFLDWHG WR DQ HGJH RI * 7KHUH LV DQ HGJH EHWZHHQ WZR QRGHV RI
/* LI WKH FRUUHVSRQGLQJ HGJHV RI * DUH DGMDFHQW *LYHQ D JUDSK * WKH KDPLOWRQLDQ
FRPSOHWLRQ QXPEHU RI LWV OLQH JUDSK /* LH +&1 /* LV FDOOHG HGJH KDPLOWRQLDQ
FRPSOHWLRQ QXPEHU RI * > @ 7KH SDUDPHWHU +&1 * ZDV nUVW VWXGLHG E\ *RRGPDQ
DQG +HGHWQLHPL >@ .XQGX >@ SURSRVHG D OLQHDU DOJRULWKP IRU WKH SUREOHP RI nQGLQJ
+&1 * ZKHQ * LV D WUHH ZKHUHDV 5D\FKDXGKXUL >@ SUHVHQWHG D SRO\QRPLDO DOJRULWKP
IRU nQGLQJ +&1 /* ZKHQ * LV D WUHH 7KH FRPSOH[LW\ RI WKH SURSRVHG DOJRULWKP LV
2Q ZKHUH Q M9 M
,Q WKLV SDSHU ZH XVH WKH IROORZLQJ GHnQLWLRQV *LYHQ D JUDSK * 9 ( D ZDON LV
D VHTXHQFH Z Y H Y H Y H HN YN ZKHUH Y Y Y YN DUH QRGHV RI *
H H H HN DUH HGJHV RI * DQG YLb DQG YL DUH WKH HQGSRLQWV RI HL IRU › L › N
7KH ZDON LV D WUDLO LI LWV HGJHV H H H HN DUH GLVWLQFW DQG D SDWK LI LWV QRGHV
Y Y Y YN DUH GLVWLQFW $ ZDON WUDLO SDWK LV FORVHG LI Y
YN $ SDWK PD\
FRQVLVW RI D VLQJOH QRGH
$ GRPLQDWLQJ WUDLO '7 LQ * LV D WUDLO VXFK WKDW HDFK HGJH RI * KDV DW OHDVW RQH HQGSRLQW
EHORQJLQJ WR LW ,Q RWKHU ZRUGV D GRPLQDWLQJ WUDLO FRYHUV DOO WKH HGJHV RI * 1RWH WKDW D
GRPLQDWLQJ WUDLO PD\ QRW H[LVW $ GRPLQDWLQJ WUDLO VHW h LV D FROOHFWLRQ RI HGJHGLVMRLQW
WUDLOV WKDW DOWRJHWKHU FRYHU DOO WKH HGJHV RI * $ PLQLPXP GRPLQDWLQJ WUDLO VHW LV D
GRPLQDWLQJ WUDLO VHW RI PLQLPXP FDUGLQDOLW\ 6LPLODU GHnQLWLRQV KROG IRU GRPLQDWLQJ SDWK
VHW DQG PLQLPXP GRPLQDWLQJ SDWK VHW $Q HGJH LV GRPLQDWHG E\ D WUDLO SDWK S LI LW
HLWKHU EHORQJV WR S RU LV DGMDFHQW WR S
,Q WKLV SDSHU D OLQHDU DOJRULWKP IRU WKH SUREOHP RI nQGLQJ +&1/* ZKHQ * LV
D WUHH LV SURSRVHG ,Q 6HFWLRQ WKH UHODWLRQVKLS EHWZHHQ +&1/* DQG D PLQLPXP
GRPLQDWLQJ WUDLO VHW RQ * LV DGGUHVVHG ,Q 6HFWLRQ WKH SURSRVHG VROXWLRQ DOJRULWKP LV
GHVFULEHG )LQDOO\ LQ 6HFWLRQ VRPH FRQFOXVLRQV DUH GUDZQ
7KHRUHWLFDO DVSHFWV
$ XVHIXO FKDUDFWHUL]DWLRQ RI KDPLOWRQLDQ OLQH JUDSKV ZDV JLYHQ E\ +DUDU\ DQG 1DVK
:LOOLDPV >@
7KHRUHP 7KH OLQH JUDSK /* RI D JUDSK * KDV D KDPLOWRQLDQ F\FOH LI DQG RQO\ LI
* KDV D GRPLQDWLQJ FORVHG WUDLO
$ VWUDLJKWIRUZDUG FRQVHTXHQFH RI 7KHRUHP LV WKDW /* KDV D KDPLOWRQLDQ SDWK LI
DQG RQO\ LI * KDV D GRPLQDWLQJ WUDLO 7KH QH[W WKHRUHP HVWDEOLVKHV D UHODWLRQVKLS EHWZHHQ
WKH FDUGLQDOLW\ RI D PLQLPXP GRPLQDWLQJ WUDLO VHW RI * DQG +&1/*
7KHRUHP *LYHQ DQ\ FRQQHFWHG JUDSK *
9 ( WKH FDUGLQDOLW\ RI D PLQLPXP
GRPLQDWLQJ WUDLO VHW h RQ * LV N LI DQG RQO\ LI +&1 /* N b 3URRI LI /HW h IW W WN J EH D GRPLQDWLQJ WUDLO VHW RQ WKH JUDSK * RI FDUGLQDOLW\
N 6LQFH HDFK HGJH RI ( LV GRPLQDWHG E\ VRPH WUDLO LW LV SRVVLEOH WR SDUWLWLRQ ( LQWR N VHWV
