Copyright 2000, Kevin Wayne 1

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1 Extnsions: Mtin Rsints to Hospitls CS 580: Aloritm Dsin n Anlysis Ex: Mn ospitls, Womn m sool rsints. Vrint 1. Som prtiipnts lr otrs s unptl. 1.2 Fiv Rprsnttiv Prolms Vrint 2. Unqul numr o mn n womn. rsint A unwillin to work in Clvln Jrmi Bloki Puru Univrsity Sprin 18 Vrint. Limit polymy. ospitl X wnts to ir rsints Gl-Sply Aloritm Still Works. Minor moiitions to o to nl vritions! Rp: Stl Mtin Prolm Extnsions: Mtin Rsints to Hospitls Dinition o Stl Mtin Stl Roomt Mtin Prolm Stl mtin os not lwys xist! Gl Sply Aloritm (Propos-An-Rjt) Proo tt Aloritm Trmints in stps Proo tt Aloritm Outputs Stl Mtin Mtin is ml-optiml I tr r multipl irnt stl mtins mn t s is st vli prtnr Mtin is ml-pssiml I tr r multipl irnt stl mtins mn t s r worst vli prtnr 2 4 Ex: Mn ospitls, Womn m sool rsints. Vrint 1. Som prtiipnts lr otrs s unptl. Vrint 2. Unqul numr o mn n womn. Vrint. Limit polymy. ospitl X wnts to ir rsints D. Mtin S unstl i tr is ospitl n rsint r su tt: n r r ptl to otr; n itr r is unmt, or r prrs to r ssin ospitl; n itr os not v ll its pls ill, or prrs r to t lst on o its ssin rsints. rsint A unwillin to work in Clvln jos on't ovrlp Tim Copyrit 00, Kvin Wyn 1

2 jos on't ovrlp jos on't ovrlp jos on't ovrlp Gry Coi. Slt jo wit rlist inis tim n limint inomptil jos. Gry Coi. Slt jo wit rlist inis tim n limint inomptil jos. Cptr 4: W will prov tt tis ry loritm lwys ins t optiml solution! Tim Tim Tim 7 9 Wit Input. St o jos wit strt tims, inis tims, n wits. Gol. Fin mximum wit sust o mutully omptil jos. jos on't ovrlp Gry Coi. Slt jo wit rlist inis tim n limint inomptil jos. jos on't ovrlp Gry Coi. Slt jo wit rlist inis tim n limint inomptil jos. Gry Aloritm No Lonr Works! Tim Tim Tim Copyrit 00, Kvin Wyn 2

3 Wit Biprtit Mtin Inpnnt St Input. St o jos wit strt tims, inis tims, n wits. Gol. Fin mximum wit sust o mutully omptil jos. Input. Biprtit rp. Gol. Fin mximum rinlity mtin. Input. Grp. Gol. Fin mximum rinlity inpnnt st. Gry Aloritm No Lonr Works! 2 A 1 1 sust o nos su tt no two join y n 2 12 B C 7 1 D 4 Brut-For Aloritm: Ck vry possil sust. RunninTim: 2 stps Tim E 5 Dirnt rom Stl Mtin Prolm! How? NP-Complt: Unlikly tt iint loritm xists! Positiv: Cn sily k tt tr is n inpnnt st o siz k Wit Biprtit Mtin Comptitiv Fility Lotion Input. St o jos wit strt tims, inis tims, n wits. Gol. Fin mximum wit sust o mutully omptil jos. Input. Biprtit rp. Gol. Fin mximum rinlity mtin. Input. Grp wit wit on no. Gm. Two omptin plyrs ltrnt in sltin nos. Not llow to slt no i ny o its niors v n slt. Prolm n solv usin tniqu ll Dynmi Prormmin A 1 Gol. Slt mximum wit sust o nos. 2 B 2 12 C 2 1 D 4 E 5 Son plyr n urnt, ut not Tim Prolm n solv usin Ntwork Flow Aloritms Copyrit 00, Kvin Wyn

