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

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1 hptr Intrvl huling Gry Algorithms lis y Kvin Wyn. opyright 00 Prson-Aison Wsly. All rights rsrv. Th originl vrsion n offiil vrsions r t Intrvl huling Intrvl huling: Gry Algorithms Intrvl shuling. Jo j strts t s j n finishs t f j. Two jos omptil if thy on't ovrlp. Gol: fin mximum sust of mutully omptil jos. Gry tmplt. onsir jos in som nturl orr. Tk h jo provi it's omptil with th ons lry tkn. [Erlist strt tim] onsir jos in sning orr of s j. [Erlist finish tim] onsir jos in sning orr of f j. [hortst intrvl] onsir jos in sning orr of f j - s j. [Fwst onflits] For h jo j, ount th numr of onfliting jos j. hul in sning orr of j. f g h Tim

2 Intrvl huling: Gry Algorithms Intrvl huling: Gry Algorithm Gry tmplt. onsir jos in som nturl orr. Tk h jo provi it's omptil with th ons lry tkn. Gry lgorithm. onsir jos in inrsing orr of finish tim. Tk h jo provi it's omptil with th ons lry tkn. ountrxmpl for rlist strt tim ountrxmpl for shortst intrvl ountrxmpl for fwst onflits ort jos y finish tims so tht f f... f n. st of jos slt A for j = to n { if (jo j omptil with A) A A {j} } rturn A Implmnttion. O(n log n). Rmmr jo j* tht ws lst to A. Jo j is omptil with A if s j f j*. 6 Intrvl huling: Anlysis Intrvl huling: Anlysis Thorm. Gry lgorithm is optiml. Thorm. Gry lgorithm is optiml. Pf. (y ontrition) Assum gry is not optiml, n lt's s wht hppns. Lt i, i,... i k not st of jos slt y gry. Lt j, j,... j m not st of jos in th optiml solution with i = j, i = j,..., i r = j r for th lrgst possil vlu of r. Pf. (y ontrition) Assum gry is not optiml, n lt's s wht hppns. Lt i, i,... i k not st of jos slt y gry. Lt j, j,... j m not st of jos in th optiml solution with i = j, i = j,..., i r = j r for th lrgst possil vlu of r. jo i r+ finishs for j r+ jo i r+ finishs for j r+ Gry: i i i r i r+ Gry: i i i r i r+ OPT: j j j r j r+... OPT: j j j r i r+... why not rpl jo j r+ with jo i r+? solution still fsil n optiml, ut ontrits mximlity of r. 7 8

3 Intrvl Prtitioning Intrvl Prtitioning Intrvl prtitioning. Ltur j strts t s j n finishs t f j. Gol: fin minimum numr of lssrooms to shul ll lturs so tht no two our t th sm tim in th sm room. Ex: This shul uss lssrooms to shul 0 lturs. j g h f i 9 9:0 0 0:0 :0 :0 :0 :0 :0 :0 Tim 0 Intrvl Prtitioning Intrvl Prtitioning: Lowr Boun on Optiml olution Intrvl prtitioning. Ltur j strts t s j n finishs t f j. Gol: fin minimum numr of lssrooms to shul ll lturs so tht no two our t th sm tim in th sm room. Df. Th pth of st of opn intrvls is th mximum numr tht ontin ny givn tim. Ky osrvtion. Numr of lssrooms n pth. Ex: This shul uss only. Ex: Dpth of shul low = shul low is optiml.,, ll ontin 9:0 Q. Dos thr lwys xist shul qul to pth of intrvls? f j f j g i g i h h 9 9:0 0 0:0 :0 :0 :0 :0 :0 :0 Tim 9 9:0 0 0:0 :0 :0 :0 :0 :0 :0 Tim

