Chapter Interval Scheduling. Greedy Algorithms. What is a Greedy Algorithm? Interval Scheduling

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1 Wht s Grdy Algorthm? Chptr Grdy Algorthms Grdy lgorthm Hrd to gv prs dfnton It ulds up soluton n smll stps, hoosng dson t h stp to optmz som undrlyng rtron Mk lol dsons, nd mglly gt glol optmum Vry ffnt It s sy to dsgn mny grdy lgorthms for sngl prolm, ut only fw n tully solv th prolm Th hrd prt s to prov th lgorthm works lds y Kvn Wyn Copyrght 005 Prson-Addson Wsly All rghts rsrvd *Adustd y Gng Tn for C8: Algorthms t Boston Collg, Fll 05 Intrvl hdulng Intrvl hdulng Intrvl shdulng Jo strts t s nd fnshs t f Two os omptl f thy don't ovrlp Gol: fnd mxmum sust of mutully omptl os d f g h Tm

2 5 A Brut-For rh Intrvl hdulng: Grdy Algorthms n os Fnd ll susts, nd tst thr omptlty n susts Grdy tmplt Consdr os n som ordr Tk h o provdd t's omptl wth th ons lrdy tkn [Erlst strt tm] Consdr os n sndng ordr of strt tm s trt usng rsours s qukly s possl Countr xmpl: d f g h Tm 6 Intrvl hdulng: Anothr Hurst Intrvl hdulng: On Mor Hurst [hortst ntrvl] Consdr os n sndng ordr of ntrvl lngth f -s o tht h o won t oupy th rsour for too long [Fwst onflts] For h o, ount th numr of onfltng os hdul n sndng ordr of onflts Countr xmpl: Countr xmpl: 7 8

3 9 Intrvl hdulng: Grdy Algorthm Th Tmplt of Workng out Grdy Algorthm [Erlst fnsh tm] Consdr os n nrsng ordr of fnsh tm Tk h o provdd t's omptl wth th ons lrdy tkn Insghts Com up wth hurst Exmpl : Exmpl : Fnd on Fgur out Countr xmpl Exmpl : Fnd on ountr xmpl Prov t Try hrdr Works for ll ths xmpls! Flng grt! 0 Intrvl hdulng: Grdy Algorthm Intrvl hdulng: Anlyss Grdy lgorthm Consdr os n nrsng ordr of fnsh tm Tk h o provdd t's omptl wth th ons lrdy tkn ort os y fnsh tms so tht f f f n os sltd A φ for = to n { f (o omptl wth A) A A {} } rturn A Grdy: Thorm Th strtgy of rlst-fnsh-tm s optml Pf (y ontrdton) Assum grdy s not optml, nd lt's s wht hppns Lt,, k dnot st of os sltd y th grdy lgorthm Lt,, m dnot st of os n th optml soluton wth =, =,, r = r for th lrgst possl vlu of r n ll optml solutons r r+ o r+ fnshs for r+ Implmntton O(n log n) Rmmr o * tht ws ddd lst to A Jo s omptl wth A f s f * OPT: r r+ why not rpl o r+ wth o r+?

4 Intrvl hdulng: Anlyss Thorm Th strtgy of rlst-fnsh-tm s optml Intrvl Prttonng Pf (y ontrdton) Assum grdy s not optml, nd lt's s wht hppns Lt,, k dnot st of os sltd y grdy Lt,, m dnot st of os n th optml soluton wth =, =,, r = r for th lrgst possl vlu of r o r+ fnshs for r+ Grdy: r r+ OPT: r r+ soluton stll fsl nd optml, ut ontrdts mxmlty of r Intrvl Prttonng Intrvl Prttonng Intrvl prttonng Ltur strts t s nd fnshs t f Gol: fnd mnmum numr of lssrooms to shdul ll lturs so tht no two our t th sm tm n th sm room Intrvl prttonng Ltur strts t s nd fnshs t f Gol: fnd mnmum numr of lssrooms to shdul ll lturs so tht no two our t th sm tm n th sm room Ex: Ths shdul uss lssrooms to shdul 0 lturs Ex: Ths shdul uss only d g d f h g f h 9 9:0 0 0:0 :0 :0 :0 :0 :0 :0 Tm 9 9:0 0 0:0 :0 :0 :0 :0 :0 :0 Tm 5 6

