Copyright 2000, Kevin Wayne 1

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1 Chptr 4 4. Intrvl hulng Gry Algorthms ls y Kvn Wyn. Copyrght 005 Prson-Ason Wsly. All rghts rsrv. Intrvl hulng Intrvl hulng: Gry Algorthms Intrvl shulng. Jo strts t s n fnshs t f. Two os omptl f thy on't ovrlp. Gol: fn mxmum sust of mutully omptl os. Gry tmplt. Consr os n som orr. Tk h o prov t's omptl wth th ons lry tkn. [Erlst strt tm] Consr os n snng orr of strt tm s. f g h Tm [Erlst fnsh tm] Consr os n snng orr of fnsh tm f. [hortst ntrvl] Consr os n snng orr of ntrvl lngth f - s. [Fwst onflts] For h o, ount th numr of onfltng os. hul n snng orr of onflts. 3 4 Copyrght 000, Kvn Wyn

2 Intrvl hulng: Gry Algorthms Intrvl hulng: Gry Algorthm Gry tmplt. Consr os n som orr. Tk h o prov t's omptl wth th ons lry tkn. Gry lgorthm. Consr os n nrsng orr of fnsh tm. Tk h o prov t's omptl wth th ons lry tkn. rks rlst strt tm rks shortst ntrvl rks fwst onflts ort os y fnsh tms so tht f f... f n. os slt A f for = to n { f (o omptl wth A) A A È {} } rturn A Implmntton. O(n log n). Rmmr o * tht ws lst to A. Jo s omptl wth A f s ³ f *. 5 6 Intrvl hulng: Anlyss Intrvl hulng: Anlyss Thorm. Gry lgorthm s optml. Thorm. Gry y fnsh tm lgorthm s optml. Pf. (y ontrton) Assum gry s not optml, n lt's s wht hppns. Lt,,... k not st of os slt y gry. Lt,,... m not st of os n th optml soluton wth =, =,..., r = r for th lrgst possl vlu of r. Pf. (y ontrton) Assum gry s not optml, n lt's s wht hppns. Lt,,... k not st of os slt y gry. Lt,,... m not st of os n th optml soluton wth =, =,..., r = r for th lrgst possl vlu of r. o r+ fnshs for r+ o r+ fnshs for r+ Gry: r r+ Gry: r r+ OPT: r r+... OPT: r r+... why not rpl o r+ wth o r+? soluton stll fsl n optml, ut ontrts mxmlty of r. 7 8 Copyrght 000, Kvn Wyn

3 Con Chngng 4. Intrvl Prttonng Gr s goo. Gr s rght. Gr works. Gr lrfs, uts through, n pturs th ssn of th volutonry sprt. - Goron Gko (Mhl Dougls) Intrvl Prttonng Intrvl Prttonng Intrvl prttonng. Ltur strts t s n fnshs t f. Gol: fn mnmum numr of lssrooms to shul ll lturs so tht no two our t th sm tm n th sm room. Intrvl prttonng. Ltur strts t s n fnshs t f. Gol: fn mnmum numr of lssrooms to shul ll lturs so tht no two our t th sm tm n th sm room. Ex: Ths shul uss 4 lssrooms to shul 0 lturs. Ex: Ths shul uss only 3. g f h g f h 9 9:30 0 0:30 :30 :30 :30 :30 3 3:30 4 4:30 Tm 9 9:30 0 0:30 :30 :30 :30 :30 3 3:30 4 4:30 Tm Copyrght 000, Kvn Wyn 3

