mincost - 0LQ&RVW)ORZ&RQWHQWV s 0LQ FRVW IORZ...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 0LQ &RVW )ORZ &RQWHQWV s 0LQ FRVW IORZ 7UDQVSRUWDWLRQ SUREOHP $VVLJQPHQW SUREOHP 0DLO FDUULHU SUREOHP .OHLQ V F\FOHFDQFHOLQJ DOJRULWKP 1HWZRUN VLPSOH[ s s s s s QvprVvrv 8PT!!% 6ytvuhq9hhTpr Tvt!" u)[email protected]!!% 0LQLPXP &RVW )ORZ 0LQLPXP FRVW IORZ SUREOHP s 'LUHFWHG QHWZRUN (DFK HGJH KDV D FRVW DQG D FDSDFLW\ (DFK YHUWH[ KDV D VXSSO\ RU GHPDQG )LQG EHVW ZD\ WR VHQG IORZ IURP VXSSO\ YHUWLFHV WR GHPDQG YHUWLFHV s s s 0LQ FRVW IORZ JHQHUDOL]HV s 7UDQVSRUWDWLRQ SUREOHP $VVLJQPHQW SUREOHP 0DLO FDUULHU SUREOHP 0D[ IORZ 6KRUWHVW SDWK s s s s 2QH VWHS FORVHU WR VLQJOH $'7 IRU JUDSK SUREOHPV 0LQLPXP &RVW )ORZ 3UREOHP 0LQ FRVW IORZ SUREOHP s 6HQG ILQLVKHG JRRG IURP SODQWV WR FXVWRPHUV VY FYZ XYZ QHW VXSSO\ GHPDQG DW YHUWH[ Y XQLW VKLSSLQJ FRVW IURP Y WR Z FDSDFLW\ RI HGJH YZ sy2sqrhq vvrrthvr vsvvx s s s s *RDO VDWLVI\ GHPDQG DW PLQLPL]H FRVW phhpv p $ '& !$ qrhq y !$ !$ "$ # # 7UDQVSRUWDWLRQ 3UREOHP 7UDQVSRUWDWLRQ SUREOHP s 6HQG ILQLVKHG JRRG IURP SODQWV WR FXVWRPHUV VY GZ FYZ DPRXQW SURGXFHG DW SODQW Y DPRXQW GHPDQGHG E\ FXVWRPHU Z XQLW VKLSSLQJ FRVW IURP SODQW Y WR FXVWRPHU Z s s s s *RDO PLQLPL]H WRWDO FRVW p $ y !$ $ !$ qrhq phhpv % !$ "$ & # # ' 7UDQVSRUWDWLRQ 3UREOHP $SSOLFDWLRQ $VVLJQ 3ULQFHWRQ XQGHUJUDGV WR ZULWLQJ VHPLQDUV s (DFK VWXGHQWV UDQNV WRS FKRLFHV 5HJLVWUDU DVVLJQV VWXGHQWV WR VHPLQDUV *RDO PD[LPL]H KDSSLQHVV RI VWXGHQWV s s 0RGHO DV D WUDQVSRUWDWLRQ SUREOHP s 6WXGHQW YHUWLFHV VXSSO\ 6HPLQDU YHUWLFHV VXSSO\ s s &RVW RI DVVLJQLQJ VWXGHQW L WR VHPLQDU M LI QRW DPRQJ WRS FKRLFHV U ZKHUH U VWXGHQW L V UDQN RI VHPLQDU M puvprsspvqrrvrhqrssrtirrr hvtvtrqrurv! q u puvprhqhururv$ $VVLJQPHQW 3UREOHP $VVLJQPHQW SUREOHP s $VVLJQ ZRUNHUV WR MREV FYZ FRVW RI DVVLJQLQJ ZRUNHU Y WR MRE Z *RDO PLQLPL]H WRWDO FRVW s s p y $ $ qrhq phhpv % & Xxr ' Ei $VVLJQPHQW 3UREOHP $SSOLFDWLRQV 0DQ\ LPSRUWDQW UHDOZRUOG DSSOLFDWLRQV /HIW MREV SHRSOH VWXGHQWV VZLPPHUV VHUYLFH SHUVRQQHO EDFKHORUV WUDQVODWRUV UDGDU EOLS