I( ( ( N J VXFK WKDW ( L FRQWDLQV DOO WKH HGJHV RI WL DQG DOO WKH RWKHU HGJHV LQ ( L
DUH GRPLQDWHG E\ WL %HFDXVH RI 7KHRUHP LQ /* WKHUH LV D VLPSOH SDWK FRQQHFWLQJ
DOO WKH QRGHV FRUUHVSRQGLQJ WR WKH HGJHV RI ( L LQ * +HQFH MRLQLQJ WKHVH N SDWKV RQ /*
E\ PHDQV RI N b HGJHV ZH JHW D KDPLOWRQLDQ SDWK LQ /*
RQO\ LI 6LQFH +&1/* Nb /* FDQ EH SDUWLWLRQHG LQWR N FRPSRQHQWV I+ +N J
HDFK KDYLQJ D KDPLOWRQLDQ SDWK %HFDXVH RI 7KHRUHP WKH QRGHV RI +L FRUUHVSRQG WR
HGJHV RI * ZKLFK DUH GRPLQDWHG E\ D WUDLO WL +HQFH h IW W WN J LV D GRPLQDWLQJ
WUDLO VHW
7KH SUREOHP RI nQGLQJ D KDPLOWRQLDQ SDWK RQ D OLQH JUDSK LV ZHOO NQRZQ WR EH 13^
FRPSOHWH >@ DQG VR LV WKHUHIRUH WKH SUREOHP RI nQGLQJ D GRPLQDWLQJ WUDLO RQ D JHQHUDO
JUDSK 3DUWLFXODU VSHFLDO FRQGLWLRQV RQ * KDYH EHHQ IRXQG WKDW HQVXUH WKH H[LVWHQFH RI
D GRPLQDWLQJ WUDLO >@ $JQHWLV HW DO >@ VKRZHG WKH 1 3 ^FRPSOHWHQHVV RI WKH SUREOHP
HYHQ ZKHQ * LV ELSDUWLWH DQG SURSRVHG IRU WKLV FDVH D KHXULVWLF DSSURDFK WR WKH SUREOHP
RI nQGLQJ WKH PLQLPXP GRPLQDWLQJ WUDLO VHW
$ OLQHDU DOJRULWKP WR nQG +&1 /7 ZKHQ 7 LV D
WUHH
,Q WKLV VHFWLRQ ZH SUHVHQW D OLQHDU DOJRULWKP IRU FRPSXWLQJ WKH HGJH KDPLOWRQLDQ FRPSOH
WLRQ QXPEHU RI D WUHH 7 9 ( RU +&1/7 ,Q YLHZ RI 7KHRUHP RXU DOJRULWKP
nQGV WKH PLQLPXP GRPLQDWLQJ WUDLO VHW ,Q SDUWLFXODU VLQFH 7 LV D WUHH LW \LHOGV WKH PLQL
PXP GRPLQDWLQJ SDWK VHW +HQFH LI e IS S SN J LV D PLQLPXP GRPLQDWLQJ SDWK
VHW RQ 7 WKHQ +&1 /7 N b ,Q RUGHU WR SURYH WKH FRUUHFWQHVV RI WKH SURSRVHG
DOJRULWKP ZH LQWURGXFH VRPH SUHOLPLQDU\ REVHUYDWLRQV
*LYHQ D WUHH 7 9 ( FRQVLGHU WZR PDUNLQJ IXQFWLRQV x 9 I J DQG y ( I J $ QRGH L DQ HGJH H VXFK WKDW xL yH LV FDOOHG PDUNHG
!
'HILQLWLRQ *LYHQ D WULSOH 7 x y D FRQVWUDLQHG GRPLQDWLQJ SDWK VHW hF LV D FRO
OHFWLRQ RI GLVMRLQW SDWKV IS S SN J VXFK WKDW LhF GRPLQDWHV DOO WKH HGJHV RI 7 ZKLFK
DUH QRW PDUNHG LLIRU HDFK PDUNHG QRGH L D SDWK S hF H[LVWV FRQWDLQLJ L
'HILQLWLRQ $ PLQLPXP FRQVWUDLQHG GRPLQDWLQJ SDWK VHW LV D FRQVWUDLQHG GRPLQDW
LQJ SDWK VHW RI PLQLPXP FDUGLQDOLW\
7KH FDUGLQDOLW\ RI D PLQLPXP FRQVWUDLQHG GRPLQDWLQJ SDWK VHW ZLOO EH GHQRWHG DV
67 x y &OHDUO\ +&1 /7 67 x y b LQ ZKLFK xL L 9 DQG yH
H (
'HILQLWLRQ *LYHQ D WULSOH 7 x y ZH FDOO FRQWUDFWLRQ RI D OHDI L RI 7 WKH WUDQVIRU
PDWLRQ IURP 7 x y WR WKH WULSOH 7 x y GHnQHG DV IROORZV
7 9 ( 9 Q L ( Q L M
x N xN N 9 Q M x M y H yH H ( Q L M ZKHUH M LV WKH DGMDFHQW QRGH WR L LQ 7 3URSRVLWLRQ *LYHQ D WULSOH 7 x y LI L LV D OHDI QRW PDUNHG LH xL DQG
L M LV WKH HGJH LQFLGHQW L ZLWK yL M WKHQ 67 x y 67 x y ZKHUH 7 x y LV WKH WULSOH REWDLQHG E\ WKH FRQWUDFWLRQ RI L
!
