kd-trees Idea: Each level of the tree compares against 1 dimension. Let s us have only two children at each node (instead of 2 d )

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1 k-trs CMSC 420

2 k-trs Invnt n 1970s Jon Bntl Nm ornll mnt 3-trs, 4-trs, t wr k ws t # o mnsons Now, popl s k-tr o mnson I: E lvl o t tr omprs nst 1 mnson. Lt s us v onl two lrn t no (nst o 2 )

3 k-trs E lvl s uttn mnson Cl trou t mnsons s ou wlk own t tr. E no ontns pont P = (,) To n (, ) ou onl ompr oornt rom t uttn mnson -.. uttn mnson s, tn ou sk: s <?

4 k-tr mpl nsrt: (30,40), (5,25), (10,12), (70,70), (50,30), (35,45) 30,40 (70,70) 5,25 70,70 (30,40) (35,45) 10,12 50,30 (5,25) (50,30) 35,45 (10,12)

5 Insrt Co nsrt(pont, KDNo t, nt ) { t == null t = nw KDNo() ls ( == t.t) // rror! uplt ls ([] < t.t[]) t.lt = nsrt(, t.lt, (+1) % DIM) ls t.rt = nsrt(, t.rt, (+1) % DIM) rturn t }

6 FnMn n k-trs FnMn(): n t pont wt t smllst vlu n t t mnson. Rursvl trvrs t tr I utm(urrnt_no) =, tn t mnmum n t n t rt sutr, so rurs on just t lt sutr - no lt sutr, tn urrnt no s t mn or tr root t ts no. I utm(urrnt_no), tn mnmum oul n tr sutr, so rurs on ot sutrs. - (unlk n 1- struturs, otn v to plor svrl pts own t tr)

7 FnMn FnMn(-mnson): (35,90) (60,80) 51,75 (51,75) (70,70) 25,40 70,70 (50,50) (25,40) 10,30 35,90 55,1 60,80 (1,10) (10,30) (55,1) 1,10 50,50

8 FnMn FnMn(-mnson): (35,90) (60,80) 51,75 (51,75) (70,70) 25,40 70,70 (50,50) (25,40) 10,30 35,90 55,1 55,1 60,80 (1,10) (10,30) (55,1) 1,10 1,10 50,50

9 FnMn FnMn(-mnson): sp sr (35,90) (60,80) 51,75 (51,75) (70,70) 25,40 70,70 (50,50) (25,40) 10,30 35,90 55,1 60,80 (1,10) (10,30) (55,1) 1,10 50,50

10 FnMn Co Pont nmn(no T, nt m, nt ): // mpt tr T == NULL: rturn NULL // T splts on t mnson w r srn // => onl vst lt sutr == m: t.lt == NULL: rturn t.t ls rturn nmn(t.lt, m, (+1)%DIM) // T splts on rnt mnson // => v to sr ot sutrs ls: rturn mnmum( nmn(t.lt, m, (+1)%DIM), nmn(t.rt, m, (+1)%DIM) T.t )

11 Dlt n k-trs Wnt to lt no A. Assum uttn mnson o A s In BST, w nmn(a.rt). A Hr, w v to nmn(a.rt, ) Evrtn n Q s -oor < B, n vrtn n P s oor B Q B P

12 Dlt n k-trs --- No Rt Sutr Wt s rt sutr s mpt? Possl : Fn t m n t lt sutr? - W mt ts not work? Suppos I nm(t.lt) n t pont (,): (,) It s possl tt T.lt ontns notr pont wt =. Now, our qul oornt nvrnt s volt! (,) Q (,)

13 No rt sutr --- Soluton Swp t sutrs o no to lt B = nmn(t.lt) Rpl lt no B (,) Now, tr s notr pont wt =, t pprs n t rt sutr, wr t soul Q (,) (,)

