Copyright 2000, Kevin Wayne 1

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1 Rp: Grphs 80: Algorthm Dsgn n Anlyss Jrmh Blok Puru Unvrsty prng 09 Dfnton of Grph Rprsnttons Any Mtrx Any Lst onntvty, yls Trs (onnt + No yls) Root Trs/Bnry Trs/Bln Bnry Trs Brth Frst rh BF Tr O(m+n) lgorthm Appltons: onnt omponnts, hortst Pth t Tstng Bprttnss Drt Grphs n trong onntvty Announmnt: Homwork rls! Du: Jnury th t :9PM (Grsop) trong onntvty: Algorthm Thorm. n trmn f G s strongly onnt n O(m + n) tm. Pf. Pk ny no s. Run BF from s n G. rvrs orntton of vry g n G Run BF from s n G rv. Rturn tru ff ll nos rh n oth BF xutons. orrtnss follows mmtly from prvous lmm.. DAGs n Topologl Orrng strongly onnt not strongly onnt Drt Ayl Grphs Prn onstrnts Df. An DAG s rt grph tht ontns no rt yls. Prn onstrnts. Eg (v, v ) mns tsk v must our for v. Ex. Prn onstrnts: g (v, v ) mns v must pr v. Df. A topologl orr of rt grph G = (V, E) s n orrng of ts nos s v, v,, v n so tht for vry g (v, v ) w hv <. Appltons. ours prrqust grph: ours v must tkn for v. omplton: moul v must ompl for v. Ppln of omputng os: output of o v n to trmn nput of o v. v v v v v v 7 v v v v v v v v 7 topologl orrng Funton F(V) W:= * V; X:=W + V; Y:=X * W; :=W *W; Z:=Y + V; rturn Z V W X Y Z DAG opyrght 000, Kvn Wyn

If G hs topologl orr, thn G s DAG. Q. Dos vry DAG hv topologl orrng? Q. If so, how o w omput on? th rt yl v v v v n th suppos topologl orr: v,, v n 7 8 Drt Ayl Grphs Drt Ayl Grphs Lmm.

2 Drt Ayl Grphs Drt Ayl Grphs Lmm. If G hs topologl orr, thn G s DAG. uppos tht G hs topologl orr v,, v n n tht G lso hs rt yl. Lt's s wht hppns. Lt v th lowst-nx no n, n lt v th no ust for v ; thus (v, v ) s n g. By our ho of, w hv <. On th othr hn, sn (v, v ) s n g n v,, v n s topologl orr, w must hv <, ontrton. Lmm. If G hs topologl orr, thn G s DAG. Q. Dos vry DAG hv topologl orrng? Q. If so, how o w omput on? th rt yl v v v v n th suppos topologl orr: v,, v n 7 8 Drt Ayl Grphs Drt Ayl Grphs Lmm. If G s DAG, thn G hs no wth no nomng gs. uppos tht G s DAG n vry no hs t lst on nomng g. Lt's s wht hppns. Pk ny no v, n gn followng gs kwr from v. n v hs t lst on nomng g (u, v) w n wlk kwr to u. Thn, sn u hs t lst on nomng g (x, u), w n wlk kwr to x. Rpt untl w vst no, sy w, tw. Lt not th squn of nos nountr twn sussv vsts to w. s yl. Lmm. If G s DAG, thn G hs topologl orrng. Pf. (y nuton on n) Bs s: tru f n =. Gvn DAG on n > nos, fn no v wth no nomng gs. G - { v } s DAG, sn ltng v nnot rt yls. By nutv hypothss, G - { v } hs topologl orrng. Pl v frst n topologl orrng; thn ppn nos of G - {v} n topologl orr. Ths s vl sn v hs no nomng gs. ply DAG v w x u v 9 0 Topologl ortng Algorthm: Runnng Tm Thorm. Algorthm fns topologl orr n O(m + n) tm. Pf. Mntn th followng nformton: ount[w] = rmnng numr of nomng gs = st of rmnng nos wth no nomng gs Intlzton: O(m + n) v sngl sn through grph. Upt: to lt v rmov v from rmnt ount[w] for ll gs from v to w, n w to f ount[w] hts 0 ths s O() pr g hptr Gry Algorthms ls y Kvn Wyn. opyrght 00 Prson-Ason Wsly. All rghts rsrv. opyrght 000, Kvn Wyn