4 Comptitiv Fility Lotion Comptitiv Fility Lotion Comptitiv Fility Lotion Input. Grp wit wit on no. Gm. Two omptin plyrs ltrnt in sltin nos. Not llow to slt no i ny o its niors v n slt. Gol. Slt mximum wit sust o nos. Input. Grp wit wit on no. Gm. Two omptin plyrs ltrnt in sltin nos. Not llow to slt no i ny o its niors v n slt. Gol. Slt mximum wit sust o nos. Input. Grp wit wit on no. Gm. Two omptin plyrs ltrnt in sltin nos. Not llow to slt no i ny o its niors v n slt. Gol. Slt mximum wit sust o nos. PSPACE-Complt: Evn rr tn NP-Complt! No sort proo tt plyr n urnt vlu B. (Unlik prvious prolm) Son plyr n urnt, ut not 25. Son plyr n urnt, ut not 25. Son plyr n urnt, ut not Comptitiv Fility Lotion Comptitiv Fility Lotion Fiv Rprsnttiv Prolms Input. Grp wit wit on no. Gm. Two omptin plyrs ltrnt in sltin nos. Not llow to slt no i ny o its niors v n slt. Gol. Slt mximum wit sust o nos. Input. Grp wit wit on no. Gm. Two omptin plyrs ltrnt in sltin nos. Not llow to slt no i ny o its niors v n slt. Gol. Slt mximum wit sust o nos. Vritions on tm: inpnnt st. Intrvl sulin: n lo n ry loritm. Wit intrvl sulin: n lo n ynmi prormmin loritm. Biprtit mtin: n k mx-low s loritm. Inpnnt st: NP-omplt. Comptitiv ility lotion: PSPACE-omplt. Son plyr n urnt, ut not 25. Son plyr n urnt, ut not Copyrit 00, Kvin Wyn 4

5 Computtionl Trtility Worst-Cs Anlysis Cptr 2 Bsis o Aloritm Anlysis As soon s n Anlyti Enin xists, it will nssrily ui t utur ours o t sin. Wnvr ny rsult is sout y its i, t qustion will ris - By wt ours o lultion n ts rsults rriv t y t min in t sortst tim? - Crls B Worst s runnin tim. Otin oun on lrst possil runnin tim o loritm on input o ivn siz N. Gnrlly pturs iiny in prti. Dronin viw, ut r to in tiv ltrntiv. Avr s runnin tim. Otin oun on runnin tim o loritm on rnom input s untion o input siz N. Hr (or impossil) to urtly mol rl instns y rnom istriutions. Aloritm tun or rtin istriution my prorm poorly on otr inputs. Slis y Kvin Wyn. Copyrit 05 Prson-Aison Wsly. All rits rsrv. Crls B (184) Anlyti Enin (smti) Polynomil-Tim Worst-Cs Polynomil-Tim 2.1 Computtionl Trtility "For m, rt loritms r t potry o omputtion. Just lik vrs, ty n trs, llusiv, ns, n vn mystrious. But on unlok, ty st rillint nw lit on som spt o omputin." - Frnis Sullivn Brut or. For mny non-trivil prolms, tr is nturl rut or sr loritm tt ks vry possil solution. Typilly tks 2 N tim or wors or inputs o siz N. Unptl in prti. n! or stl mtin wit n mn n n womn Dsirl slin proprty. Wn t input siz ouls, t loritm soul only slow own y som onstnt tor C. Tr xists onstnts > 0 n > 0 su tt on vry input o siz N, its runnin tim is oun y N stps. D. An loritm is poly-tim i t ov slin proprty ols. oos C = 2 D. An loritm is iint i its runnin tim is polynomil. Justiition: It rlly works in prti! Altou N is tnilly poly-tim, it woul uslss in prti. In prti, t poly-tim loritms tt popl vlop lmost lwys v low onstnts n low xponnts. Brkin trou t xponntil rrir o rut or typilly xposs som ruil strutur o t prolm. Exptions. Som poly-tim loritms o v i onstnts n/or xponnts, n r uslss in prti. Som xponntil-tim (or wors) loritms r wily us us t worst-s instns sm to rr. simplx mto Unix rp 28 0 Copyrit 00, Kvin Wyn 5