3URRI /HW e IS S SN J EH D PLQLPXP FRQVWUDLQHG GRPLQDWLQJ SDWK VHW 67 x y
N DQG OHW SK e EH WKH SDWK GRPLQDWLQJ WKH LQFLGHQW HGJH L M RI WKH OHDI L /HW h
EH REWDLQHG IURP he E\ HOLPLQDWLQJ WKH OHDI L DQG WKH HGJH L M RI 7 IURP WKH SDWK SK ,I L M SK WKHQ LQ h WKHUH H[VLVWV D SDWK SK
SK Q ILJ VXFK WKDW M SK DQG KHQFH
GRPLQDWLQJ L M +HQFH h LV D FRQVWUDLQHG GRPLQDWLQJ SDWK VHW IRU 7 x y LQ ZKLFK
7
7 Q L x M
ZKLOH x N
xN N M DQG y H
yH H ( 6LQFH
Mh M Mhe M N WKHQ 67 x y 67 x y ,I L M SK WKHQ M SK LQ e 7KHQ
s e DQG WKH 3URSRVLWLRQ REYLRXVO\ KROGV
!
!
! !
%HFDXVH RI 3URSRVLWLRQ LI L LV D OHDI RI 7 LQ RUGHU WR nQG D PLQLPXP FRQVWUDLQHG
GRPLQDWLQJ SDWK VHW RQ 7 x y LW LV SRVVLEOH WR FRQWUDFW L DQG nQG D PLQLPXP FRQ
VWUDLQHG GRPLQDWLQJ SDWK VHW RQ 7 x y 'HILQLWLRQ *LYHQ D WULSOH 7 x y ZH FDOO GHOHWLRQ RI D OHDI L RI 7 WKH WUDQVIRUPD
WLRQ IURP 7 x y WR WKH WULSOH 7 x y GHnQHG DV IROORZV
7 9 ( 9 Q L ( Q L M
x 9 I J x N xN N 9 y ( I J y H yH H ( 3URSRVLWLRQ *LYHQ D WULSOH 7 x y OHW L EH D OHDI RI 7 VXFK WKDW xL L M LV
LWV LQFLGHQW HGJH DQG yL M 7KHQ 67 x y 67 x y ZKHUH 7 x y LV WKH
WULSOH REWDLQHG E\ WKH GHOHWLRQ RI L
3URRI 6LQFH HGJH L M LV PDUNHG LW GRHV QRW QHHG WR EH GRPLQDWHG DQG WKHUHIRUH DQ\
FRQVWUDLQHG GRPLQDWLQJ SDWK VHW IRU 7 x y LV D FRQVWUDLQHG GRPLQDWLQJ SDWK VHW DOVR
IRU 7 x y 'HILQLWLRQ *LYHQ D WUHH 7
9 ( FRQVLGHU D SDWK 3
93 (3 DQG OHW 93
IY 93 GHJUHHY › J ,I WKH WUHH 7 9 Q 93 ( Q (3 LV FRQQHFWHG SRVVLEO\ 7 3 LV FDOOHG &SDWK
1RWH WKDW D &SDWK PD\ FRQWDLQ DW PRVW RQH QRGH Q& RI GHJUHH JUHDWHU WKDQ WZR
%HFDXVH Q& FRQQHFWV WKH &SDWK ZLWK WKH UHVW RI WKH WUHH LH WKH WUHH 7 LQ WKH IROORZLQJ
ZH UHIHU WR Q& LI LW H[LVWV DV WKH FRQQHFWLQJ QRGH RI WKH &SDWK
3URSRVLWLRQ ,Q DQ\ WUHH WKHUH DOZD\V H[LVWV DW OHDVW RQH &SDWK
3URRI 1RWH WKDW LI QR QRGH KDYLQJ GHJUHH JUHDWHU WKDQ H[LVWV LQ 7 WKHQ 7 LV D SDWK
DQG KHQFH D &SDWK $SSO\ WKH IROORZLQJ WUDQVIRUPDWLRQ WR 7 L PDUN WKH QRGHV RI 7
KDYLQJ GHJUHH JUHDWHU WKDQ LL IRU HDFK OHDI L FRQVLGHU WKH SDWK SL IURP L WR WKH QHDUHVW
PDUNHG QRGH /HW 3
ISL L LV D OHDIJ DQG OHW 7 EH WKH WUHH REWDLQHG E\ UHPRYLQJ
IURP 7 DOO WKH SDWKV RI 3 %\ FRQVWUXFWLRQ D OHDI T RI 7 LV D QRGH RI 7 ZLWK DW OHDVW WZR
SDWKV RI 3 HQGLQJ LQ T $ SDLU RI WKHVH SDWKV LV D &SDWK RI 7 7KHRUHP &RQVLGHU D WULSOH 7 x y ZKHUH 7 9 ( LV D WUHH DQG xO IRU HDFK
