4.1 Interval Scheduling. Chapter 4. Greedy Algorithms. Interval Scheduling: Greedy Algorithms. Interval Scheduling. Interval scheduling.

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1 Cptr 4 4 Intrvl Suln Gry Alortms Sls y Kvn Wyn Copyrt 005 Prson-Ason Wsly All rts rsrv Intrvl Suln Intrvl Suln: Gry Alortms Intrvl suln! Jo strts t s n nss t! Two os omptl ty on't ovrlp! Gol: n mxmum sust o mutully omptl os Gry tmplt Consr os n som orr Tk o prov t's omptl wt t ons lry tkn! [Erlst strt tm] Consr os n snn orr o strt tm s! [Erlst ns tm] Consr os n snn orr o ns tm! [Sortst ntrvl] Consr os n snn orr o ntrvl lnt - s! [Fwst onlts] For o, ount t numr o onltn os Sul n snn orr o onlts Tm 4

2 Intrvl Suln: Gry Alortms Intrvl Suln: Gry Alortm Gry tmplt Consr os n som orr Tk o prov t's omptl wt t ons lry tkn Gry lortm Consr os n nrsn orr o ns tm Tk o prov t's omptl wt t ons lry tkn rks rlst strt tm rks sortst ntrvl rks wst onlts Sort os y ns tms so tt " " " n os slt A # $ or = to n { (o omptl wt A) A # A % {} } rturn A Implmntton O(n lo n)! Rmmr o * tt ws lst to A! Jo s omptl wt A s! * 5 6 Intrvl Suln: Anlyss Intrvl Suln: Anlyss Torm Gry lortm s optml Torm Gry lortm s optml P (y ontrton)! Assum ry s not optml, n lt's s wt ppns! Lt,, k not st o os slt y ry! Lt,, m not st o os n t optml soluton wt =, =,, r = r or t lrst possl vlu o r P (y ontrton)! Assum ry s not optml, n lt's s wt ppns! Lt,, k not st o os slt y ry! Lt,, m not st o os n t optml soluton wt =, =,, r = r or t lrst possl vlu o r o r+ nss or r+ o r+ nss or r+ Gry: r r+ Gry: r r+ OPT: r r+ OPT: r r+ wy not rpl o r+ wt o r+? soluton stll sl n optml, ut ontrts mxmlty o r 7 8

3 Intrvl Prttonn 4 Intrvl Prttonn Intrvl prttonn! Ltur strts t s n nss t! Gol: n mnmum numr o lssrooms to sul ll lturs so tt no two our t t sm tm n t sm room Ex: Ts sul uss 4 lssrooms to sul 0 lturs 9 9:0 0 0:0 :0 :0 :0 :0 :0 4 4:0 Tm 0 Intrvl Prttonn Intrvl Prttonn: Lowr Boun on Optml Soluton Intrvl prttonn! Ltur strts t s n nss t! Gol: n mnmum numr o lssrooms to sul ll lturs so tt no two our t t sm tm n t sm room D T pt o st o opn ntrvls s t mxmum numr tt ontn ny vn tm Ky osrvton Numr o lssrooms n! pt Ex: Ts sul uss only Ex: Dpt o sul low = & sul low s optml,, ll ontn 9:0 Q Dos tr lwys xst sul qul to pt o ntrvls? 9 9:0 0 0:0 :0 :0 :0 :0 :0 4 4:0 Tm 9 9:0 0 0:0 :0 :0 :0 :0 :0 4 4:0 Tm

4 Intrvl Prttonn: Gry Alortm Intrvl Prttonn: Gry Anlyss Gry lortm Consr lturs n nrsn orr o strt tm: ssn ltur to ny omptl lssroom Osrvton Gry lortm nvr suls two nomptl lturs n t sm lssroom Sort ntrvls y strtn tm so tt s " s " " s n # 0 numr o llot lssrooms or = to n { (ltur s omptl wt som lssroom k) sul ltur n lssroom k ls llot nw lssroom + sul ltur n lssroom + # + } Torm Gry lortm s optml P! Lt = numr o lssrooms tt t ry lortm llots! Clssroom s opn us w n to sul o, sy, tt s nomptl wt ll - otr lssrooms! Sn w sort y strt tm, ll ts nomptlts r us y lturs tt strt no ltr tn s! Tus, w v lturs ovrlppn t tm s + '! Ky osrvton & ll suls us! lssrooms! Implmntton O(n lo n)! For lssroom k, mntn t ns tm o t lst o! Kp t lssrooms n prorty quu 4 Suln to Mnmzn Ltnss 4 Suln to Mnmz Ltnss Mnmzn ltnss prolm! Snl rsour prosss on o t tm! Jo rqurs t unts o prossn tm n s u t tm! I strts t tm s, t nss t tm = s + t! Ltnss: l = mx { 0, - }! Gol: sul ll os to mnmz mxmum ltnss L = mx l Ex: t ltnss = ltnss = 0 mx ltnss = 6 = 9 = 8 6 = 5 = 6 5 = 4 4 =

5 Mnmzn Ltnss: Gry Alortms Mnmzn Ltnss: Gry Alortms Gry tmplt Consr os n som orr Gry tmplt Consr os n som orr! [Sortst prossn tm rst] Consr os n snn orr o prossn tm t! [Sortst prossn tm rst] Consr os n snn orr o prossn tm t! [Erlst ln rst] Consr os n snn orr o ln t ountrxmpl! [Smllst slk] Consr os n snn orr o slk - t! [Smllst slk] Consr os n snn orr o slk - t t 0 0 ountrxmpl 7 8 Mnmzn Ltnss: Gry Alortm Mnmzn Ltnss: No Il Tm Gry lortm Erlst ln rst Osrvton Tr xsts n optml sul wt no l tm Sort n os y ln so tt " " " n = 4 = = t # 0 or = to n Assn o to ntrvl [t, t + t ] s # t, # t + t t # t + t output ntrvls [s, ] = 4 = 6 = Osrvton T ry sul s no l tm mx ltnss = = 6 = 8 = 9 4 = 9 5 = 4 6 =

6 Mnmzn Ltnss: Invrsons Mnmzn Ltnss: Invrsons D An nvrson n sul S s pr o os n su tt: < ut sul or nvrson D An nvrson n sul S s pr o os n su tt: < ut sul or nvrson or swp or swp tr swp Osrvton Gry sul s no nvrsons Osrvton I sul (wt no l tm) s n nvrson, t s on wt pr o nvrt os sul onsutvly Clm Swppn two nt, nvrt os rus t numr o nvrsons y on n os not nrs t mx ltnss P Lt l t ltnss or t swp, n lt l ' t trwrs! l' k = l k or ll k (, '! l' " l! I o s lt: l# = =!! # " " " l (nton) ( nss t tm ) ( < ) (nton) Mnmzn Ltnss: Anlyss o Gry Alortm Gry Anlyss Strts Torm Gry sul S s optml P Dn S* to n optml sul tt s t wst numr o nvrsons, n lt's s wt ppns! Cn ssum S* s no l tm! I S* s no nvrsons, tn S = S*! I S* s n nvrson, lt - n nt nvrson swppn n os not nrs t mxmum ltnss n strtly rss t numr o nvrsons ts ontrts nton o S*! Gry lortm stys Sow tt tr stp o t ry lortm, ts soluton s t lst s oo s ny otr lortm's Exn rumnt Grully trnsorm ny soluton to t on oun y t ry lortm wtout urtn ts qulty Struturl Dsovr smpl "struturl" oun ssrtn tt vry possl soluton must v rtn vlu Tn sow tt your lortm lwys vs ts oun 4