5 7 Intrvl Prttonng: Lowr Bound on Optml oluton Intrvl Prttonng: Grdy Algorthm Df Th dpth of st of opn ntrvls s th mxmum numr tht pss ovr ny sngl pont on th tm-ln Grdy lgorthm Consdr lturs n nrsng ordr of strt tm: ssgn ltur to ny omptl lssroom Ky osrvton Numr of lssrooms ndd dpth Ex: Dpth of shdul low = shdul low s optml,, ll ontn 9:0 Q Dos thr lwys xst shdul qul to dpth of ntrvls? d f g 9 9:0 0 0:0 :0 :0 :0 :0 h :0 :0 Tm Implmntton O(n log n) For h lssroom k, mntn th fnsh tm of th lst o ddd Kp th lssrooms n prorty quu 8 Intrvl Prttonng: Grdy Anlyss Osrvton Grdy lgorthm nvr shduls two nomptl lturs n th sm lssroom hdulng to Mnmz Ltnss Thorm Evry ntrvl wll ssgnd ll Pf Consdr n ntrvl I uppos thr r k ntrvls rlr tht ovrlp t k+ d I Thus k d - n w hv d lssrooms, w hv t lst on lssroom vll 9

6 hdulng to Mnmzng Ltnss Mnmzng Ltnss: Grdy Algorthms Mnmzng ltnss prolm ngl rsour prosss on o t tm Jo rqurs t unts of prossng tm Jo hs ddln: d If strts t tm s, t fnshs t tm f = s + t Allow ltnss: l = mx { 0, f -d } Gol: shdul ll os to mnmz mxmum ltnss L = mx l 5 6 Grdy tmplt Consdr os n som ordr [hortst prossng tm frst] Consdr os n sndng ordr of prossng tm t [Erlst ddln frst] Consdr os n sndng ordr of ddln d Ex: t d [mllst slk] Consdr os n sndng ordr of slk d -t Jo: 6 5 d = 9 d = 8 d 6 = 5 d = 6 d 5 = d = ltnss = ltnss = 0 mx ltnss = 6 Mnmzng Ltnss: Grdy Algorthms Mnmzng Ltnss: Grdy Algorthms Grdy tmplt Consdr os n som ordr Grdy tmplt Consdr os n som ordr [hortst prossng tm frst] Consdr os n sndng ordr of prossng tm t t 0 d 00 0 [mllst slk] Consdr os n sndng ordr of slk d -t t 0 d 0 slk 0 Aordng to th hurst Ltnss: 0 Aordng to th hurst 0 0 Ltnss: 9 OPT Ltnss: OPT 0 Ltnss:

7 5 urprs Mnmzng Ltnss: No Idl Tm [Erlst ddln frst] Consdr os n sndng ordr of ddln d Ths on works! Although th hurst ompltly gnors t Nxt, w try to prov t Assum os r sortd n sndng ordr of ddln Osrvton Thr xsts n optml shdul wth no dl tm d = d = 6 d = d = d = 6 d = Tht s, f <, thn d d Osrvton Th grdy shdul hs no dl tm 6 Mnmzng Ltnss: Invrsons Mnmzng Ltnss: Invrsons Df An nvrson n shdul s pr of os nd suh tht: < ut shduld for nvrson Df An nvrson n shdul s pr of os nd suh tht: < ut shduld for nvrson f for swp ftr swp f' Osrvton Grdy shdul hs no nvrsons Osrvton If shdul (wth no dl tm) hs n nvrson, t hs on wth pr of nvrtd os shduld onsutvly Why? d d Clm wppng two dnt, nvrtd os rdus th numr of nvrsons y on nd dos not nrs th mx ltnss Pf Lt l th ltnss for th swp, nd lt l ' t ftrwrds l' k = l k for ll k, l' l If o s lt: l = = f d f d f d l (dfnton) ( fnshs t tm f ) ( < ) (dfnton) 7 8

8 9 Mnmzng Ltnss: Anlyss of Grdy Algorthm Mnmzng Ltnss: Grdy Algorthm Thorm Grdy shdul s optml Pf hs no nvrsons Dfn * to n optml shdul tht hs th fwst numr of nvrsons, nd lt's s wht hppns Cn ssum * hs no dl tm If * hs no nvrsons, thn = * If * hs n nvrson, lt - n dnt nvrson swppng nd dos not nrs th mxmum ltnss nd strtly drss th numr of nvrsons ths ontrdts dfnton of * Grdy lgorthm Erlst ddln frst ort n os y ddln so tht d d d n t 0 for = to n Assgn o to ntrvl [t, t + t ] s t, f t + t t t + t output ntrvls [s, f ] mx ltnss = d = 6 d = 8 d = 9 d = 9 d 5 = d 6 = Grdy Anlyss trtgs Grdy lgorthm stys hd how tht ftr h stp of th grdy lgorthm, ts soluton s t lst s good s ny othr lgorthm's Exmpl: Intrvl hdulng Optml Chng Exhng rgumnt Grdully trnsform ny soluton to th on found y th grdy lgorthm wthout hurtng ts qulty Exmpl: hdulng to mnmz ltnss Th most usful strtgy truturl Dsovr smpl "struturl" ound ssrtng tht vry possl soluton must hv rtn vlu Thn show tht your lgorthm lwys hvs ths ound Exmpl: Intrvl prttonng