OHDI O 7 /HW O DQG O EH WZR OHDYHV RI 7 VXFK WKDW D &SDWK &3O O H[LVWV EHWZHHQ WKHP
7KHQ WKHUH H[LVWV D PLQLPXP FRQVWUDLQHG GRPLQDWLQJ SDWK VHW RQ 7 x y FRQWDLQLQJ
&3O O 3URRI /HW h IS SN J EH D FRQVWUDLQHG GRPLQDWLQJ SDWK VHW RQ 7 x y RI FDU
GLQDOLW\ N /HW Q& EH WKH FRQQHFWLQJ QRGH RI &3O O ,I D SDWK SL h H[LVWV VXFK WKDW
SL ? &3O O DQG SL ? 7 Q &3O O ZH FDOO LW LQWHUVHFWLQJ SDWK &OHDUO\ WKHUH DUH
QR PRUH WKDQ WZR GLVWLQFW LQWHUVHFWLQJ SDWKV LQ h VLQFH HDFK LQWHUVHFWLQJ SDWK PXVW SDVV
WKURXJK DQ HGJH RI &3O O LQFLGHQW Q& +HQFH RQO\ WKUHH FDVHV DUH SRVVLEOH L WKHUH
DUH QR LQWHUVHFWLQJ SDWKV LQ h LL WKHUH LV H[DFWO\ RQH LQWHUVHFWLQJ SDWK LQ h LLL WKHUH
DUH H[DFWO\ WZR LQWHUVHFWLQJ SDWKV LQ h ,Q HDFK FDVH ZH ZLOO VKRZ WKDW D FRQVWUDLQHG
GRPLQDWLQJ SDWK VHW h H[LVWV FRQWDLQLQJ &3O O DQG VXFK WKDW Mh M › MhM
,Q WKH FDVH L VLQFH xO
IRU HDFK OHDI O ZH VLPSO\ REVHUYH WKDW LQ h WKHUH PXVW
EH DW OHDVW RQH SDWK FRQWDLQHG LQ &3O O 5HPRYLQJ IURP h DOO SDWKV FRQWDLQHG LQ
&3O O DQG DGGLQJ &3O O ZH REWDLQ D QHZ FRQVWUDLQHG GRPLQDWLQJ SDWK VHW h ZLWK
FDUGLQDOLW\ QRW JUHDWHU WKDQ WKH FDUGLQDOLW\ RI h ,Q WKH FDVH LL WKH LQWHUVHFWLQJ SDWK
SK ILS Q& LT J ZKHUH LT &3O O DQG LS &3O O FDQQRW FRQWDLQ ERWK O DQG
O /HW O EH WKH OHDI QRW FRQWDLQHG LQ SK 6LQFH HJ xO LQ h WKHUH H[LVWV DW
OHDVW RQH SDWK VWULFWO\ FRQWDLQHG LQ &3O O 5HPRYLQJ DOO WKH SDWKV VWULFWO\ FRQWDLQHG LQ
&3O O DQG SDWK SK IURP h DQG DGGLQJ &3O O DQG WKH SDWK ILS Q& J WR h ZH REWDLQ
D QHZ FRQVWUDLQHG GRPLQDWLQJ SDWK VHW ZLWK FDUGLQDOLW\ QRW JUHDWHU WKDQ N ,Q WKH FDVH
LLL OHW SK
ILS Q& LT J DQG SM
ILU Q& LW J ZKHUH LT LW &3O O DQG
LU LS &3O O EH WKH WZR LQWHUVHFWLQJ SDWKV 5HPRYLQJ SK SM DQG DOO SDWKV LI DQ\ FRQ
WDLQHG LQ &3O O IURP h DQG DGGLQJ &3O O DQG WKH SDWK ILS Q& LU J ZH REWDLQ
D QHZ FRQVWUDLQHG GRPLQDWLQJ SDWK VHW h ZLWK FDUGLQDOLW\ QRW JUHDWHU WKDQ N +HQFH
JLYHQ DQ\ h ZH FDQ DOZD\V nQG D h VXFK WKDW Mh M › MhM DQG h FRQWDLQV &3O O 7KLV LV
WUXH DOVR ZKHQ h LV D PLQLPXP FDUGLQDOLW\ GRPLQDWLQJ SDWK VHW DQG WKH WKHVLV IROORZV 'HILQLWLRQ *LYHQ D WULSOH 7 x y ZH FDOO UHPRYDO RI D &SDWK 3
93 (3 RI
7 WKH WUDVIRUPDWLRQ IURP 7 x y WR WKH WULSOH 7 x y GHnQHG DV IROORZV
7 9 ( 9 QI93 QQ3 J ( Q(3 ZKHUH Q3 LV WKH RQO\ QRGH RI 3 ZLWK GHJUHHQ3 !