4 Intrvl Prttonng: Lowr Boun on Optml oluton Intrvl Prttonng: Gry Algorthm Df. Th pth of st of opn ntrvls s th mxmum numr tht ontn ny gvn tm. Gry lgorthm. Consr lturs n nrsng orr of strt tm: ssgn ltur to ny omptl lssroom. Ky osrvton. Numr of lssrooms n ³ pth. Ex: Dpth of shul low = 3 Þ shul low s optml.,, ll ontn 9:30 Q. Dos thr lwys xst shul qul to pth of ntrvls? f ort ntrvls y strtng tm so tht s s... s n. 0 numr of llot lssrooms for = to n { f (ltur s omptl wth som lssroom k) shul ltur n lssroom k ls llot nw lssroom + shul ltur n lssroom + + } g :30 9 9:30 0 0:30 :30 :30 :30 h 3 3:30 4 4:30 Tm Implmntton. O(n log n). For h lssroom k, mntn th fnsh tm of th lst o. Kp th lssrooms n prorty quu. 3 4 Intrvl Prttonng: Gry Anlyss Osrvton. Gry lgorthm nvr shuls two nomptl lturs n th sm lssroom. 4. hulng to Mnmz Ltnss Thorm. Gry lgorthm s optml. Pf. Lt = numr of lssrooms tht th gry lgorthm llots. Clssroom s opn us w n to shul o, sy, tht s nomptl wth ll - othr lssrooms. n w sort y strt tm, ll ths nomptlts r us y lturs tht strt no ltr thn s. Thus, w hv lturs ovrlppng t tm s +. Ky osrvton Þ ll shuls us ³ lssrooms. 5 Copyrght 000, Kvn Wyn 4

5 Announmnts /0/5 UIC Engnrng Crr Fr Tusy, /4/5, :00-4:00, n CE Illnos Rooms UIC Engnrng Wk Atvts: F -8 Wht: Unrgrut Rsum Rvw sson (oph, Jun, nors) Whn: Thursy, Frury 9th, 05 :00m-:00pm Whr: tunt Cntr Est- Room 30 hulng to Mnmzng Ltnss Mnmzng ltnss prolm. ngl rsour prosss on o t tm. Jo rqurs t unts of prossng tm n s u t tm. If strts t tm s, t fnshs t tm f = s + t. Ltnss:! = mx { 0, f - }. Gol: shul ll os to mnmz mxmum ltnss L = mx!. Wht: COE tunt Orgnztons Fr Whn: Thursy, Frury 6th, 05 :00pm-3:00pm Drop y ny tm urng ths hours! Whr: tunt Cntr Est- Room 30 Ex: t Wht: Bowlng & Foo for COE tunts Whn: Fry, Frury 7th, 05 5:00pm-8:00pm Whr: Bowlng Ally n tunt Cntr Est 3 = 9 ltnss = ltnss = 0 = 8 6 = 5 = 6 5 = 4 4 = 9 mx ltnss = Mnmzng Ltnss: Gry Algorthms Mnmzng Ltnss: Gry Algorthms Gry tmplt. Consr os n som orr. Gry tmplt. Consr os n som orr. [hortst prossng tm frst] Consr os n snng orr of prossng tm t. [hortst prossng tm frst] Consr os n snng orr of prossng tm t. [Erlst ln frst] Consr os n snng orr of ln. t ountrxmpl [mllst slk] Consr os n snng orr of slk - t. [mllst slk] Consr os n snng orr of slk - t. t 0 0 ountrxmpl 9 0 Copyrght 000, Kvn Wyn 5

6 Mnmzng Ltnss: Gry Algorthm Mnmzng Ltnss: No Il Tm Gry lgorthm. Erlst ln frst. Osrvton. Thr xsts n optml shul wth no l tm. ort n os y ln so tht n = 4 = = t 0 for = to n Assgn o to ntrvl [t, t + t ] s t, f t + t t t + t output ntrvls [s, f ] = 4 = 6 = Osrvton. Th gry shul hs no l tm. mx ltnss = = 6 = 8 3 = 9 4 = 9 5 = 4 6 = Mnmzng Ltnss: Invrsons Mnmzng Ltnss: Invrsons Df. An nvrson n shul s pr of os n suh tht: < ut shul for. nvrson Df. An nvrson n shul s pr of os n suh tht: < ut shul for. nvrson f for swp for swp ftr swp Osrvton. Gry shul hs no nvrsons. Osrvton. If shul (wth no l tm) hs n nvrson, t hs on wth pr of nvrt os shul onsutvly. f' Clm. wppng two nt, nvrt os rus th numr of nvrsons y on n os not nrs th mx ltnss. Pf. Lt! th ltnss for th swp, n lt! ' t ftrwrs.!' k =! k for ll k ¹,!'! If o s lt: l# = = f # f f l (fnton) ( fnshs t tm f ) ( < ) (fnton) 3 4 Copyrght 000, Kvn Wyn 6