9 Optml Offln Chng Optml Offln Chng: Frthst-In-Futur Chng Ch wth pty to stor k tms qun of m tm rqusts d, d,, d m Ch ht: tm lrdy n h whn rqustd Ch mss: tm not lrdy n h whn rqustd: must rng rqustd tm nto h, nd vt som xstng tm, f full Frthst-n-futur Evt tm n th h tht s not rqustd untl frthst n th futur urrnt h: d f futur qurs: g d d f d f g h Gol Evton shdul tht mnmzs numr of h msss h mss t ths on Ex: k =, ntl h =, rqusts:,,,,,,, Optml vton shdul: h msss Thorm [Blldy, 960s] FF s optml vton shdul Pf Algorthm nd thorm r ntutv; proof s sutl rqusts h Rdud Evton hduls Rdud Evton hduls Df A rdud shdul s shdul tht only nsrts n tm nto th h n stp n whh tht tm s rqustd Intuton Cn trnsform n unrdud shdul nto rdud on wth no mor h msss x d d d d d d x d n unrdud shdul rdud shdul Clm Gvn ny unrdud shdul, n trnsform t nto rdud shdul ' wth no mor h msss Pf (y nduton on numr of unrdud tms) uppos rngs d nto th h t tm t, wthout rqust Lt th tm vts whn t rngs d nto th h Cs : d vtd t tm t', for nxt rqust for d Cs : d rqustd t tm t' for d s vtd t t' d t t' d vtd t tm t', for nxt rqust ' t t' d dosn't ntr h t rqustd tm t t' d rqustd t tm t' Cs Cs ' d 5 6

10 7 Frthst-In-Futur: Anlyss Frthst-In-Futur: Anlyss Thorm FF s optml vton lgorthm Pf (y nduton on numr or rqusts ) Invrnt: Thr xsts n optml rdud shdul tht mks th sm vton shdul s FF through th frst + rqusts Lt rdud shdul tht stsfs nvrnt through rqusts W produ ' tht stsfs nvrnt ftr + rqusts Consdr (+) st rqust d = d + n nd FF hv grd up untl now, thy hv th sm h ontnts for rqust + Cs : (d s lrdy n th h) ' = stsfs nvrnt Cs : (d s not n th h nd nd FF vt th sm lmnt) ' = stsfs nvrnt Pf (ontnud) Cs : (d s not n th h; FF vts ; vts f ) gn onstruton of ' from y vtng nstd of f sm f sm f ' + sm d sm d f ' now ' grs wth FF on frst + rqusts; w show tht hvng lmnt f n h s no wors thn hvng lmnt 8 Frthst-In-Futur: Anlyss Frthst-In-Futur: Anlyss Lt ' th frst tm ftr + tht nd ' tk dffrnt ton, nd lt g tm rqustd t tm ' must nvolv or f (or oth) Lt ' th frst tm ftr + tht nd ' tk dffrnt ton, nd lt g tm rqustd t tm ' must nvolv or f (or oth) ' sm sm f ' sm sm f ' ' Cs : g = Cn't hppn wth Frthst-In-Futur sn thr must rqust for f for Cs : g = f Elmnt f n't n h of, so lt ' th lmnt tht vts f ' =, ' sss f from h; now nd ' hv sm h f ', ' vts ' nd rngs nto th h; now nd ' hv th sm h Cs : g, f must vt Mk ' vt f; now nd ' hv th sm h ' othrws ' would tk th sm ton sm g sm g ' Not: ' s no longr rdud, ut n trnsformd nto rdud shdul tht grs wth FF through stp + 9 0

11 Chng Prsptv Onln vs offln lgorthms Offln: full squn of rqusts s known pror Onln (rlty): rqusts r not known n dvn Chng s mong most fundmntl onln prolms n C hortst Pths n Grph LIFO Evt pg rought n most rntly LRU Evt pg whos most rnt ss ws rlst FF wth drton of tm rvrsd! Thorm FF s optml offln vton lgorthm Provds ss for undrstndng nd nlyzng onln lgorthms LRU s k-ompttv [ton 8] LIFO s rtrrly d hortst Pth Prolm Dkstr's Algorthm s hortst pth ntwork Drtd grph G = (V, E) Eh dg hs ost l our s, dstnton t hortst pth prolm: fnd shortst drtd pth from s to t ost of pth = sum of dg osts n pth Cost of pth s---5-t = = 8 Hgh lvl Comput th shortst pth from s to ll othr nods ffntly Mntn st of xplord nods for whh w hv dtrmnd th shortst pth For ny u, th dstn d(u) s th ost of th shortst pth from s to u Intlz = { s }, d(s) = 0 Rptd hoos n unxplord nod v to dd to, ordng to som undrlyng rtron---th ommon hrtrsts of Grdy lgorthms 7 t