x 9 I J VXFK WKDW x N xN N 3 DQG x Q3 y ( I J VXFK WKDW y H yH H ( DQG y Q3 M IRU HDFK QRGH M 9 DGMDFHQW WR Q3 &RUROODU\ *LYHQ D WULSOH 7 x y ZKHUH 7 9 ( LV D WUHH DQG xO IRU HDFK
OHDI O 7 OHW 3
93 (3 EH D &SDWK RI 7 DQG OHW 7 x y EH WKH WULSOH REWDLQHG
DIWHU WKH UHPRYDO RI 3 7KHQ 67 x y 67 x y 3URRI )URP 7KHRUHP LW IROORZV WKDW D PLQLPXP FDUGLQDOLW\ FRQVWUDLQHG GRPLQDWLQJ
SDWK VHW h FRQWDLQLQJ 3 H[LVWV LQ 7 &OHDUO\ 3 SDVVHV WKURXJK Q3 DQG WKH SDWKV GRP
LQDWLQJ 7 GR QRW QHHG WR SDVV WKURXJK Q3 0RUHRYHU 3 GRPLQDWHV DOVR DOO WKH HGJHV RI
7 LQFLGHQW QRGH Q3 +HQFH h Q 3 LV D PLQLPXP FDUGLQDOLW\ FRQVWUDLQHG GRPLQDWLQJ SDWK
VHW IRU 7 x y 7KHRUHP DQG &RUROODU\ VXJJHVW DQ LWHUDWLYH DOJRULWKP IRU nQGLQJ D PLQLPXP
FRQVWUDLQHG GRPLQDWLQJ SDWK VHW RQ D JHQHUDO WULSOH 7 x y ,Q )LJXUH WKH SURSRVHG
DOJRULWKP UHIHUUHG WR DV '2075(( LV UHSRUWHG
7KHRUHP '2075(( nQGV D PLQLPXP FDUGLQDOLW\ FRQVWUDLQHG GRPLQDWLQJ SDWK
VHW RI D WUHH LQ 2Q ZKHUH Q M9 M
3URRI 7KH 67(3 RI WKH DOJRULWKP LV D SUHSURFHVVLQJ SKDVH LQ ZKLFK DFFRUGLQJ WR
3URSRVLWLRQ '2075(( DSSOLHV FRQWUDFWLRQ WR WKH OHDYHV RI 7 ,Q 67(3 D &SDWK
LV IRXQG DQG UHPRYHG IURP WKH FXUUHQW WULSOH DFFRUGLQJ WR 7KHRUHP DQG &RUROODU\
7KH H[LVWHQFH RI D &SDWK LV HQVXUHG E\ 3URSRVLWLRQ 67(3 XSGDWHV WKH
WULSOH &RQVLGHU WKH FRQQHFWLQJ QRGH Q& RI WKH &SDWK MXVW UHPRYHG 1RWH WKDW Q& LV
QRW PDUNHG 6WDUWLQJ IURP Q& WKH RSHUDWLRQV RI FRQWUDFWLRQ DQG GHOHWLRQ GHVFULEHG LQ
3URSRVLWLRQV DQG DUH DSSOLHG WR WKH FXUUHQW WULSOH 6LQFH 67(3 LV H[HFXWHG RQO\
ZKHQ M9 M ! LW UHWXUQV D WULSOH 7 x y LQ ZKLFK HLWKHU DOO WKH OHDYHV RI 7 DUH PDUNHG
RU M7 M
DQG WKH RQO\ QRGH LV PDUNHG 7KH DOJRULWKP HQGV HLWKHU ZKHQ LQ 67(3
WKH FXUUHQW QRGH VHW LV HPSW\ RU ZKHQ DW WKH HQG RI 67(3 7 FRQWDLQV RQO\ RQH
PDUNHG QRGH ,Q WKLV FDVH WKH UHPDLQLQJ PDUNHG QRGH LV WKH ODVW GHJHQHUDWH SDWK RI
WKH GRPLQDWLQJ SDWK VHW 7KH FRUUHFWQHVV RI WKH DOJRULWKP WKHQ IROORZV IURP 3URSRVLWLRQV
7KHRUHP DQG &RUROODU\ /HW XV QRZ WXUQ WR FRPSOH[LW\ 67(3 LV
2Q DQG LV H[HFXWHG RQO\ RQFH LQ WKH DOJRULWKP $OVR HDFK GHOHWLRQ DQG FRQWUDFWLRQ WDNHV
FRQVWDQW WLPH DQG HDFK QRGH FRPHV XS DW PRVW RQFH IRU GHOHWLRQFRQWUDFWLRQ RSHUDWLRQV
+HQFH DOO WKH RFFXUUHQFHV RI 67(3 UHTXLUH DQ RYHUDOO WLPH RI 2Q )RU ZKDW FRQFHUQV