6 Wy It Mttrs Asymptoti Orr o Growt Proprtis Uppr ouns. T(n) is O((n)) i tr xist onstnts > 0 n n 0 0 su tt or ll n n 0 w v T(n) (n). Lowr ouns. T(n) is ((n)) i tr xist onstnts > 0 n n 0 0 su tt or ll n n 0 w v T(n) (n). Tit ouns. T(n) is ((n)) i T(n) is ot O((n)) n ((n)). Ex: T(n) = 2n n + 2. T(n) is O(n 2 ), O(n ), (n 2 ), (n), n (n 2 ). T(n) is not O(n), (n ), (n), or (n ). Trnsitivity. I = O() n = O() tn = O(). I = () n = () tn = (). I = () n = () tn = (). Aitivity. I = O() n = O() tn + = O(). I = () n = () tn + = (). I = () n = O() tn + = (). 1 5 Nottion Asymptoti Bouns or Som Common Funtions 2.2 Asymptoti Orr o Growt Slit us o nottion. T(n) = O((n)). Not trnsitiv: (n) = 5n ; (n) = n 2 (n) = O(n ) = (n) ut (n) (n). Bttr nottion: T(n) O((n)). Mninlss sttmnt. Any omprison-s sortin loritm rquirs t lst O(n lo n) omprisons. Sttmnt osn't "typ-k." Us or lowr ouns. Polynomils n + + n is (n ) i > 0. Polynomil tim. Runnin tim is O(n ) or som onstnt inpnnt o t input siz n. Loritms. O(lo n) = O(lo n) or ny onstnts, > 0. n voi spiyin t s Loritms. For vry x > 0, lo n = O(n x ). lo rows slowr tn vry polynomil Exponntils. For vry r > 1 n vry > 0, n = O(r n ). vry xponntil rows str tn vry polynomil 4 Copyrit 00, Kvin Wyn

7 Linr Tim: O(n) Qurti Tim: O(n 2 ) 2.4 A Survy o Common Runnin Tims Mr. Comin two sort lists A = 1, 2,, n wit B = 1, 2,, n into sort wol. Qurti tim. Enumrt ll pirs o lmnts. Closst pir o points. Givn list o n points in t pln (x 1, y 1 ),, (x n, y n ), in t pir tt is losst. i = 1, j = 1 wil (ot lists r nonmpty) { i ( i j ) ppn i to output list n inrmnt i ls( i j )ppn j to output list n inrmnt j ppn rminr o nonmpty list to output list Clim. Mrin two lists o siz n tks O(n) tim. P. Atr omprison, t lnt o output list inrss y 1. O(n 2 ) solution. Try ll pirs o points. min (x 1 - x 2 ) 2 + (y 1 - y 2 ) 2 or i = 1 to n { or j = i+1 to n { (x i - x j ) 2 + (y i - y j ) 2 i ( < min) min Rmrk. (n 2 ) sms invitl, ut tis is just n illusion. s ptr 5 on't n to tk squr roots 9 41 Linr Tim: O(n) O(n lo n) Tim Cui Tim: O(n ) Linr tim. Runnin tim is proportionl to input siz. Computin t mximum. Comput mximum o n numrs 1,, n. mx 1 or i = 2 to n { i ( i > mx) mx i O(n lo n) tim. Ariss in ivi-n-onqur loritms. lso rrr to s linritmi tim Sortin. Mrsort n psort r sortin loritms tt prorm O(n lo n) omprisons. Lrst mpty intrvl. Givn n tim-stmps x 1,, x n on wi opis o il rriv t srvr, wt is lrst intrvl o tim wn no opis o t il rriv? O(n lo n) solution. Sort t tim-stmps. Sn t sort list in orr, intiyin t mximum p twn sussiv tim-stmps. Cui tim. Enumrt ll tripls o lmnts. St isjointnss. Givn n sts S 1,, S n o wi is sust o 1, 2,, n, is tr som pir o ts wi r isjoint? O(n ) solution. For pirs o sts, trmin i ty r isjoint. or st S i { or otr st S j { or lmnt p o S i { trmin wtr p lso lons to S j i (no lmnt o S i lons to S j ) rport tt S i n S j r isjoint Copyrit 00, Kvin Wyn 7