14 Pont lt(pont, No T, nt ): T == NULL: rror pont not oun! nt_ = (+1)%DIM // Ts s t pont to lt: = T.t: // us mn() rom rt sutr: t.rt!= NULL: t.t = nmn(t.rt,, nt_) t.rt = lt(t.t, t.rt, nt_) // swp sutrs n us mn() rom nw rt: ls T.lt!= NULL: t.t = nmn(t.lt,, nt_) t.rt = lt(t.t, t.lt, nt_) ls t = null // w r l: just rmov // ts s not t pont, so sr or t: ls [] < t.t[]: t.lt = lt(, t.lt, nt_) ls t.rt = lt(, t.rt, nt_) rturn t

15 Nrst Nor Srn n k-trs Nrst Nor Qurs r vr ommon: vn pont Q n t pont P n t t st tt s losst to Q. Dosn t work: n ll tt woul ontn Q n rturn t pont t ontns. - Rson: t nrst pont to P n sp m r rom P n t tr: - E.. NN(52,52): (35,90) (60,80) 51,75 (51,75) 25,40 70,70 (70,70) (50,50) (25,40) 10,30 35,90 55,1 60,80 (1,10) (10,30) (55,1) 1,10 50,50

16 k-trs Nrst Nor I: trvrs t wol tr, BUT mk two motons to prun to sr sp: 1. Kp vrl o losst pont C oun so r. Prun sutrs on tr ounn os s tt t n t ontn n pont losr tn C 2. Sr t sutrs n orr tt mmzs t n or prunn

17 Nrst Nor: Is, ontnu Qur Pont Q Bounn o o sutr root t T T I > st(c, Q), tn no pont n BB(T) n losr to Q tn C. Hn, no rson to sr sutr root t T. Upt t st pont so r, T s ttr: st(c, Q) > st(t.t, Q), C := T.t Rurs, ut strt wt t sutr losr to Q: Frst sr t sutr tt woul ontn Q w wr nsrtn Q low T.

18 Nrst Nor, Co st, st_st r lol vr (n lso pss nto unton lls) NN(Pont Q, ktr T, nt, Rt BB): // ts ounn o s too r, o notn T == NULL or stn(q, BB) > st_st: rturn // ts pont s ttr tn t st: st = stn(q, T.t) st < st_st: st = T.t st_st = st // vst sutrs s most promsn orr: Q[] < T.t[]: NN(Q, T.lt, nt_, BB.trmLt(, t.t)) NN(Q, T.rt, nt_, BB.trmRt(, t.t)) ls: NN(Q, T.rt, nt_, BB.trmRt(, t.t)) NN(Q, T.lt, nt_, BB.trmLt(, t.t)) Follown Dv Mount s Nots (p 77)

19 Nrst Nor Fts Mt v to sr los to t wol tr n t worst s. [O(n)] In prt, runtm s losr to: - O(2 + lo n) - lo n to n lls nr t qur pont - 2 to sr roun lls n tt noroo Tr mportnt onpts tt rour n rn / nrst nor srn: - storn prtl rsults: kp st so r, n upt - prunn: ru sr sp lmntn rrlvnt trs. - trvrsl orr: vst t most promsn sutr rst.

20 Ftur-Bs Dt Fuson or 3D Potorp Pro. Gor Wolr Dpt. o Computr Sn Ct Coll o Nw York

21 Ftur Mtn (1) Two ms n tr mt tur ponts 58

22 Ftur Mtn (2) To mt turs trt twn two potos, w n to sr or t smllst Euln stn mon turs. Et soluton rqurs n ustv O(n 2 ) sr, wr s t tur srptor mnson n n s t numr o trt turs rom poto. Tpll, s 128 or 64 n n s 5,000-30,000. Ts rutor sr s vr pnsv. For mpl, wn n=5000 n =128, t # omputtons or ll mts s 3.2 llon. 59

23 Ftur Mtn (3) To sp up t ustv sr, w us t Appromt Nrst Nor (ANN) mto. - I w llow smll rror to m, t sr tm n snntl ru. - Sn t nput t ontns rrors nw, ts wll not rtl mpr mtn qult. - W us t ANN-lrr Dv M. Mount n Sunl Ar. T unrln sr mto s s on t K-tr. - T qur tm s n orr o O( lo n). For mpl, wn n = 5000 n = 128, t # omputtons or ll mts s T mtn prormn s mprov