nQGLQJ WKH &SDWKV LW LV HDV\ WR VHH WKDW LW LV SRVVLEOH WR LPSOHPHQW WKH VHDUFK LQ VXFK D
ZD\ WKDW HDFK HGJH RI WKH WUHH LV QHYHU YLVLWHG PRUH WKDQ RQFH WKURXJKRXW WKH DOJRULWKP
+HQFH WKH ZKROH DOJRULWKP UXQV LQ 2Q
,Q WKH DSSHQGL[ D SURFHGXUH IRU nQGLQJ &SDWKV VXLWDEOH IRU D OLQHDU LPSOHPHQWDWLRQ
RI WKH '2075(( DOJRULWKP LV GHVFULEHG
)LJXUH E VKRZV WKH RXWSXW RI WKH SUHSURFHVVLQJ ZKHQ DSSOLHG WR WKH WUHH 7 RI )LJXUH
D )LJXUH VKRZV WKH DSSOLFDWLRQ RI VWHSV DQG WR WKH WUHH RI )LJXUH E ,Q )LJXUH
D WKH &SDWK 3
I J Q3
LV VKRZQ DQG LQ )LJXUH E WKH WUHH DIWHU
WKH UHPRYDO RI 3 LV UHSRUWHG )LJXUHV F DQG G LOOXVWUDWH WKH RSHUDWLRQV GHOHWLRQ DQG
FRQWUDFWLRQ RQ QRGHV DQG UHVSHFWLYHO\ ,Q )LJXUH H WKURXJK O WKH QH[W LWHUDWLRQV
RI WKH DOJRULWKP DUH VKRZQ GDVKHG OLQHV LQGLFDWH &SDWKV ,Q )LJXUH D WKH PLQLPXP
FDUGLQDOLW\ GRPLQDWLQJ SDWK VHW LV UHSRUWHG 1RWH WKDW WKH VLQJOH QRGH LV DOVR D &SDWK
7KH HGJHV WR EH DGGHG WR /7 WR PDNH LW KDPLOWRQLDQ DUH IRU H[DPSOH DQG +&1 /7 7KH UHVXOWLQJ KDPLOWRQLDQ SDWK RQ /7 LV VKRZQ LQ )LJXUH
E
&RQFOXVLRQV
,Q WKLV SDSHU ZH LOOXVWUDWHG D OLQHDUWLPH DOJRULWKP IRU WKH HGJH KDPLOWRQLDQ FRPSOHWLRQ
QXPEHU RI D WUHH 'LmHUHQW GLUHFWLRQV FDQ EH IRUHVHHQ IRU IXUWKHU UHVHDUFK
:HLJKWHG FDVH $QDO\]H WKH SUREOHP LQ ZKLFK HDFK HGJH DGGHG WR /7 KDV D FHUWDLQ
ZHLJKW DQG ZH ZLVK WR PLQLPL]H WKH WRWDO ZHLJKW RI WKH HGJHV ZKLFK PXVW EH DGGHG 7KH
SUREOHP LV PRUH GLpFXOW VLQFH 7KHRUHP LV QR ORQJHU YDOLG
*HQHUDO JUDSKV 7KH SUREOHP RQ JHQHUDO JUDSKV LV NQRZQ WR EH 13KDUG DQG QR HmHFWLYH
$OJRULWKP '2075((
,QSXW $ WULSOH 7 x y VXFK WKDW xL L 9 yH H (
2XWSXW $ PLQLPXP FDUGLQDOLW\ FRQVWUDLQHG SDWK VHW h RI FDUGLQDOLW\ N
%(*,1
67(3 SUHSURFHVVLQJ
h QXP SDWK ZKLOH D OHDI L H[LVWV LQ 7 VXFK WKDW xL GR
7 x y FRQWUDFWLRQ RI L IURP 7 x y
3URSRVLWLRQ 67(3 UHPRYDO RI D &SDWK IURP WKH FXUUHQW WULSOH
)LQG D &SDWK 3 93 (3 RQ 7 QXP SDWK QXP SDWK h h > I3 J
7 x y UHPRYDO RI 3 IURP 7 x y
LI M9 M WKHQ
6723
67(3 XSGDWLQJ RI WKH FXUUHQW WULSOH
L DGMDFHQWQ3 LI Q3 7 LV D OHDI WKHQ
I 7 x y GHOHWLRQ RI Q3 IURP 7 x y
3URSRVLWLRQ LI xL WKHQ
JRWR 67(3 ZKLOH L 7 LV D OHDI ZKLFK LV QRW PDUNHG GR
I LI L DGMDFHQWL LV PDUNHG WKHQ
7 x y GHOHWLRQ RI L IURP 7 x y
3URSRVLWLRQ HOVH
7 x y FRQWUDFWLRQ RI L IURP 7 x y