3 Intrvl hulng. Intrvl hulng 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. f g h Tm Intrvl hulng: Gry Algorthms Intrvl hulng: Gry Algorthms Gry tmplt. onsr os n som nturl orr. Tk h o prov t's omptl wth th ons lry tkn. Gry tmplt. onsr os n som nturl orr. Tk h o prov t's omptl wth th ons lry tkn. [Erlst strt tm] onsr os n snng orr of s. [Erlst fnsh tm] onsr os n snng orr of f. ountrxmpl for rlst strt tm [hortst ntrvl] onsr os n snng orr of f -s. [Fwst onflts] For h o, ount th numr of onfltng os. hul n snng orr of. ountrxmpl for shortst ntrvl ountrxmpl for fwst onflts Intrvl hulng: Gry Algorthm Intrvl hulng: Anlyss Gry lgorthm. onsr os n nrsng orr of fnsh tm. Tk h o prov t's omptl wth th ons lry tkn. ort os y fnsh tms so tht f f... f n. st of os slt A for = to n { f (o omptl wth A) A A {} } rturn A ply Gry: Thorm. Gry lgorthm s optml. 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+ r r+... Implmntton. O(n log n). Rmmr o * tht ws lst to A. Jo s omptl wth A f s f *. OPT: r r+ why not rpl o r+ wth o r+? opyrght 000, Kvn Wyn

4 Intrvl hulng: Anlyss Thorm. Gry lgorthm s optml. 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.. Intrvl Prttonng o r+ fnshs for r+ Gry: r r+ OPT: r r+... soluton stll fsl n optml, ut ontrts mxmlty of r. 9 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. Ex: Ths shul uss lssrooms to shul 0 lturs. 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 only. f g h g h f 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 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. Ky osrvton. Numr of lssrooms n pth. Ex: Dpth of shul low = shul low s optml. Q. Dos thr lwys xst shul qul to pth of ntrvls?,, ll ontn 9:0 9 9:0 0 0:0 :0 :0 :0 :0 f g h :0 :0 Tm Gry lgorthm. onsr lturs n nrsng orr of strt tm: ssgn ltur to ny omptl lssroom. 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 + + } Implmntton. O(n log n). For h lssroom k, mntn th fnsh tm of th lst o. Kp th lssrooms n prorty quu. opyrght 000, Kvn Wyn

Intrvl Prttonng: Gry Anlyss Osrvton. Gry lgorthm nvr shuls two nomptl lturs n th sm lssroom.. hortst Pths n Grph Thorm. Gry lgorthm s optml. Pf. Lt = numr of lssrooms tht th gry lgorthm llots.

Thus, w hv lturs ovrlppng t tm s +. Ky osrvton ll shuls us lssrooms. shortst rout from Wng Hll to Mm Bh hortst Pth Prolm Dkstr's Algorthm hortst pth ntwork. Drt grph G = (V, E). our s, stnton t.

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 ( v) mn ( u), ( u, v) : u v to, n st (v) = (v).

5 Intrvl Prttonng: Gry Anlyss Osrvton. Gry lgorthm nvr shuls two nomptl lturs n th sm lssroom.. hortst Pths n Grph Thorm. Gry lgorthm s optml. Pf. Lt = numr of lssrooms tht th gry lgorthm llots. lssroom s opn us w n to shul o, sy, tht s nomptl wth ll - othr lssrooms. Ths os (nlung ) h n ftr s. 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 ll shuls us lssrooms. shortst rout from Wng Hll to Mm Bh hortst Pth Prolm Dkstr's Algorthm 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 s----t = = 0. ost of pth = sum of g osts n pth 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 ( v) mn ( u), ( u, v) : u v to, n st (v) = (v). shortst pth to som u n xplor prt, follow y sngl g (u, v) s s (u) u v 7 t 7 8 Dkstr's Algorthm Dkstr's Algorthm: Proof of orrtnss 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 ( v) mn ( u), ( u, v) : u v to, n st (v) = (v). shortst pth to som u n xplor prt, follow y sngl g (u, v) Invrnt. For h no u, (u) s th lngth of th shortst s-u pth. Pf. (y nuton on ) Bs s: = s trvl. Inutv hypothss: Assum tru for = k. 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 (v). onsr ny s-v pth P. W'll s tht t's no shortr thn (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' x P y s (u) u v s (P) (P') + (x,y) (x) + (x, y) (y) (v) u v nonngtv wghts nutv hypothss fn of (y) Dkstr hos v nst of y 9 0 opyrght 000, Kvn Wyn