24 K-D Tr Construton Empl

25 K-D Tr Construton Empl

26 K-D Tr Construton Empl

27 K-D Tr Construton Empl

28 K-D Tr Construton Empl

29 K-D Tr Construton Empl

30 K-D Tr Construton Empl

31 K-D Tr Construton Empl

32 K-D Tr Construton Empl

33 K-D Tr Construton Empl

34 K-D Tr Construton Empl

35 K-D Tr Construton Empl

36 K-D Tr Construton Empl

37 K-D Tr Construton Empl

38 K-D Tr Construton Empl

39 K-D Tr Construton Empl s8 s8

40 K-D Tr Construton Empl s8 s8

41 K-D Tr Construton Empl s8 s8

42 K-D Tr Construton Frst sort t ponts n mnson: O(n lo n) tm n n stor. Ts r stor n A[1..,1..n] Fnn t wst spr n qull vn nto two susts n on n O(n) tm. Construtn t k- tr n on n O(n lon) n n stor

43 K-D Tr Construton A no s 5 ls s (splttn s) vlu (splttn vlu) lt (lt sutr) rt (rt sutr) pont (l no lt n rt lrn r null)

44 K-D Tr Nrst Nor Sr Sr lt Sr rt r r no.vlu q(no.s) q(no.s) no.vlu q(no.s) r < no.vlu Mns t rl ovrlps t lt sutr. q(no.s) + r > no.vlu Mns t rl ovrlps t rt sutr.

45 K-D Tr Nrst Nor Sr s8 s8 Emn nr ponts rst: Eplor t rn o t tr tt s losst to t qur pont rst.

46 K-D Tr Nrst Nor Sr s8 s8

47 K-D Tr Nrst Nor Sr s8 s8

48 K-D Tr Nrst Nor Sr r s8 s8 Wn w r l no: omput t stn to pont n t no.

49 K-D Tr Nrst Nor Sr r s8 s8 Tn w n ktrk n tr t otr rn t no vst.

50 K-D Tr Nrst Nor Sr s8 s8 r

51 K-D Tr Nrst Nor Sr r s8 s8 E tm nw losst no s oun, w n upt t stn ouns.

52 K-D Tr Nrst Nor Sr s8 s8 r

53 K-D Tr Nrst Nor Sr s8 s8 r

54 K-D Tr Nrst Nor Sr s8 s8 r

55 K-D Tr Nrst Nor Sr s8 s8 r

56 K-D Tr Nrst Nor Sr s8 s8 r

57 K-D Tr Nrst Nor Sr s8 s8 r

58 K-D Tr Nrst Nor Sr s8 s8 r

59 s8 s8 r K-D Tr Nrst Nor Sr

60 K-D Tr Nrst Nor Sr s8 s8 r

61 K-D Tr Nrst Nor Sr r s8 s8 Usn t stn ouns n t ouns o t t low no, w n prun prts o t tr tt oul NOT nlu t nrst nor.

62 Appromt Nrst Nor (ANN) Gvn t qur pont q, w n lotn t l ll ontnn t qur pont n O(lo n) tm smpl snt trou t tr. Nt, w n numrtn t l lls n nrsn orr o stn rom t qur pont. W ll ts prort sr. Wn ll s vst, t stn rom q to t pont ssot wt ts ll s omput. W kp trk o t losst pont sn so r. E ll s n numr orn to ts stn rom t qur pont.

63 Appromt Nrst Nor (ANN) Lt p not t losst pont sn so r. As soon s t stn rom q to t urrnt l ll s st(q, p)=(1 + ε ) (ott rl) t ollows tt t sr n trmnt, n p n rport s n ppromt nrst nor to q. T rson s tt n pont lot n susquntl vst ll nnot los nou to q to volt p's lm to n ppromt nrst nor. In t mpl sown, t sr trmnts just pror to vstn ll 9. T prort sr n prorm n tm O( lo n) tms usn n ulr p.