4 Intrvl Prtitioning: Gry Algorithm Intrvl Prtitioning: Gry Anlysis Gry lgorithm. onsir lturs in inrsing orr of strt tim: ssign ltur to ny omptil lssroom. ort intrvls y strting tim so tht s s... s n. 0 numr of llot lssrooms for j = to n { if (ltur j is omptil with som lssroom k) shul ltur j in lssroom k ls llot nw lssroom + shul ltur j in lssroom + + } Implmnttion. O(n log n). For h lssroom k, mintin th finish tim of th lst jo. Kp th lssrooms in priority quu. Osrvtion. Gry lgorithm nvr shuls two inomptil lturs in th sm lssroom. Thorm. Gry lgorithm is optiml. Pf. Lt = numr of lssrooms tht th gry lgorithm llots. lssroom is opn us w n to shul jo, sy j, tht is inomptil with ll - othr lssrooms. Ths jos h n ftr s j. in w sort y strt tim, ll ths inomptiilitis r us y lturs tht strt no ltr thn s j. Thus, w hv lturs ovrlpping t tim s j +. Ky osrvtion ll shuls us lssrooms. huling to Minimizing Ltnss huling to Minimiz Ltnss Minimizing ltnss prolm. ingl rsour prosss on jo t tim. Jo j rquirs t j units of prossing tim n is u t tim j. If j strts t tim s j, it finishs t tim f j = s j + t j. Ltnss: j = mx { 0, f j - j }. Gol: shul ll jos to minimiz mximum ltnss L = mx j. Ex: t j 6 j ltnss = ltnss = 0 mx ltnss = 6 = 9 = 8 6 = = 6 = =

5 Minimizing Ltnss: Gry Algorithms Minimizing Ltnss: Gry Algorithms Gry tmplt. onsir jos in som orr. Gry tmplt. onsir jos in som orr. [hortst prossing tim first] onsir jos in sning orr of prossing tim t j. [hortst prossing tim first] onsir jos in sning orr of prossing tim t j. [Erlist lin first] onsir jos in sning orr of lin j. t j 00 j 0 0 ountrxmpl [mllst slk] onsir jos in sning orr of slk j - t j. [mllst slk] onsir jos in sning orr of slk j - t j. t j j 0 0 ountrxmpl 7 8 Minimizing Ltnss: Gry Algorithm Minimizing Ltnss: No Il Tim Gry lgorithm. Erlist lin first. Osrvtion. Thr xists n optiml shul with no il tim. ort n jos y lin so tht n = = = t 0 for j = to n Assign jo j to intrvl [t, t + t j ] s j t, f j t + t j t t + t j output intrvls [s j, f j ] = = 6 = Osrvtion. Th gry shul hs no il tim. mx ltnss = = 6 = 8 = 9 = 9 = 6 =

6 Minimizing Ltnss: Invrsions Minimizing Ltnss: Invrsions Df. Givn shul, n invrsion is pir of jos i n j suh tht: i < j ut j shul for i. invrsion f i Df. Givn shul, n invrsion is pir of jos i n j suh tht: i < j ut j shul for i. invrsion f i for swp j i for swp j i [ s for, w ssum jos r numr so tht n ] ftr swp i j f' j Osrvtion. Gry shul hs no invrsions. Osrvtion. If shul (with no il tim) hs n invrsion, it hs on with pir of invrt jos shul onsutivly. lim. wpping two onsutiv, invrt jos rus th numr of invrsions y on n os not inrs th mx ltnss. Pf. Lt th ltnss for th swp, n lt ' it ftrwrs. ' k = k for ll k i, j ' i i If jo j is lt: j f j j f i j (finition) (j finishs t tim f i ) f i i (i j) i (finition) Minimizing Ltnss: Anlysis of Gry Algorithm Gry Anlysis trtgis Thorm. Gry shul is optiml. Pf. Dfin * to n optiml shul tht hs th fwst numr of invrsions, n lt's s wht hppns. n ssum * hs no il tim. If * hs no invrsions, thn = *. If * hs n invrsion, lt i-j n jnt invrsion. swpping i n j os not inrs th mximum ltnss n stritly rss th numr of invrsions this ontrits finition of * Gry lgorithm stys h. how tht ftr h stp of th gry lgorithm, its solution is t lst s goo s ny othr lgorithm's. truturl. Disovr simpl "struturl" oun ssrting tht vry possil solution must hv rtin vlu. Thn show tht your lgorithm lwys hivs this oun. Exhng rgumnt. Grully trnsform ny solution to th on foun y th gry lgorithm without hurting its qulity. Othr gry lgorithms. Kruskl, Prim, Dijkstr, Huffmn,