8 Polynomil Tim: O(n k ) Tim Hp Dt Strutur Hp Insrtion Inpnnt st o siz k. Givn rp, r tr k nos su tt no two r join y n? k is onstnt Hp.Insrt() O(n k ) solution. Enumrt ll susts o k nos. or sust S o k nos { k wtr S in n inpnnt st i (S is n inpnnt st) rport S is n inpnnt st Ck wtr S is n inpnnt st = O(k 2 ). Numr o k lmnt susts = n O(k 2 n k / k!) = O(n k ). n (n 1) (n 2) (n k 1) k k (k 1) (k 2) (2) (1) poly-tim or k=17, ut not prtil nk k! Nxt Min Hp Orr: For no v in t tr Prnt v. Vlu v. Vlu Mx Hp Orr: For no v in t tr Prnt v. Vlu v. Vlu Min Hp Orr: For no v in t tr Prnt v. Vlu v. Vlu Exponntil Tim Hp Insrtion Hp Insrtion Inpnnt st. Givn rp, wt is mximum siz o n inpnnt st? Hp.Insrt() O(n 2 2 n ) solution. Enumrt ll susts. Hp.Insrt() 9 S* or sust S o nos { k wtr S in n inpnnt st i (S is lrst inpnnt st sn so r) upt S* S Min Hp Orr: For no v in t tr Prnt v. Vlu v. Vlu Min Hp Orr: For no v in t tr Prnt v. Vlu v. Vlu Copyrit 00, Kvin Wyn 8

9 Hp Insrtion Hp Extrt Minimum Hp Extrt Minimum Hp.Insrt() Hp.ExtrtMin() Hp.ExtrtMin() Nxt Min Hp Orr: For no v in t tr Prnt v. Vlu v. Vlu Torm 2.12 [KT]: T prour Hpiy-up ixs t p proprty n llows us to insrt nw lmnt into p o n lmnts in O(lo n) tim. Nxt Min Hp Orr: For no v in t tr Prnt v. Vlu v. Vlu Torm 2.1 [KT]: T prour Hpiy-own ixs t p proprty n llows us to lt n lmnt in p o n lmnts in O(lo n) tim. Nxt Min Hp Orr: For no v in t tr Prnt v. Vlu v. Vlu Torm 2.1 [KT]: T prour Hpiy-own ixs t p proprty n llows us to lt n lmnt in p o n lmnts in O(lo n) tim Hp Extrt Minimum Hp Extrt Minimum Hp Extrt Minimum Hp.ExtrtMin() Hp.ExtrtMin() Hp.ExtrtMin() Nxt Min Hp Orr: For no v in t tr Prnt v. Vlu v. Vlu Torm 2.1 [KT]: T prour Hpiy-own ixs t p proprty n llows us to lt n lmnt in p o n lmnts in O(lo n) tim. Nxt Min Hp Orr: For no v in t tr Prnt v. Vlu v. Vlu Torm 2.1 [KT]: T prour Hpiy-own ixs t p proprty n llows us to lt n lmnt in p o n lmnts in O(lo n) tim. Nxt Min Hp Orr: For no v in t tr Prnt v. Vlu v. Vlu Torm 2.1 [KT]: T prour Hpiy-own ixs t p proprty n llows us to lt n lmnt in p o n lmnts in O(lo n) tim Copyrit 00, Kvin Wyn 9

10 Hp Extrt Minimum Hp.ExtrtMin() Nxt Min Hp Orr: For no v in t tr Prnt v. Vlu v. Vlu Torm 2.1 [KT]: T prour Hpiy-own ixs t p proprty n llows us to lt n lmnt in p o n lmnts in O(lo n) tim. 55 Hp Summry Insrt: O(lo n) FinMin: O(1) Dlt: O(lo n) tim ExtrtMin: O(lo n) tim Tout Qustion: O(n lo n) tim sortin loritm usin ps? 5 Copyrit 00, Kvin Wyn 10