DW WLPH W 5LJKW PDFKLQHV SURMHFWV GRUP URRPV HYHQWV PLOLWDU\ SRVWLQJV EDFKHORUHWWHV GLSORPDWLF PHHWLQJV UDGDU EOLS DW WLPH W 2SWLPL]H FRVW FRVW KDSSLQHVV FKDQFH RI ZLQQLQJ UHORFDWLRQ FRVW FRPSDWLELOLW\ FRVW DFFXUDF\ 0DLO &DUULHU 3UREOHP 0DLO FDUULHU SUREOHP s 3RVW RIILFH ORFDWHG DW QRGH V )LQG PLQLPXP OHQJWK URXWH WKDW VWDUWV DQG HQGV DW V DQG YLVLWV HDFK URDG DW OHDVW RQFH 1HHG WR WUDYHUVH URDGV PRUH WKDQ RQFH XQOHVV JUDSK LV (XOHULDQ s s yrtu yriqriq $ V 5HVLGXDO *UDSK 2ULJLQDO JUDSK s )ORZ IH $UF H YZ Y p s phhpv Z sy 5HVLGXDO DUFV s YZ DQG ZY 8QGR IORZ VHQW Y p s p rvqhyphhpv 5HVLGXDO JUDSK s $OO UHVLGXDO DUFV ZLWK SRVLWLYH FDSDFLW\ Z )LQGLQJ D )HDVLEOH )ORZ )HDVLEOH IORZ SUREOHP *LYHQ D FDSDFLWDWHG QHWZRUN ZLWK VXSSOLHV DQG GHPDQGV ILQG D IHDVLEOH IORZ LI RQH H[LVWV y ! $ ' " qrhq ! % ( # phhpv ( & ! # " !! )LQGLQJ D )HDVLEOH )ORZ )HDVLEOH IORZ SUREOHP *LYHQ D FDSDFLWDWHG QHWZRUN ZLWK VXSSOLHV DQG GHPDQGV ILQG D IHDVLEOH IORZ LI RQH H[LVWV 2QH VROXWLRQ 6ROYH D PD[LPXP IORZ SUREOHP LQ D UHODWHG QHWZRUN y ! Y $ ' Y " qrhq phhpv ! ! " V % Y % ( # ( & Y & W # ! !! Y # " !! Y )LQGLQJ D )HDVLEOH )ORZ )HDVLEOH IORZ SUREOHP *LYHQ D FDSDFLWDWHG QHWZRUN ZLWK VXSSOLHV DQG GHPDQGV ILQG D IHDVLEOH IORZ LI RQH H[LVWV 2QH VROXWLRQ 6ROYH D PD[LPXP IORZ SUREOHP LQ D UHODWHG QHWZRUN $ sy phhpv ! ! % $ $ ! ! ! " ' " # & V % # # & ( # ( & !! W ! !! ! " )LQGLQJ D )HDVLEOH )ORZ )HDVLEOH IORZ SUREOHP *LYHQ D FDSDFLWDWHG QHWZRUN ZLWK VXSSOLHV DQG GHPDQGV ILQG D IHDVLEOH IORZ LI RQH H[LVWV 2QH VROXWLRQ 6ROYH D PD[LPXP IORZ SUREOHP LQ D UHODWHG QHWZRUN $ y ! $ ! ! ' " qrhq sy $ ! phhpv # % ( # ( & & ! # ! " !! 0LQ &RVW )ORZ $VVXPSWLRQV 8VHIXO DVVXPSWLRQ IRU PLQ FRVW IORZ SUREOHPV s 8QGHUO\LQJ JUDSK LV FRQQHFWHG 1R VXSSO\ RU GHPDQG YHUWLFHV ILQG D IHDVLEOH VROXWLRQ VROYH SUREOHP LQ UHVLGXDO JUDSK DQG WKHQ WUDQVODWH EDFN s $ % $ % ! rvqhy phhpv ! & ! # ' ! ! &\FOH &DQFHOLQJ $OJRULWKP +RZ WR LPSURYH WKH FXUUHQW IHDVLEOH IORZ ZKLOH PDLQWDLQLQJ IHDVLELOLW\" s $8*0(17,1* &<&/( QHJDWLYH FRVW F\FOH LQ UHVLGXDO JUDSK &DQ VHQG IORZ DURXQG F\FOH VWULFWO\ GHFUHDVHV FRVW SUHVHUYHV IHDVLELOLW\ # # $ s # # ! % p % # $ $ ppyr82 !"