7 Optiml Offlin hing Optiml hing hing. h with pity to stor k itms. qun of m itm rqusts,,, m. h hit: itm lry in h whn rqust. h miss: itm not lry in h whn rqust: must ring rqust itm into h, n vit som xisting itm, if full. Gol. Evition shul tht minimizs numr of h misss. Ex: k =, initil h =, rqusts:,,,,,,,. Optiml vition shul: h misss. r = h miss rqusts h 6 Optiml Offlin hing: Frthst-In-Futur Ru Evition huls Frthst-in-futur. Evit itm in th h tht is not rqust until frthst in th futur. Df. A ru shul is shul tht only insrts n itm into th h in stp in whih tht itm is rqust. urrnt h: f Intuition. n trnsform n unru shul into ru on with no mor h misss. futur quris: g f f g h... h miss jt this on Thorm. [Blly, 960s] FF is optiml vition shul. Pf. Algorithm n thorm r intuitiv; proof is sutl. x x n unru shul ru shul 7 8

8 Ru Evition huls Frthst-In-Futur: Anlysis lim. Givn ny unru shul, n trnsform it into ru shul ' with no mor h misss. Pf. (y inution on numr of unru itms) uppos rings into th h t tim t, without rqust. Lt th itm vits whn it rings into th h. s : vit t tim t', for nxt rqust for. s : rqust t tim t' for is vit. t t' t t' vit t tim t', for nxt rqust ' t t' osn't ntr h t rqust tim t t' rqust t tim t' ' Thorm. FF is optiml vition lgorithm. Pf. (y inution on numr or rqusts j) Invrint: Thr xists n optiml ru shul tht mks th sm vition shul s FF through th first j+ rqusts. Lt ru shul tht stisfis invrint through j rqusts. W prou ' tht stisfis invrint ftr j+ rqusts. onsir (j+) st rqust = j+. in n FF hv gr up until now, thy hv th sm h ontnts for rqust j+. s : ( is lry in th h). ' = stisfis invrint. s : ( is not in th h n n FF vit th sm lmnt). ' = stisfis invrint. s s 9 0 Frthst-In-Futur: Anlysis Frthst-In-Futur: Anlysis Pf. (ontinu) s : ( is not in th h; FF vits ; vits f ). gin onstrution of ' from y viting inst of f j sm f sm f j+ j sm sm f now ' grs with FF on first j+ rqusts; w show tht hving lmnt f in h is no wors thn hving lmnt ' ' Lt j' th first tim ftr j+ tht n ' tk iffrnt tion, n lt g itm rqust t tim j'. j' s : g =. n't hppn with Frthst-In-Futur sin thr must rqust for f for. s : g = f. Elmnt f n't in h of, so lt ' th lmnt tht vits. if ' =, ' sss f from h; now n ' hv sm h if ', ' vits ' n rings into th h; now n ' hv th sm h sm sm f must involv or f (or oth) ' Not: ' is no longr ru, ut n trnsform into ru shul tht grs with FF through stp j+

9 Frthst-In-Futur: Anlysis hing Prsptiv Lt j' th first tim ftr j+ tht n ' tk iffrnt tion, n lt g itm rqust t tim j'. j' sm sm f must involv or f (or oth) Onlin vs. offlin lgorithms. Offlin: full squn of rqusts is known priori. Onlin (rlity): rqusts r not known in vn. hing is mong most funmntl onlin prolms in. ' s : g, f. must vit. Mk ' vit f; now n ' hv th sm h. j' othrwis ' woul tk th sm tion sm g sm g ' LIFO. Evit pg rought in most rntly. LRU. Evit pg whos most rnt ss ws rlist. Thorm. FF is optiml offlin vition lgorithm. Provis sis for unrstning n nlyzing onlin lgorithms. LRU is k-omptitiv. [tion.8] LIFO is ritrrily. FF with irtion of tim rvrs! lting Brkpoints lting Brkpoints lting rkpoints. Ro trip from Printon to Plo Alto long fix rout. Rfuling sttions t rtin points long th wy. Ful pity =. Gol: mks s fw rfuling stops s possil. Gry lgorithm. Go s fr s you n for rfuling. Printon Plo Alto 6 7 6