3URSRVLWLRQ L DGMDFHQWLJ
J
HOVH
PDUN DOO HGJHV LQFLGHQW Q3 &RUROODU\ 67(3 VWRSSLQJ FULWHULRQ
LI M9 M ! WKHQ
JRWR 67(3 LI M9 M WKHQ
IQXP SDWK QXP SDWK h h>9J
6723
(1'
)LJXUH $OJRULWKP '2075((
4
11
5
3
1
12
10
6
a)
5
15
12
3
16
10
15
14
9
7
2
marked nodes
13
2
17
8
9
7
b)
18
8
14
17
)LJXUH D $ WUHH 7 E 7KH WUHH 7 DIWHU WKH SUHSURFHVVLQJ SKDVH
marked nodes
3
5
10
12
7
14
9
2
17
8
a)
10
10
10
15
10
d)
12
10
12
15
9
8
14
14
9
17
8
e)
9
10
10
9
8
f)
17
8
c)
12
14
9
17
8
17
8
12
14
9
2
14
9
2
12
10
15
15
b)
12
12
7
10
marked edges
C-path
15
8
g)
h)
9
i)
l)
)LJXUH $ JHQHULF VWHS RI WKH $OJRULWKP D $ &SDWK & E 7KH WUHH DIWHU WKH
WUDVIRUPDWLRQ UHSRUWHG LQ &RUROODU\ F $ GHOHWLRQ G $ FRQWUDFWLRQ H^O 1H[W
LWHUDWLRQV RI WKH DOJRULWKP
4
11
5
3
1
12
10
6
2
7
,
D-paths
13
15
16
14
9
17
8
18
)LJXUH D $ PLQLPXP FDUGLQDOLW\ GRPLQDWLQJ SDWK VHW IRU 7 E 7KH FRUUHVSRQGLQJ
KDPLOWRQLDQ SDWK RQ /7 DOJRULWKPLF DSSURDFK LV NQRZQ IRU LWV VROXWLRQ VR IDU $ WRSLF IRU IXWXUH UHVHDUFK ZLOO EH
WR SURYLGH HmHFWLYH LPSOLFLW HQXPHUDWLRQ DOJRULWKPV IRU VXFK JHQHUDO FDVH
5HIHUHQFHV
>@ $JQHWLV $ 'HWWL 3 0HORQL & 3DFFLDUHOOL ' 6HWXS FRRUGLQDWLRQ EHWZHHQ
WZR VWDJHV RI D VXSSO\ FKDLQ 7HFK 5HS Q ^ 'LS GL ,QIRUPDWLFD H 6LVWHPLVWLFD
8QLYHUVLWtD /D 6DSLHQ]D 5RPD
>@ %HUWRVVL $$ 7KH HGJH KDPLOWRQLDQ SUREOHP LV 13FRPSOHWH ,QIRUPDWLRQ
3URFHVVLQJ /HWWHUV ^
>@ *DUH\ 0* -RKQVRQ '6 7DUMDQ 5( 7KH SODQDU +DPLOWRQLDQ FLUFXLW SURE
OHP LV 13FRPSOHWH 6,$0 -RXUQDO RQ &RPSXWLQJ ^
>@ *RRGPDQ 6 +HGHWQLHPL 6 2Q WKH KDPLOWRQLDQ FRPSOHWLRQ SUREOHPV ,Q $
'ROG DQG % (FNPDQ HGV *UDSKV DQG &RPELQDWRULFV /HFWXUH QRWHV LQ PDWHPDWKLFV
^ 6SULQJHU %HUOLQ
>@ +DUDU\ ) *UDSK 7KHRU\ $GGLVRQ:HVOH\ 5HDGLQJ 0$
>@ +DUDU\ ) 1DVK:LOOLDPV &6W-$ 2Q (XOHULDQ DQG +DPLOWRQLDQ JUDSKV DQG
OLQHJUDSKV &DQDGLDQ 0DWKHPDWLFV %XOOHWLQ ^
>@ .DUS 50 5HGXFLELOLW\ DPRQJ FRPELQDWRULDO SUREOHPV ,Q 5 ( 0LOOHU DQG
-: 7KDWFKHU HGV &RPSOH[LW\ RI &RPSXWHU FRPSXWDWLRQ ^3OHQXP 3UHVV 1<
>@ .XQGX 6 $ OLQHDU DOJRULWKP IRU WKH +DPLOWRQLDQ FRPSOHWLRQ QXPEHU RI D WUHH
,QIRUPDWLRQ 3URFHVVLQJ /HWWHUV ^
>@ 5D\FKDXGKXUL $ 7KH WRWDO LQWHUYDO QXPEHU RI D WUHH DQG WKH +DPLOWRQLDQ
FRPSOHWLRQ QXPEHU RI LWV OLQH JUDSK ,QIRUPDWLRQ 3URFHVVLQJ /HWWHUV ^
>@ 9HOGPDQ +- $ 5HVXOW RQ +DPLOWRQLDQ /LQH *UDSKV ,QYROYLQJ 5HVWULFWLRQV