%$# p82#%" $ # $2 % " " SrvqhyBhu &\FOH &DQFHOLQJ $OJRULWKP +RZ WR LPSURYH WKH FXUUHQW IHDVLEOH IORZ ZKLOH PDLQWDLQLQJ IHDVLELOLW\" s $8*0(17,1* &<&/( QHJDWLYH FRVW F\FOH LQ UHVLGXDO JUDSK &DQ VHQG IORZ DURXQG F\FOH VWULFWO\ GHFUHDVHV FRVW SUHVHUYHV IHDVLELOLW\ $ s # # % $ # # ! rvqhyphhpv ! % p % ! # $ $ ppyr82 !"%$# p82#%" $ # $2 ph82v%! !$2 % " " SrvqhyBhu &\FOH &DQFHOLQJ $OJRULWKP +RZ WR LPSURYH WKH FXUUHQW IHDVLEOH IORZ ZKLOH PDLQWDLQLQJ IHDVLELOLW\" s $8*0(17,1* &<&/( QHJDWLYH FRVW F\FOH LQ UHVLGXDO JUDSK $ # # $ # # rvqhyphhpv ! p ' % % # ! % $ ppyr82!%" p82% " %2 " ph82v!!!'2! " !! " SrvqhyBhu &\FOH &DQFHOLQJ $OJRULWKP +RZ WR LPSURYH WKH FXUUHQW IHDVLEOH IORZ ZKLOH PDLQWDLQLQJ IHDVLELOLW\" s $8*0(17,1* &<&/( QHJDWLYH FRVW F\FOH LQ UHVLGXDO JUDSK $ # # $ ! ! & # # p % % # $ ppyr82 #$ p82$# !2 " ph82v &!2 % " " SrvqhyBhu &\FOH &DQFHOLQJ $OJRULWKP +RZ WR LPSURYH WKH FXUUHQW IHDVLEOH IORZ ZKLOH PDLQWDLQLQJ IHDVLELOLW\" s $8*0(17,1* &<&/( QHJDWLYH FRVW F\FOH LQ UHVLGXDO JUDSK ,V IORZ RSWLPDO ZKHQ QR PRUH DXJPHQWLQJ F\FOHV" # # $ ! ! # # p % % # $ % " " SrvqhyBhu &\FOH &DQFHOLQJ $OJRULWKP .OHLQ V F\FOH FDQFHOLQJ DOJRULWKP s *HQHULF PHWKRG IRU VROYLQJ PLQ FRVW IORZ SUREOHP $QDORJ RI )RUG)XONHUVRQ DXJPHQWLQJ SDWK DOJRULWKP IRU PD[ IORZ .OHLQ V &\FOH &DQFHOLQJ $OJRULWKP Start with a feasible flow f. REPEAT (until no augmenting cycles) Find an augmenting cycle C. Augment flow along C. s 4XHVWLRQV s 'RHV WKLV OHDG WR D PLQ FRVW IORZ" +RZ GR ZH ILQG DQ DXJPHQWLQJ F\FOH" +RZ PDQ\ DXJPHQWLQJ F\FOHV GRHV LW WDNH" s s 0LQ &RVW )ORZ 2SWLPDOLW\ &RQGLWLRQV 2EVHUYDWLRQ ,I DOO UHVLGXDO DUFV KDYH FRVW WKHQ IORZ LV RSWLPDO s &XUUHQW IORZ LV DOZD\V IHDVLEOH $Q\ FKDQJH LQ IORZ FDQ RQO\ LQFUHDVH FRVW s % $ ! ! p # ! SrvqhyBhu 0LQ &RVW )ORZ 5HGXFHG &RVW 5HGXFHG FRVW JLYHQ YHUWH[ SRWHQWLDOV Y WKH UHGXFHG FRVW RI HGJH YZ LV FY Z Y Z ,QWXLWLRQ Y PDUNHW SULFH IRU RQH XQLW RI IORZ D Y rqprqp rr rvhy ' ! Y $ $ p 2EVHUYDWLRQ &RVW RI F\FOH