10 lting Brkpoints: Gry Algorithm lting Brkpoints: orrtnss Truk rivr's lgorithm. Thorm. Gry lgorithm is optiml. ort rkpoints so tht: 0 = 0 < < <... < n = L {0} x 0 rkpoints slt urrnt lotion whil (x n ) lt p lrgst intgr suh tht p x + if ( p = x) rturn "no solution" x p {p} rturn Gry: Pf. (y ontrition) Assum gry is not optiml, n lt's s wht hppns. Lt 0 = g 0 < g <... < g p = L not st of rkpoints hosn y gry. Lt 0 = f 0 < f <... < f q = L not st of rkpoints in n optiml solution with f 0 = g 0, f = g,..., f r = g r for lrgst possil vlu of r. Not: g r+ > f r+ y gry hoi of lgorithm. g 0 g g g r g r+ Implmnttion. O(n log n) Us inry srh to slt h rkpoint p. OPT: f 0 f f f r f q why osn't optiml solution riv littl furthr? f r lting Brkpoints: orrtnss Thorm. Gry lgorithm is optiml. Pf. (y ontrition) Assum gry is not optiml, n lt's s wht hppns. Lt 0 = g 0 < g <... < g p = L not st of rkpoints hosn y gry. Lt 0 = f 0 < f <... < f q = L not st of rkpoints in n optiml solution with f 0 = g 0, f = g,..., f r = g r for lrgst possil vlu of r. Not: g r+ > f r+ y gry hoi of lgorithm. oin hnging Gry: g 0 g g g r g r+ OPT:... f 0 f f f r f q nothr optiml solution hs on mor rkpoint in ommon ontrition 9

11 oin hnging oin-hnging: Gry Algorithm Gol. Givn urrny nomintions:,, 0,, 00, vis mtho to py mount to ustomr using fwst numr of oins. shir's lgorithm. At h itrtion, oin of th lrgst vlu tht os not tk us pst th mount to pi. Ex:. ort oins nomintions y vlu: < < < n. oins slt shir's lgorithm. At h itrtion, oin of th lrgst vlu tht os not tk us pst th mount to pi. Ex: $.89. whil (x 0) { lt k lrgst intgr suh tht k x if (k = 0) rturn "no solution foun" x x - k {k} } rturn Q. Is shir's lgorithm optiml? Proprtis of optiml solution oin-hnging: Anlysis of Gry Algorithm Proprty. Pf. Rpl pnnis with nikl. Proprty. Proprty. Proprty. Pf. Rpl ims n 0 nikls with qurtr n nikl; Rpl ims n nikl with qurtr. Rll: t most nikl. pnny= nikl= im=0 qurtr= Thorm. Gry lgorithm is optiml for U.. oing:,, 0,, 00. Pf. (y inution on x) onsir optiml wy to hng k x < k+ : gry tks oin k. W lim tht ny optiml solution must lso tk oin k. if not, it ns nough oins of typ,, k- to up to x tl low inits no optiml solution n o this Prolm rus to oin-hnging x - k nts, whih, y inution, is optimlly solv y gry lgorithm. k k All optiml solutions must stisfy P Mx vlu of oins,,, k- in ny OPT - N 0 N + D + = 9 Q 0 + = 00 no limit 7 + = 99

12 Is shir's lgorithm for ny st of nomintions? Osrvtion. Gry lgorithm is su-optiml for U postl nomintions:, 0,,, 70, 00, 0,, 00. Minimum pnning Tr ountrxmpl. 0. Gry: 00,,,,,,,. Optiml: 70, 70. Osrvtion. It my not vn l to fsil solution if > : 7, 8, 9. shir's lgorithm: = 9 +???. Optiml: = Minimum pnning Tr Applitions Minimum spnning tr. Givn onnt grph G = (V, E) with rlvlu g wights, n MT is sust of th gs T E suh tht T is spnning tr whos sum of g wights is minimiz. MT is funmntl prolm with ivrs pplitions. Ntwork sign. tlphon, ltril, hyruli, TV l, omputr, ro Approximtion lgorithms for NP-hr prolms. trvling slsprson prolm, tinr tr G = (V, E) T, T = Inirt pplitions. mx ottlnk pths LDP os for rror orrtion img rgistrtion with Rnyi ntropy lrning slint fturs for rl-tim f vrifition ruing t storg in squning mino is in protin mol lolity of prtil intrtions in turulnt flui flows utoonfig protool for Ethrnt riging to voi yls in ntwork yly's Thorm. Thr r n n- spnning trs of K n. lustr nlysis. n't solv y rut for 7 8