12 5 Dkstr's Algorthm Dkstr's Algorthm Dkstr's lgorthm Mntn st of xplord nods for whh w hv dtrmnd th shortst pth dstn d(u) from s to u Intlz = { s }, d(s) = 0 Rptdly hoos unxplord nod v whh mnmzs π ( v) = mn = ( u, v) : u dd v to, nd st d(v) = π(v) s ( d( u) + l ), u d(u) u d(u ) shortst pth to v through th xplord prt l l v Dkstr's lgorthm Mntn st of xplord nods for whh w hv dtrmnd th shortst pth dstn d(u) from s to u Intlz = { s }, d(s) = 0 Rptdly hoos unxplord nod v whh mnmzs π ( v) = mn = ( u, v) : u dd v to, nd st d(v) = π(v) s d(u) u l ( d( u) + l ), v Assrtng tht th ost of th shortst pth from s to v s π(v); ths nds proof π(v) = mn {d(u)+ l, d(u )+ l } 6 Dkstr's Algorthm: Proof of Corrtnss Dkstr's Algorthm: Implmntton Invrnt For h nod u, d(u) s th lngth of th shortst s-u pth Pf (y nduton on ) Bs s: = s trvl Indutv hypothss: Assum tru for = k Lt v nxt nod ddd to, nd lt u-v th hosn dg Th shortst s-u pth plus (u, v) s n s-v pth of lngth π(v) Consdr ny s-v pth P W'll s tht t's no shortr thn π(v) Lt x-y th frst dg n P tht lvs, nd lt P' th supth to x P s lrdy too long s soon s t lvs l (P) l (P') + l (x,y) d(x) + l (x, y) π(y) π(v) s P' u x P v y For h unxplord nod, xpltly mntn How to mntn t ffntly? Intlly: = {s}, If thr s n dg (s,w), st π(w)=l (s,w) If thr s no dg (s,w), st π(w)= s u π ( v) = mn ( d( u) + l ) = ( u, v): u v π(w) w nonngtv wghts ndutv hypothss dfn of π(y) Dkstr hos v nstd of y 7 8

13 9 Dkstr's Algorthm: Implmntton Edsgr W Dkstr For h unxplord nod, xpltly mntn How to mntn t ffntly? s d(u) u v π(v) = w π(w) mn = (u,v):u d(u) + l Th quston of whthr omputrs n thnk s lk th quston of whthr sumrns n swm Do only wht only you n do In thr pty s tool, omputrs wll ut rppl on th surf of our ultur In thr pty s ntlltul hllng, thy r wthout prdnt n th ulturl hstory of mnknd d(u) u v Th us of COBOL rppls th mnd; ts thng should, thrfor, rgrdd s rmnl offn l s w Nd to updt π(w) π ( w) = mn d( u) + l = ( u, w): u APL s mstk, rrd through to prfton It s th lngug of th futur for th progrmmng thnqus of th pst: t rts nw gnrton of odng ums Whn ddng v, for h ndnt dg = (v, w), updt π w) = mn{ π ( w), d( v) + l } ( Progrmmng s on of th most dffult rnhs of ppld mthmts; th poorr mthmtns hd ttr rmn pur mthmtns 50 Dkstr's Algorthm: Implmntton lt th smllst π(v): Usng Prorty quus Dkstr's lgorthm Mntn st of xplord nods for whh w hv dtrmnd Rd Ch5 th shortst pth dstn d(u) from s to u Intlz = { s }, d(s) = 0 Rptdly hoos unxplord nod v wth th smllst π(v) Bnry hp-sd prorty quus lt mn log n Chng Ky π ( v) = mn = ( u, v) : u ( d( u) + l ), dd v to, nd st d(v) = π(v) Prorty Quu PQ Oprton Dkstr Arry Bnry hp d-wy Hp F hp W know how to omput π(v) lrdy But How to slt nod v wth th smllst π(v) Insrt ExtrtMn ChngKy n n m n n log n log n log n d log d n d log d n log d n log n IsEmpty n Totl n m log n m log m/n n m + n log n Indvdul ops r mortzd ounds 5 5