UHGXFHG FRVW RI F\FOH # Y Y # Y p2$ " # rqprqp2$ " Y ( # 0LQ &RVW )ORZ 2SWLPDOLW\ &RQGLWLRQV 7KHRUHP $ IHDVLEOH IORZ I LV RSWLPDO LI DQG RQO\ LI WKHUH DUH QR DXJPHQWLQJ F\FOHV &RUROODU\ ,I .OHLQ V DOJRULWKP WHUPLQDWHV LW WHUPLQDWHV ZLWK DQ RSWLPDO IORZ 3URRI s ,I DXJPHQWLQJ F\FOH GHFUHDVH FRVW E\ VHQGLQJ IORZ DURXQG F\FOH ,I QR DXJPHQWLQJ F\FOH FRPSXWH VKRUWHVW SDWK Y IURP V WR HYHU\ QRGH Y LQ UHVLGXDO JUDSK Z Y FY Z XVLQJ DV YHUWH[ SRWHQWLDOV DOO DUFV KDYH UHGXFHG FRVW WKXV FXUUHQW IORZ LV RSWLPDO s 5XQQLQJ 7LPH $VVXPSWLRQ DOO FDSDFLWLHV DUH LQWHJHUV EHWZHHQ DQG 8 DOO FRVWV DUH LQWHJHUV EHWZHHQ & DQG & ,QYDULDQW HYHU\ IORZ YDOXH DQG HYHU\ UHVLGXDO FDSDFLW\ UHPDLQ LQWHJUDO WKURXJKRXW .OHLQ V DOJRULWKP 7KHRUHP .OHLQ V DOJRULWKP WHUPLQDWHV DIWHU DW PRVW ( 8 & LWHUDWLRQV s (DFK DXJPHQWLQJ F\FOH GHFUHDVH FRVW E\ DW OHDVW yvhyv vvr ,QWHJUDOLW\ WKHRUHP LI DOO DUF FDSDFLWLHV VXSSOLHV DQG GHPDQGV DUH LQWHJHUV WKHQ WKHUH H[LVWV DQ LQWHJUDO PLQ FRVW IORZ s $VVLJQPHQW SUREOHP IRUPXODWLRQ UHOLHV RQ WKLV IDFW &DQ W URXWH DLUSODQH IURP 3ULQFHWRQ WR 3DOR $OWR s )LQGLQJ $ 1HJDWLYH &RVW &\FOH +RZ WR ILQG DQ DXJPHQWLQJ F\FOH" s 5XQ %HOOPDQ)RUG LQ UHVLGXDO JUDSK 2( 9 WLPH SHU F\FOH s +RZ PDQ\ F\FOHV ZLOO ZH QHHG WR FDQFHO" s 6RPH UXOHV OHDG WR H[SRQHQWLDO DOJRULWKPV &OHYHU UXOHV OHDG WR SRO\QRPLDO DOJRULWKPV JHQHUDOL]H VKRUWHVW DXJPHQWLQJ SDWK JHQHUDOL]H IDWWHVW DXJPHQWLQJ SDWK s &DQ ZH UHGXFH WKH WLPH QHHGHG WR ILQG D QHJDWLYH F\FOH" s 1R XQOHVV ZH VROYH D PDMRU RSHQ UHVHDUFK SUREOHP <HV VLQFH ZH FDQ UHXVH LQIRUPDWLRQ IURP LWHUDWLRQ WR LWHUDWLRQ 5HVXOW QHWZRUN VLPSOH[ PHWKRG s s 1HWZRUN 6LPSOH[ 0DLQWDLQ D VSDQQLQJ WUHH DQG YHUWH[ SRWHQWLDOV VXFK WKDW s $OO QRQWUHH DUFV H HLWKHU KDYH IORZH $OO WUHH DUFV KDYH UHGXFHG FRVW RU IORZH FDSH s $OZD\V SRVVLEOH VLQFH LW V D WUHH $1< UHVLGXDO DUF ZLWK QHJ UHGXFHG FRVW FRPSOHWHV D QHJ FRVW F\FOH s rr rqtr $ # rqprq p # rqtr$ uhrqprqp# ppyr82 !