6 Dkstr's Algorthm: Implmntton For h unxplor no, xpltly mntn (v) mn (u). (u,v):u Extr ls Nxt no to xplor = no wth mnmum (v). Whn xplorng v, for h nnt g = (v, w), upt (w) mn { (w), (v) }. Effnt mplmntton. Mntn prorty quu of unxplor nos, prortz y (v). ply Prorty Quu PQ Oprton Dkstr Arry Bnry hp -wy Hp F hp Insrt ExtrtMn hngky n n m n n log n log n log n log n log n log n log n IsEmpty n Totl n m log n m log m/n n m + n log n Invul ops r mortz ouns hulng to Mnmzng Ltnss. hulng to Mnmz 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. Ex: t ltnss = ltnss = 0 mx ltnss = = 9 = 8 = = = = Mnmzng Ltnss: Gry Algorthms Mnmzng Ltnss: Gry Algorthms Gry tmplt. onsr os n som orr. Gry tmplt. onsr os n som orr. [hortst prossng tm frst] onsr os n snng orr of prossng tm t. [hortst prossng tm frst] onsr os n snng orr of prossng tm t. [Erlst ln frst] onsr os n snng orr of ln. t ountrxmpl [mllst slk] onsr os n snng orr of slk -t. [mllst slk] onsr os n snng orr of slk -t. t 0 0 ountrxmpl opyrght 000, Kvn Wyn

7 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 = = 0 = t 0 for = to n Assgn o to ntrvl [t, t + t ] s t, f t + t t t + t output ntrvls [s, f ] = = = Osrvton. Th gry shul hs no l tm. mx ltnss = = = 8 = 9 = 9 = = Mnmzng Ltnss: Invrsons Mnmzng Ltnss: Invrsons Df. Gvn shul, n nvrson s pr of os n suh tht: < ut shul for. nvrson f Df. Gvn shul, n nvrson s pr of os n suh tht: < ut shul for. nvrson f for swp for swp [ s for, w ssum os r numr so tht n ] ftr swp f' 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. lm. wppng two onsutv, 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, ' If o s lt: f f (fnton) ( fnshs t tm f ) f ( ) (fnton) 9 0 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. n 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. truturl. Dsovr smpl "struturl" oun ssrtng tht vry possl soluton must hv rtn vlu. Thn show tht your lgorthm lwys hvs ths oun. Exhng rgumnt. Grully trnsform ny soluton to th on foun y th gry lgorthm wthout hurtng ts qulty. Othr gry lgorthms. Kruskl, Prm, Dkstr, Huffmn, opyrght 000, Kvn Wyn 7

8 Optml Offln hng. Optml hng hng. h wth pty to stor k tms. qun of m tm rqusts,,, m. h ht: tm lry n h whn rqust. h 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 =, r = h mss rqusts:,,,,,,,. Optml vton shul: h msss. rqusts h Optml Offln hng: 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 futur qurs: g f f g h... h mss t ths on Thorm. [Blly, 90s] FF s optml vton shul. Pf. Algorthm n thorm r ntutv; proof s sutl. Intuton. n trnsform n unru shul nto ru on wth no mor h msss. x x n unru shul ru shul Ru Evton huls Frthst-In-Futur: Anlyss lm. Gvn ny unru shul, n trnsform t nto ru shul ' wth no mor h msss. osn't ntr h t rqust Pf. (y nuton on numr of unru tms) tm uppos rngs nto th h t tm t, wthout rqust. Lt th tm vts whn t rngs nto th h. s : vt t tm t', for nxt rqust for. s : rqust t tm t' for s vt. t t' t t' vt t tm t', for nxt rqust ' t t' 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. onsr (+) st rqust = +. n n FF hv gr up untl now, thy hv th sm h ontnts for rqust +. s : ( s lry n th h). ' = stsfs nvrnt. s : ( s not n th h n n FF vt th sm lmnt). ' = stsfs nvrnt. s s 7 8 opyrght 000, Kvn Wyn 8