14 5 Bnry-Hp Bsd Prorty Quus Bnry-Hp Bsd Prorty Quus: lt Mn A mn hp Th vlu of ny nod s no grtr thn th vlus of ts hldrn Rturn Bnry-Hp Bsd Prorty Quus: lt Mn Bnry-Hp Bsd Prorty Quus: lt Mn Rturn Rturn

15 57 Bnry-Hp Bsd Prorty Quus: lt Mn Bnry-Hp Bsd Prorty Quus: lt Mn Rturn Rturn Bnry-Hp Bsd Prorty Quus: Drs Ky Bnry-Hp Bsd Prorty Quus: Drs Ky 9 -> 9 ->

16 6 Bnry-Hp Bsd Prorty Quus: Drs Ky Bnry-Hp Bsd Prorty Quus: Drs Ky 9 -> 9 -> Extr lds Con Chngng Grd s good Grd s rght Grd works Grd lrfs, uts through, nd pturs th ssn of th volutonry sprt - Gordon Gko (Mhl Dougls)

17 65 Con Chngng Con-Chngng: Grdy Algorthm Gol Gvn urrny dnomntons:, 5, 0, 5, 00, dvs mthod to py mount to ustomr usng fwst numr of ons Cshr's lgorthm At h trton, dd on of th lrgst vlu tht dos not tk us pst th mount to pd Ex: Cshr's lgorthm At h trton, dd on of th lrgst vlu tht dos not tk us pst th mount to pd Ex: $89 ort ons dnomntons y vlu: < < < n ons sltd φ whl (x 0) { lt k lrgst ntgr suh tht k x f (k = 0) rturn "no soluton found" x x - k {k} } rturn Q Is shr's lgorthm optml? 66 Con-Chngng: Anlyss of Grdy Algorthm Con-Chngng: Anlyss of Grdy Algorthm Thorm Grd s optml for U ong:, 5, 0, 5, 00 Pf (y nduton on x) Consdr optml wy to hng k x < k+ : grdy tks on k W lm tht ny optml soluton must lso tk on k f not, t nds nough ons of typ,, k- to dd up to x tl low ndts no optml soluton n do ths Prolm rdus to on-hngng x - k nts, whh, y nduton, s optmlly solvd y grdy lgorthm Osrvton Grdy lgorthm s su-optml for U postl dnomntons:, 0,,, 70, 00, 50, 5, 500 Countrxmpl 0 Grdy: 00,,,,,,, Optml: 70, 70 k 5 k All optml solutons must stsfy P 5 N N + D Q no lmt Mx vlu of ons,,, k- n ny OPT = = 75 + =

18 ltng Brkponts ltng Brkponts ltng rkponts Rod trp from Prnton to Plo Alto long fxd rout Rfulng sttons t rtn ponts long th wy Ful pty = C Gol: mks s fw rfulng stops s possl Grdy lgorthm Go s fr s you n for rfulng C C C C Prnton C C C Plo Alto ltng Brkponts: Grdy Algorthm ltng Brkponts: Corrtnss Truk drvr's lgorthm ort rkponts so tht: 0 = 0 < < < < n = L {0} x 0 rkponts sltd urrnt loton whl (x n ) lt p lrgst ntgr suh tht p x + C f ( p = x) rturn "no soluton" x p {p} rturn Grdy: Thorm Grdy lgorthm s optml Pf (y ontrdton) Assum grdy s not optml, nd lt's s wht hppns Lt 0 = g 0 < g < < g p = L dnot st of rkponts hosn y grdy Lt 0 = f 0 < f < < f q = L dnot st of rkponts n n optml soluton wth f 0 = g 0, f = g,, f r = g r for lrgst possl vlu of r Not: g r+ > f r+ y grdy ho of lgorthm g 0 g g g r g r+ Implmntton O(n log n) Us nry srh to slt h rkpont p OPT: f 0 f f f r f r+ f q why dosn't optml soluton drv lttl furthr? 7 7

19 7 ltng Brkponts: Corrtnss Thorm Grdy lgorthm s optml Pf (y ontrdton) Assum grdy s not optml, nd lt's s wht hppns Lt 0 = g 0 < g < < g p = L dnot st of rkponts hosn y grdy Lt 0 = f 0 < f < < f q = L dnot st of rkponts n n optml soluton wth f 0 = g 0, f = g,, f r = g r for lrgst possl vlu of r Not: g r+ > f r+ y grdy ho of lgorthm Grdy: g 0 g g g r g r+ OPT: f 0 f f f r f q nothr optml soluton hs on mor rkpont n ommon ontrdton