%$ rqp82p82# 1HWZRUN 6LPSOH[ +RZ WR XSGDWH VSDQQLQJ WUHH" s )LQG ERWWOHQHFN FDSDFLW\ rr rqtr $ $ % $ % ! # & rqprq p # & ! # % " ' rvqhyphhpv 1HWZRUN 6LPSOH[ +RZ WR XSGDWH VSDQQLQJ WUHH" s )LQG ERWWOHQHFN FDSDFLW\ 'HFUHDVH IORZ RQ VRPH HGJHV E\ LQFUHDVH LW E\ RQ RWKHUV 'HOHWH D ERWWOHQHFN HGJH IURP VSDQQLQJ WUHH LQVHUW QHZ HGJH s s rr rqtr $ ' " $ % ! # # rqprq p # & ! # ( $ rvqhyphhpv 1HWZRUN 6LPSOH[ +RZ WR XSGDWH VSDQQLQJ WUHH" s )LQG ERWWOHQHFN FDSDFLW\ 'HFUHDVH IORZ RQ VRPH HGJHV E\ LQFUHDVH LW E\ RQ RWKHUV 'HOHWH D ERWWOHQHFN HGJH IURP VSDQQLQJ WUHH LQVHUW QHZ HGJH 5HFRPSXWH YHUWH[ SRWHQWLDOV $ s s s Y & rr rqtr # ! rqprq p # ' # % ; 1HWZRUN 6LPSOH[ ,PSOHPHQWDWLRQ +RZ WR ILQG F\FOH" s &DQ ILQG LQ 29 WLPH XVLQJ ')6 *RDO ILQG LQ WLPH SURSRUWLRQDO WR OHQJWK RI F\FOH ,Q SUDFWLFH OHQJWK RI F\FOH YHU\ VKRUW s s 8VH SDUHQWOLQN UHSUHVHQWDWLRQ RI WUHH s &OLPE WUHH IURP WZR HQGSRLQWV XQWLO \RX KLW OHDVW FRPPRQ DQFHVWRU rrvtrqtr 1HWZRUN 6LPSOH[ ,PSOHPHQWDWLRQ +RZ WR ILQG F\FOH" s &DQ ILQG LQ 29 WLPH XVLQJ ')6 *RDO ILQG LQ WLPH SURSRUWLRQDO WR OHQJWK RI F\FOH ,Q SUDFWLFH OHQJWK RI F\FOH YHU\ VKRUW s s 8VH SDUHQWOLQN UHSUHVHQWDWLRQ RI WUHH s &OLPE WUHH IURP WZR HQGSRLQWV XQWLO \RX KLW OHDVW FRPPRQ DQFHVWRU rrvtrqtr 1HWZRUN 6LPSOH[ ,PSOHPHQWDWLRQ +RZ WR ILQG F\FOH" s &DQ ILQG LQ 29 WLPH XVLQJ ')6 *RDO ILQG LQ WLPH SURSRUWLRQDO WR OHQJWK RI F\FOH ,Q SUDFWLFH OHQJWK RI F\FOH YHU\ VKRUW s s 8VH SDUHQWOLQN UHSUHVHQWDWLRQ RI WUHH s &OLPE WUHH IURP WZR HQGSRLQWV XQWLO \RX KLW OHDVW FRPPRQ DQFHVWRU rrvtrqtr 1HWZRUN 6LPSOH[ ,PSOHPHQWDWLRQ +RZ WR ILQG F\FOH" s &DQ ILQG LQ 29 WLPH XVLQJ ')6 *RDO ILQG LQ WLPH SURSRUWLRQDO WR OHQJWK RI F\FOH ,Q SUDFWLFH OHQJWK RI F\FOH YHU\ VKRUW yph s s 8VH SDUHQWOLQN UHSUHVHQWDWLRQ RI WUHH s &OLPE WUHH IURP WZR HQGSRLQWV XQWLO \RX KLW OHDVW FRPPRQ DQFHVWRU rrvtrqtr 1HWZRUN 6LPSOH[ ,PSOHPHQWDWLRQ +RZ WR ILQG F\FOH" s &DQ ILQG LQ 29 WLPH XVLQJ ')6 *RDO ILQG LQ WLPH SURSRUWLRQDO WR OHQJWK RI F\FOH ,Q SUDFWLFH OHQJWK RI F\FOH YHU\ VKRUW yph s s 8VH SDUHQWOLQN UHSUHVHQWDWLRQ RI WUHH s &OLPE WUHH IURP WZR HQGSRLQWV XQWLO \RX KLW OHDVW FRPPRQ DQFHVWRU 'HOHWH ERWWOHQHFN HGJH iyrrpx s rrvtrqtr 1HWZRUN 6LPSOH[ ,PSOHPHQWDWLRQ +RZ WR ILQG F\FOH" s &DQ ILQG LQ 29 WLPH XVLQJ ')6 *RDO ILQG LQ WLPH SURSRUWLRQDO WR OHQJWK RI F\FOH ,Q SUDFWLFH OHQJWK RI F\FOH YHU\ VKRUW yph s s 8VH SDUHQWOLQN UHSUHVHQWDWLRQ RI WUHH s &OLPE WUHH IURP WZR HQGSRLQWV XQWLO \RX KLW OHDVW FRPPRQ DQFHVWRU 'HOHWH ERWWOHQHFN HGJH 8SGDWH WUHH E\ UHYHUVLQJ VXESDWK s s rrvtrqtr 1HWZRUN 6LPSOH[ ,PSOHPHQWDWLRQ +RZ WR ILQG F\FOH" s &DQ ILQG LQ 29 WLPH XVLQJ ')6 *RDO ILQG LQ WLPH SURSRUWLRQDO WR OHQJWK RI F\FOH ,Q SUDFWLFH OHQJWK RI F\FOH YHU\ VKRUW yph s s 8VH SDUHQWOLQN UHSUHVHQWDWLRQ RI WUHH s &OLPE WUHH IURP WZR HQGSRLQWV XQWLO \RX KLW OHDVW FRPPRQ DQFHVWRU 'HOHWH ERWWOHQHFN HGJH 8SGDWH WUHH E\ UHYHUVLQJ VXESDWK s s rrvtrqtr 1HWZRUN 6LPSOH[ ,VVXHV :KLFK HGJH VKRXOG , DGG WR WUHH" s $Q\ RQH ZLWK QHJDWLYH UHGXFHG FRVW ZRUNV 8VH ILUVW RQH OHVV WLPH VHDUFKLQJ IRU F\FOH 8VH PRVW QHJDWLYH RQH PD[LPL]H UDWH DW ZKLFK FRVW GHFUHDVHV &DQGLGDWH OLVW SUDFWLFDO WUDGHRII s s s 'HJHQHUDF\ ZKHQ ERWWOHQHFN FDSDFLW\ s s &DQ KDSSHQ LI WUHH DUF LV DW XSSHU RU ORZHU ERXQG &DQ VWLOO PDNH SURJUHVV VLQFH VSDQQLQJ WUHH FKDQJHV &RPPRQ LQ SUDFWLFH VORZV GRZQ DOJRULWKP XS WR GHJHQHUDWH SLYRWV s &DQ GHJHQHUDF\ OHDG WR LQILQLWH ORRS" <HV EXW F\FOLQJ UDUH LQ SUDFWLFH &DQ WKLV EH DYRLGHG" <HV FKRRVH OHDYLQJ HGJH XVLQJ &XQQLQJKDP V UXOH 'HYHORSPHQW RI 0LQ &RVW )ORZ $OJRULWKPV 6v s 6pphhpvvrirrr hqVpirrrò8hq8 DtrytWshp s `rh ($ (% ($' (%! (%& (&! (&" (' ('$ ('' 9vprr 9hvt HvAyxr Erryy AqAyxr Fyrv @qqFh9vv 9vvBhi Spx7yhqErr Uhq Pyv Hruq Irxvyr Psxvyr Tpprvrurhu Qvhyqhy 8pyrphpryvt 8hhpvphyvt Drqphhpvphyvt 8phyvt 7vtPu @ ! W ! V @WV @WV @W @ @ ! ! V 8V ! ytV @W ytV @W ! yt8 vhyv @uhprqphhpvphyvt [email protected] @ ! Chqirhvvrqrxvyrvhpvpr 7shrhytvuruvvphrqphyvtrpuvr &RQFOXVLRQV 0LQ FRVW IORZ LV LPSRUWDQW EHFDXVH s ,W V D YHU\ JHQHUDO SUREOHP VROYLQJ PRGHO 7KHUH DUH PDQ\ IDVW DQG SUDFWLFDO DOJRULWKPV s 0LQ FRVW IORZV UHOLHV RQ DOJRULWKPLF PDFKLQHU\ ZH YH EHHQ EXLOGLQJ XS s *UDSK 6KRUWHVW SDWK SUREOHP 0D[ IORZ SUREOHP 3DUHQWOLQN UHSUHVHQWDWLRQ s s s ...
View Full Document

Ask a homework question - tutors are online