13 Gry Algorithms Gry Algorithms Kruskl's lgorithm. trt with T =. onsir gs in sning orr of ost. Insrt g in T unlss oing so woul rt yl. Rvrs-Dlt lgorithm. trt with T = E. onsir gs in sning orr of ost. Dlt g from T unlss oing so woul isonnt T. Prim's lgorithm. trt with som root no s n grily grow tr T from s outwr. At h stp, th hpst g to T tht hs xtly on npoint in T. implifying ssumption. All g osts r istint. ut proprty. Lt ny sust of nos, n lt th min ost g with xtly on npoint in. Thn th MT ontins. yl proprty. Lt ny yl, n lt f th mx ost g longing to. Thn th MT os not ontin f. f Rmrk. All thr lgorithms prou n MT. is in th MT f is not in th MT 9 0 yls n uts yl-ut Intrstion yl. t of gs th form -, -, -,, y-z, z-. lim. A yl n utst intrst in n vn numr of gs. 6 yl = -, -, -, -, -6, 6-6 yl = -, -, -, -, -6, 6- utst D = -, -, -6, -7, 7-8 Intrstion = -, utst. A ut is sust of nos. Th orrsponing utst D is th sust of gs with xtly on npoint in. 6 ut = {,, 8 } utst D = -6, -7, -, -, 7-8 Pf. (y pitur) V - 7 8

14 Gry Algorithms Gry Algorithms implifying ssumption. All g osts r istint. implifying ssumption. All g osts r istint. ut proprty. Lt ny sust of nos, n lt th min ost g with xtly on npoint in. Thn th MT T* ontins. yl proprty. Lt ny yl in G, n lt f th mx ost g longing to. Thn th MT T* os not ontin f. Pf. (xhng rgumnt) uppos os not long to T*, n lt's s wht hppns. Aing to T* rts yl in T*. Eg is oth in th yl n in th utst D orrsponing to thr xists nothr g, sy f, tht is in oth n D. T' = T* { } - { f } is lso spnning tr. in < f, ost(t') < ost(t*). This is ontrition. f Pf. (xhng rgumnt) uppos f longs to T*, n lt's s wht hppns. Dlting f from T* rts ut in T*. Eg f is oth in th yl n in th utst D orrsponing to thr xists nothr g, sy, tht is in oth n D. T' = T* { } - { f } is lso spnning tr. in < f, ost(t') < ost(t*). This is ontrition. f T* T* Prim's Algorithm: Proof of orrtnss Implmnttion: Prim's Algorithm Prim's lgorithm. [Jrník 90, Dijkstr 97, Prim 99] Initiliz = ny no. Apply ut proprty to. A min ost g in utst orrsponing to to T, n on nw xplor no u to. Implmnttion. Us priority quu. Mintin st of xplor nos. For h unxplor no v, mintin tthmnt ost [v] = ost of hpst g v to no in. O(n ) with n rry; O(m log n) with inry hp. Prim(G, ) { forh (v V) [v] Initiliz n mpty priority quu Q forh (v V) insrt v onto Q Initiliz st of xplor nos } whil (Q is not mpty) { u lt min lmnt from Q { u } forh (g = (u, v) inint to u) if ((v ) n ( < [v])) rs priority [v] to 6