7 Mnmzng Ltnss: Anlyss of Gry Algorthm Gry Anlyss trtgs Thorm. Gry shul s optml. Pf. Dfn * to n optml shul tht hs th fwst numr of nvrsons, n lt's s wht hppns. Cn ssum * hs no l tm. If * hs no nvrsons, thn = *. If * hs n nvrson, lt - n nt nvrson. swppng n os not nrs th mxmum ltnss n strtly rss th numr of nvrsons ths ontrts fnton of * Gry lgorthm stys h. how tht ftr h stp of th gry lgorthm, ts soluton s t lst s goo s ny othr lgorthm's. Exhng rgumnt. Grully trnsform ny soluton to th on foun y th gry lgorthm wthout hurtng ts qulty. truturl. Dsovr smpl "struturl" oun ssrtng tht vry possl soluton must hv rtn vlu. Thn show tht your lgorthm lwys hvs ths oun. 5 6 Optml Offln Chng 4.3 Optml Chng Chng. Ch wth pty to stor k tms. qun of m tm rqusts,,, m. Ch ht: tm lry n h whn rqust. Ch mss: tm not lry n h whn rqust: must rng rqust tm nto h, n vt som xstng tm, f full. Gol. Evton shul tht mnmzs numr of h msss. Ex: k =, ntl h =, rqusts:,,,,,,,. Optml vton shul: h msss. rqusts h 8 Copyrght 000, Kvn Wyn 7

8 Optml Offln Chng: Frthst-In-Futur Ru Evton huls Frthst-n-futur. Evt tm n th h tht s not rqust untl frthst n th futur. Df. A ru shul s shul tht only nsrts n tm nto th h n stp n whh tht tm s rqust. urrnt h: f Intuton. Cn trnsform n unru shul nto ru on wth no mor h msss. futur qurs: g f f g h... h mss t ths on Thorm. [Blly, 960s] FF s optml vton shul. Pf. Algorthm n thorm r ntutv; proof s sutl. x x n unru shul ru shul 9 30 Ru Evton huls Frthst-In-Futur: Anlyss Clm. Gvn ny unru shul, n trnsform t nto ru shul ' wth no mor h msss. Pf. (y nuton on numr of unru tms) uppos rngs nto th h t tm t, wthout rqust. Lt th tm vts whn t rngs nto th h. Cs : vt t tm t', for nxt rqust for. Cs : rqust t tm t' for s vt. t t' t t' vt t tm t', for nxt rqust ' t t' osn't ntr h t rqust tm t t' rqust t tm t' ' Thorm. FF s optml vton lgorthm. Pf. (y nuton on numr or rqusts ) Invrnt: Thr xsts n optml ru shul tht mks th sm vton shul s FF through th frst + rqusts. Lt ru shul tht stsfs nvrnt through rqusts. W prou ' tht stsfs nvrnt ftr + rqusts. Consr (+) st rqust = +. n n FF hv gr up untl now, thy hv th sm h ontnts for rqust +. Cs : ( s lry n th h). ' = stsfs nvrnt. Cs : ( s not n th h n n FF vt th sm lmnt). ' = stsfs nvrnt. Cs Cs 3 3 Copyrght 000, Kvn Wyn 8

9 Frthst-In-Futur: Anlyss Frthst-In-Futur: Anlyss Pf. (ontnu) Cs 3: ( s not n th h; FF vts ; vts f ¹ ). gn onstruton of ' from y vtng nst of f sm f sm f + sm sm f now ' grs wth FF on frst + rqusts; w show tht hvng lmnt f n h s no wors thn hvng lmnt ' ' Lt ' th frst tm ftr + tht n ' tk ffrnt ton, n lt g tm rqust t tm '. must nvolv or f (or oth) ' sm sm f ' Cs 3: g =. Cn't hppn wth Frthst-In-Futur sn thr must rqust for f for. Cs 3: g = f. Elmnt f n't n h of, so lt ' th lmnt tht vts. f ' =, ' sss f from h; now n ' hv sm h f ' ¹, ' vts ' n rngs nto th h; now n ' hv th sm h Not: ' s no longr ru, ut n trnsform nto ru shul tht grs wth FF through stp Frthst-In-Futur: Anlyss Chng Prsptv Lt ' th frst tm ftr + tht n ' tk ffrnt ton, n lt g tm rqust t tm '. ' sm sm f must nvolv or f (or oth) Onln vs. offln lgorthms. Offln: full squn of rqusts s known pror. Onln (rlty): rqusts r not known n vn. Chng s mong most funmntl onln prolms n C. ' othrws ' woul tk th sm ton Cs 3: g ¹, f. must vt. Mk ' vt f; now n ' hv th sm h. ' sm g sm g ' LIFO. Evt pg rought n most rntly. LRU. Evt pg whos most rnt ss ws rlst. FF wth rton of tm rvrs! Thorm. FF s optml offln vton lgorthm. Provs ss for unrstnng n nlyzng onln lgorthms. LRU s k-ompttv. [ton 3.8] LIFO s rtrrly Copyrght 000, Kvn Wyn 9