9 Frthst-In-Futur: Anlyss Frthst-In-Futur: Anlyss Pf. (ontnu) s : ( s not n th h; FF vts ; vts f ). 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 ' s : g =. n't hppn wth Frthst-In-Futur sn thr must rqust for f for. s : 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 ', ' 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 hng Prsptv Lt ' th frst tm ftr + tht n ' tk ffrnt ton, n lt g tm rqust t tm '. must nvolv or f (or oth) ' s : g, f. must vt. Mk ' vt f; now n ' hv th sm h. ' sm sm f ' othrws ' woul tk th sm ton sm g sm g ' Onln vs. offln lgorthms. Offln: full squn of rqusts s known pror. Onln (rlty): rqusts r not known n vn. hng s mong most funmntl onln prolms n. LIFO. Evt pg rought n most rntly. LRU. Evt pg whos most rnt ss ws rlst. Thorm. FF s optml offln vton lgorthm. Provs ss for unrstnng n nlyzng onln lgorthms. LRU s k-ompttv. [ton.8] LIFO s rtrrly. FF wth rton of tm rvrs! on hngng on hngng Gol. Gvn urrny nomntons:,, 0,, 00, vs mtho to py mount to ustomr usng fwst numr of ons. Ex:. Gr s goo. Gr s rght. Gr works. Gr lrfs, uts through, n pturs th ssn of th volutonry sprt. - Goron Gko (Mhl Dougls) shr's lgorthm. At h trton, on of th lrgst vlu tht os not tk us pst th mount to p. Ex: $.89. opyrght 000, Kvn Wyn 9

10 on-hngng: Gry Algorthm on-hngng: Anlyss of Gry Algorthm shr'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 whl (x 0) { lt k lrgst ntgr suh tht k x f (k = 0) rturn "no soluton foun" x x - k {k} } rturn Q. Is shr's lgorthm optml? Thorm. Gr s optml for U.. ong:,, 0,, 00. Pf. (y nuton on x) onsr 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 All optml solutons k must stsfy P N 0 N + D 00 Q no lmt Mx vlu of ons,,, k- n ny OPT - + = = 7 + = 99 on-hngng: Anlyss of Gry Algorthm Osrvton. Gry lgorthm s su-optml for U postl nomntons:, 0,,, 70, 00, 0,, 00. ltng Brkponts ountrxmpl. 0. Gry: 00,,,,,,,. 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 =. Gol: mks s fw rfulng stops s possl. Gry lgorthm. Go s fr s you n for rfulng. Prnton Plo Alto Truk rvr's lgorthm. ort rkponts so tht: 0 = 0 < < <... < n = L {0} rkponts slt x 0 urrnt loton whl (x n ) lt p lrgst ntgr suh tht p x + f ( p = x) rturn "no soluton" x p {p} rturn 7 Implmntton. O(n log n) Us nry srh to slt h rkpont p. 9 0 opyrght 000, Kvn Wyn 0

ltng Brkponts: orrtnss ltng Brkponts: orrtnss Thorm. Gry lgorthm s optml. 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 <.

11 ltng Brkponts: orrtnss ltng Brkponts: orrtnss Thorm. Gry lgorthm s optml. 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. 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 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 OBOL 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. opyrght 000, Kvn Wyn

Copyright 2000, Kevin Wayne 1

Copyright 2000, Kevin Wayne 1 Chptr. Intrvl Shulng Gry Algorthms Sls y Kvn Wyn. Copyrght 005 Prson-Ason Wsly. All rghts rsrv. Intrvl Shulng Intrvl Shulng: Gry Algorthms Intrvl shulng. Jo strts t s n fnshs t f. Two os omptl f thy on't