15 Kruskl's Algorithm: Proof of orrtnss Implmnttion: Kruskl's Algorithm Kruskl's lgorithm. [Kruskl, 96] onsir gs in sning orr of wight. s : If ing to T rts yl, isr oring to yl proprty. s : Othrwis, insrt = (u, v) into T oring to ut proprty whr = st of nos in u's onnt omponnt. Implmnttion. Us th union-fin t strutur. Buil st T of gs in th MT. Mintin st for h onnt omponnt. O(m log n) for sorting n O(m (m, n)) for union-fin. m n log m is O(log n) ssntilly onstnt Kruskl(G, ) { ort gs wights so tht... m. T forh (u V) mk st ontining singlton u s v u s } for i = to m (u,v) = i if (u n v r in iffrnt sts) { T T { i } mrg th sts ontining u n v } mrg two omponnts rturn T r u n v in iffrnt onnt omponnts? 7 8 Lxiogrphi Tirking MT Algorithms: Thory To rmov th ssumption tht ll g osts r istint: prtur ll g osts y tiny mounts to rk ny tis. Impt. Kruskl n Prim only intrt with osts vi pirwis omprisons. If prturtions r suffiintly smll, MT with prtur osts is MT with originl osts..g., if ll g osts r intgrs, prturing ost of g i y i / n Implmnttion. n hnl ritrrily smll prturtions impliitly y rking tis lxiogrphilly, oring to inx. ooln lss(i, j) { if (ost( i ) < ost( j )) rturn tru ls if (ost( i ) > ost( j )) rturn fls ls if (i < j) rturn tru ls rturn fls } Dtrministi omprison s lgorithms. O(m log n) [Jrník, Prim, Dijkstr, Kruskl, Boruvk] O(m log log n). [hriton-trjn 976, Yo 97] O(m (m, n)). [Frmn-Trjn 987] O(m log (m, n)). [Gow-Glil-pnr-Trjn 986] O(m (m, n)). [hzll 000] Holy gril. O(m). Notl. O(m) rnomiz. [Krgr-Klin-Trjn 99] O(m) vrifition. [Dixon-Ruh-Trjn 99] Eulin. -: O(n log n). omput MT of gs in Dluny k-: O(k n ). ns Prim 9 60

16 lustring lustring lustring. Givn st U of n ojts ll p,, p n, lssify into ohrnt groups. photos, oumnts. miro-orgnisms Distn funtion. Numri vlu spifying "losnss" of two ojts. numr of orrsponing pixls whos intnsitis iffr y som thrshol Funmntl prolm. Divi into lustrs so tht points in iffrnt lustrs r fr prt. Routing in moil ho ntworks. Intify pttrns in gn xprssion. Doumnt tgoriztion for w srh. imilrity srhing in mil img tss kyt: lustr 0 9 sky ojts into strs, qusrs, glxis. 6 lustring of Mximum ping Gry lustring Algorithm k-lustring. Divi ojts into k non-mpty groups. Distn funtion. Assum it stisfis svrl nturl proprtis. (p i, p j ) = 0 iff p i = p j (intity of inisrnils) (p i, p j ) 0 (nonngtivity) (p i, p j ) = (p j, p i ) (symmtry) ping. Min istn twn ny pir of points in iffrnt lustrs. ingl-link k-lustring lgorithm. Form grph on th vrtx st U, orrsponing to n lustrs. Fin th losst pir of ojts suh tht h ojt is in iffrnt lustr, n n g twn thm. Rpt n-k tims until thr r xtly k lustrs. Ky osrvtion. This prour is prisly Kruskl's lgorithm (xpt w stop whn thr r k onnt omponnts). lustring of mximum sping. Givn n intgr k, fin k-lustring of mximum sping. Rmrk. Equivlnt to fining n MT n lting th k- most xpnsiv gs. sping k = 6 6

17 Gry lustring Algorithm: Anlysis Thorm. Lt * not th lustring *,, * k form y lting th k- most xpnsiv gs of MT. * is k-lustring of mx sping. Gr is goo. Pf. Lt not som othr lustring,, k. Th sping of * is th lngth * of th (k-) st most xpnsiv g. Lt p i, p j in th sm lustr in *, sy * r, ut iffrnt lustrs in, sy s n t. om g (p, q) on p i -p j pth in * r spns two iffrnt lustrs in. All gs on p i -p j pth hv lngth * sin Kruskl hos thm. ping of is * sin p n q r in iffrnt lustrs. * r s t Gr is right. Gr works. Gr lrifis, uts through, n pturs th ssn of th volutionry spirit. - Goron Gko (Mihl Dougls) p i p q p j 6 66