10 hortst Pth Prolm 4.4 hortst Pths n Grph hortst pth ntwork. Drt grph G = (V, E). our s, stnton t. Lngth! = lngth of g. hortst pth prolm: fn shortst rt pth from s to t. ost of pth = sum of g osts n pth shortst pth from Prnton C prtmnt to Enstn's hous s Cost of pth s--3-5-t = = t 38 Dkstr's Algorthm Dkstr's Algorthm Dkstr's lgorthm. Mntn st of xplor nos for whh w hv trmn th shortst pth stn (u) from s to u. Intlz = { s }, (s) = 0. Rptly hoos unxplor no v whh mnmzs π ( v) = mn = ( u, v) : u v to, n st (v) = p(v). ( u) + l, shortst pth to som u n xplor prt, follow y sngl g (u, v) Dkstr's lgorthm. Mntn st of xplor nos for whh w hv trmn th shortst pth stn (u) from s to u. Intlz = { s }, (s) = 0. Rptly hoos unxplor no v whh mnmzs π ( v) = mn = ( u, v) : u v to, n st (v) = p(v). ( u) + l, shortst pth to som u n xplor prt, follow y sngl g (u, v) (u) u! v (u) u! v s s Copyrght 000, Kvn Wyn 0

11 Dkstr's Algorthm: Proof of Corrtnss Dkstr's Algorthm: Implmntton Invrnt. For h no u Î, (u) s th lngth of th shortst s-u pth. Pf. (y nuton on ) Bs s: = s trvl. Inutv hypothss: Assum tru for = k ³. Lt v nxt no to, n lt u-v th hosn g. Th shortst s-u pth plus (u, v) s n s-v pth of lngth p(v). Consr ny s-v pth P. W'll s tht t's no shortr thn p(v). Lt x-y th frst g n P tht lvs, n lt P' th supth to x. P s lry too long s soon s t lvs.! (P) ³! (P') +! (x,y) ³ (x) +! (x, y) ³ p(y) ³ p(v) nonngtv wghts nutv hypothss fn of p(y) s Dkstr hos v nst of y P' u x P v y For h unxplor no, xpltly mntn π(v) = mn. = (u,v) : u Nxt no to xplor = no wth mnmum p(v). Whn xplorng v, for h nnt g = (v, w), upt π(w) = mn { π(w), π(v)+ l }. Effnt mplmntton. Mntn prorty quu of unxplor nos, prortz y p(v). Prorty Quu PQ Oprton Insrt ExtrtMn ChngKy Dkstr n n m Arry n n Bnry hp log n log n log n -wy Hp log n log n log n F hp log n IsEmpty n Totl n m log n m log m/n n m + n log n Invul ops r mortz ouns 4 4 Esgr W. Dkstr Th quston of whthr omputrs n thnk s lk th quston of whthr sumrns n swm. Extr ls Do only wht only you n o. In thr pty s tool, omputrs wll ut rppl on th surf of our ultur. In thr pty s ntlltul hllng, thy r wthout prnt n th ulturl hstory of mnkn. Th us of COBOL rppls th mn; ts thng shoul, thrfor, rgr s rmnl offn. APL s mstk, rr through to prfton. It s th lngug of th futur for th progrmmng thnqus of th pst: t rts nw gnrton of ong ums. 43 Copyrght 000, Kvn Wyn