RQ ,QGXFHG 6XEJUDSKV -RXUQDO RI *UDSK 7KHRU\ ^
$
$SSHQGL[
,W LV SRVVLEOH HPSOR\ WKH IROORZLQJ GDWD VWUXFWXUH LQ D OLQHDU WLPH LPSOHPHQWDWLRQ RI WKH
'2075(( DOJRULWKP
:H DVVRFLDWH WR HDFK QRGH N RI WKH WUHH WKH IROORZLQJ LQIRUPDWLRQ
&OHDI FRQWDLQV WKH ODEHO RI D OHDI O RI WKH FXUUHQW WULSOH 7 x y VXFK WKDW N O
DUH FRQWDLQHG LQ WKH VDPH &SDWK
GHJUHH FRQWDLQV WKH GHJUHH RI N
YLVLW PDUNV WKH QRGH DV YLVLWHG
OLVW FRQWDLQV D QRGH VHTXHQFH
SURFHGXUH &3$7+ ),1'(5
,QSXW 7UHH 7 9 ( ZLWK 9 1 > / ZKHUH 1 DUH QRGHV ZLWK GHJUHH ! DQG / DUH OHDYHV
2XWSXW & b 3 DWK 9& (& RI 7
,QL]LDOL]DWLRQ 7 9 ( VWRS ‚OL L QM M & b 3 DWK OLVWOL L
EHJLQ
ZKLOH VWRS $1' D OHDI OL 7 VXFK WKDW ‚OL GR
I‚OL QFXUU OL ZKLOH GHJUHHQFXUU GR
IQFXUU L
DGG WR OLVWOL QFXUU QFXUU WKH QH[W QRGH DGMDFHQW WR QFXUU J
LI GHJUHHQFXUU IQFXUU L
& b 3 DWK
VWRS
J
WKHQ
OLVWOL HOVH
LI QFXUU IQFXUU WKHQ
L
DGG WR OLVWOL QFXUU J
HOVH
J
& b 3 DWK
/,1.83 OLVWOL OLVWOQ.:77 HQG
)LJXUH 3URFHGXUH &3$7+ ),1'(5
,Q VWHS D &SDWK LV IRXQG E\ WKH SURFHGXUH &3$7+ ),1'(5 UHSRUWHG LQ )LJXUH
XVLQJ WKH SUHYLRXV GDWD VWUXFWXUH 7KH LQSXW IRU &3$7+ ),1'(5 LV D WULSOH 7 x y
DQG D VWDUWLQJ QRGH 7KH LGHD LV WR FRQVWUXFW D SDWK VWDUWLQJ RQFH IURP HDFK OHDI OL RI
WKH FXUUHQW WUHH 7 DQG HQGLQJ LQ WKH nUVW HQFRXQWHUHG QRGH QT ZLWK GHJUHH GLmHUHQW
IURP 'XULQJ WKH FRQVWUXFWLRQ SURFHVV RI D SDWK WKH OHDI OL LV PDUNHG DV YLVLWHG HDFK
HQFRXQWHUHG QRGH N LV ODEHOHG ZLWK WKH QXPEHU L RI WKH OHDI DQG DGGHG WR WKH OLVW RI WKH
OHDI 7KH SURFHGXUH HQGV ZKHQ D QRGH QT DOUHDG\ ODEHOHG LV HQFRXQWHUHG 7KLV IDFW LQ
YLHZ RI WKH SURRI RI 3URSRVLWLRQ PHDQV WKDW WZR SDWKV VWDUWLQJ IURP GLmHUHQW OHDYHV
RI 7 KDYH D FRPPRQ HQGSRLQW LQ QT +HQFH D &SDWK PD\ EH FRQVWUXFWHG OLQNLQJ XS WKHVH
SDWKV LH OLQNLQJ WRJHWKHU WKH QRGH VHTXHQFHV FRQWDLQHG LQ WKH OLVWV RI WKH WZR OHDYHV
%HFDXVH WKH HGJHV DUH YLVLWHG RQFH &3$7+ ),1'(5 GHWHUPLQHV D &SDWK LQ OLQHDU
WLPH
7KH FRUH LGHD FRQVLVWV LQ VDYLQJ WKH VWDWH LH WKH SDWKV O QT ZLWK O 7 D OHDI
DQG QT 9 WKH nUVW HQFRXQWHUHG QRGH ZLWK GHJUHH GLmHUHQW IURP RI WKH SURFHGXUH
&3$7+ ),1'(5 ZKHQHYHU LW LV FDOOHG E\ WKH PDLQ '2075(( DOJRULWKP ,Q WKLV ZD\
LW LV SRVVLEOH YLVLW HDFK HGJH RI WKH WUHH 7 RQFH +HQFH '2075(( ZLOO EH JOREDOO\ OLQHDU
LQ WLPH