12 Con Chngng Con Chngng Gol. Gvn urrny nomntons:, 5, 0, 5, 00, vs mtho to py mount to ustomr usng fwst numr of ons. Ex: 34. Gr s goo. Gr s rght. Gr works. Gr lrfs, uts through, n pturs th ssn of th volutonry sprt. - Goron Gko (Mhl Dougls) Cshr's lgorthm. At h trton, on of th lrgst vlu tht os not tk us pst th mount to p. Ex: $ Con-Chngng: Gry Algorthm Con-Chngng: Anlyss of Gry Algorthm Cshr's lgorthm. At h trton, on of th lrgst vlu tht os not tk us pst th mount to p. ort ons nomntons y vlu: < < < n. ons slt f whl (x ¹ 0) { lt k lrgst ntgr suh tht k x f (k = 0) rturn "no soluton foun" x x - k È {k} } rturn Q. Is shr's lgorthm optml? Thorm. Gr s optml for U.. ong:, 5, 0, 5, 00. Pf. (y nuton on x) Consr optml wy to hng k x < k+ : gry tks on k. W lm tht ny optml soluton must lso tk on k. f not, t ns nough ons of typ,, k- to up to x tl low nts no optml soluton n o ths Prolm rus to on-hngng x - k nts, whh, y nuton, s optmlly solv y gry lgorthm. k k All optml solutons must stsfy P 4 5 N N + D Q 3 no lmt Mx vlu of ons,,, k- n ny OPT = = = Copyrght 000, Kvn Wyn

13 9/8/7 Con-Chngng: Anlyss of Gry Algorthm ltng Brkponts Osrvton. Gry lgorthm s su-optml for U postl nomntons:, 0,, 34, 70, 00, 350, 5, 500. Countrxmpl. 40. Gry: 00, 34,,,,,,. Optml: 70, ltng Brkponts ltng Brkponts: Gry Algorthm ltng rkponts. Ro trp from Prnton to Plo Alto long fx rout. Rfulng sttons t rtn ponts long th wy. Ful pty = C. Truk rvr's lgorthm. ort rkponts so tht: 0 = 0 < < <... < n = L {0} x 0 Gol: mks s fw rfulng stops s possl. Gry lgorthm. Go s fr s you n for rfulng. C Prnton C C C C whl (x ¹ n) lt p lrgst ntgr suh tht p x + C f (p = x) rturn "no soluton" x p È {p} rturn C C 6 rkponts slt urrnt loton Plo Alto Implmntton. O(n log n) Us nry srh to slt h rkpont p. 7 5 Copyrght 000, Kvn Wyn 5 3

14 ltng Brkponts: Corrtnss ltng Brkponts: Corrtnss Thorm. Gry lgorthm s optml. Pf. (y ontrton) Assum gry s not optml, n lt's s wht hppns. Lt 0 = g 0 < g <... < g p = L not st of rkponts hosn y gry. Lt 0 = f 0 < f <... < f q = L not 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 gry ho of lgorthm. Thorm. Gry lgorthm s optml. Pf. (y ontrton) Assum gry s not optml, n lt's s wht hppns. Lt 0 = g 0 < g <... < g p = L not st of rkponts hosn y gry. Lt 0 = f 0 < f <... < f q = L not 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 gry ho of lgorthm. g 0 g g g r g r+ g 0 g g g r g r+ Gry: Gry: OPT:... OPT:... f 0 f f f r f r+ f q why osn't optml soluton rv lttl furthr? f 0 f f f r f q nothr optml soluton hs on mor rkpont n ommon Þ ontrton Esgr W. Dkstr Th quston of whthr omputrs n thnk s lk th quston of whthr sumrns n swm. Do only wht only you n o. In thr pty s tool, omputrs wll ut rppl on th surf of our ultur. In thr pty s ntlltul hllng, thy r wthout prnt n th ulturl hstory of mnkn. Th us of COBOL rppls th mn; ts thng shoul, thrfor, rgr s rmnl offn. APL s mstk, rr through to prfton. It s th lngug of th futur for th progrmmng thnqus of th pst: t rts nw gnrton of ong ums. 55 Copyrght 000, Kvn Wyn 4