64 CS106L Sprn 2014 Hnout #04 M 15, 2014 Assnmnt 3: KDTr Du Jun 4, 11:59 PM Ovr t pst svn wks, w'v plor w rr o STL ontnr lsss. You'v sn t lnr vtor n qu, lon wt t ssotv mp n st. On proprt ommon to ll ts ontnrs s tt t r t. An lmnt s tr n st or t sn't. A vlu tr pprs t prtulr poston n vtor or t os not. For most ppltons, ts s tl wt w wnt. Howvr, n som ss w m ntrst not n t quston s X n ts ontnr, ut rtr wt vlu n t ontnr s X most smlr to? Qurs o ts sort otn rs n t mnn, mn lrnn, n omputtonl omtr. In ts ssnmnt, ou wll mplmnt spl t strutur ll k-tr (sort or k-mnsonl tr ) tt ntl supports ts oprton. At lvl, k-tr s nrlzton o nr sr tr tt stors ponts n k-mnsonl sp. Tt s, ou oul us k-tr to stor ollton o ponts n t Crtsn pln, n tr-mnsonl sp, t. You oul lso us k-tr to stor omtr t, or mpl, rprsntn t t s n orr tupl, prps (t, wt, loo prssur, olstrol). Howvr, k-tr nnot us to stor olltons o otr t tps, su s strns. Also not tt wl t's possl to ul k-tr to ol t o n mnson, ll o t t stor n k-tr must v t sm mnson. Tt s, ou n't stor ponts n two-mnsonl sp n t sm k-tr s ponts n our-mnsonl sp. It's sst to unrstn ow k-tr works sn n mpl. Blow s k-tr tt stors ponts n tr-mnsonl sp: (3, 1, 4) (2, 3, 7) (4, 3, 4) (2, 1, 3) (2, 4, 5) (6, 1, 4) (1, 4, 4) (0, 5, 7) (5, 2, 5) (4, 0, 6) (7, 1, 6) Not tt n lvl o t k-tr, rtn omponnt o no s n ol. I w zro-n t omponnts (.. t rst omponnt s omponnt zro, t son omponnt s omponnt on, t.), n lvl n o t tr, t (n % 3)r omponnt o no s sown n ol. T rson tt ts vlus r ol s us no ts lk nr sr tr no tt srmnts onl lon t ol omponnt. For mpl, t rst omponnt o vr no n t lt sutr s lss tn t rst omponnt o t root o t tr, wl t rst om

65 ponnt o vr no n t rt sutr s rst omponnt t lst s lr s t root no's. Smlrl, onsr t k-tr's lt sutr. T root o ts tr s t vlu (2, 3, 7), wt t tr n ol. I ou look t ll t nos n ts lt sutr, ou'll not tt t son omponnt s vlu strtl lss tn tr. Smlrl, n t rt sutr t son omponnt o no s t lst tr. Ts trn ontnus trouout t tr. Gvn ow k-trs stor tr t, w n ntl qur wtr vn pont s stor n k-tr s ollows. Gvn pont P, strt t t root o t tr. I t root no s P, rturn t root no. I t rst omponnt o P s strtl lss tn t rst omponnt o t root no, tn look or P n t lt sutr, ts tm omprn t son omponnt o P. Otrws, tn t rst omponnt o P s t lst s lr s t rst omponnt o t root no, n w sn nto t rt sutr n nt tm ompr t son omponnt o P. W ontnu ts pross, ln trou w omponnt s onsr t stp, untl w ll o t tr or n t no n quston. Insrtn nto k-tr s smlrl nloous to nsrtn nto rulr BST, pt tt lvl onl onsrs on prt o t pont. T Gomtr Intuton Bn k-trs You mt wonrn w k-trs stor tr t s t o. Atr ll, t's not mmtl o - vous w ou' ompr rnt oornt t lvl o t tr. It turns out tt tr s utul omtr mnn n ts stup, n plotn ts strutur t's possl to prorm nrst-nor lookups trml ntl (n tm ttr tn O(n)) usn k-tr. In orr to mk t ntuton n t oornt--oornt omprson lr, w'll qukl rturn to t stnr nr sr tr ormulton ou'r mlr wt to plor n spt o BSTs tt ou m not v mmtl not. Consr BST wr no ols rl numr. In ts susson, w'll us ts tr s rrn: Bus t BST ols ollton o rl numrs, w n ovrl ts BST wt t numr ln. Ts s sown low:

66 Now, suppos tt w trvrs t BST lookn or zro. W n t t root n k wtr t root no s t vlu w'r lookn or. Sn t osn't, w trmn w o t two sutrs to sn nto, tn rursvl look n tt sutr or zro. Mtmtll, ts s quvlnt to splttn t rl numr ln nto two rons numrs lss tn two n numrs rtr tn to two. Ts s sown r: Vlus lss tn two Vlus rtr tn two Not tt ll o t nos n t lt sutr r n t lt prtton n ll t nos n t rt sutr r n t rt prtton. Sn 0 < 2, w know tt zro s ontn n ts tr t ll, t must n t lt prtton. Ts mmtl ruls out t posslt tt zro s n t rt su - tr, n so w n rursvl sn nto t lt sutr wtout worrn out mssn t no or zro. T ov susson lts t k nst tt mks nr sr trs possl. E no ns som prtton o t rl ln nto two smnts, n o t no's sutrs s ull ontn wtn on o t smnts. Srn BST n tus tout o s ontnuousl splttn sp n l, tn ontnun t sr onl n t l tt ontns t vlu n quston. T mn rson or mntonn ts ln o rsonn s tt t s possl to sl ts up to t o r mnsons. Suppos, or mpl, tt w v t ollown ollton o ponts n t pln: Suppos tt w wnt to ul nr sr tr out o ts ponts. I w us t mlr n - ton o BST, w woul pk som no s t root, tn ul sutr out o t rmnn nos tt r lss tn t root no n on sutr out o t vlus tt r rtr tn t root no. Unortuntl, tr sn't prtulrl oo nton o wt t mns or pont n sp - 3 -

67 to lss tn notr. But lt's nst onsr t vw o BST w suss ov. In BST, pont nturll splt t ntr rl ln nto two rons. In two mnsons, w n splt t pln nto two rons roun pont rwn ln trou tt pont. For mpl, w rw t ollown ln trou t nt pont: Tn w'v splt t pln nto two stnt rons, on ov t ln n on low ts ln. Ts osrvton vs us w to ul nr sr tr n multpl mnsons. Frst, pk n rtrr pont n sp n rw ln (osn owvr ou' lk) trou t. Nt, sprt t rmnn ponts nto ponts to on s o t ln n ponts on t otr. Fnll, rursvl onstrut nr sr trs out o tos ponts. Ts tnqu s known s nr sp prttonn (sn stp splts sp nto two rons), n trs nrt ts w r known s nr sp prttonn trs or BSP trs. But BSP trs r not rstrt to just t two-mnsonl pln; t sm tnqu works n rtrrl mn mnsons. In tr mnsons, w oul prtton sp nto two rons rwn pln trou pont, tn tkn t rons ov n low t pln s t two l-rons. Wn workn wt BSP trs, on otn uss t trm splttn prpln to rr to t ojt pssn trou pont tt splts sp n l. In two mnsons, prpln s ln, wl n tr t's pln. In stnr nr sr tr, ts prpln s just pont. Wt os n o ts susson v to o wt k-trs? To nswr tt quston, lt's rturn to our ornl ollton o ponts n two-mnsonl sp, s sown r: Suppos tt w wnt to ul k-tr out o ts t ponts. W n oosn som no (w w'll s s t ( 0, 0 ) or nottonl smplt) n splttn t t st nto two roups, on o ponts wos omponnts r lss tn t splttn no's, n on o ponts wos omponnts r t lst s lr s t splttn no's. W n vsulz t splt lk ts: - 4 -

68 < 0 0 Not tt ts s ssntll quvlnt to runnn splttn prpln trou on o t ponts. In tt sns, k-tr s spl s o BSP tr wt spl rul tt trmns w splttn prplns to us. Howvr, w'v on so wtout nn to wrt n o tt mnpults prplns or l-sps. All o t ompl omtr s tkn r o mpltl. Lt's ontnu uln ts k-tr. W rursvl ul k-tr n t rt l-sp (t ponts to t rt o t ntrl no) pkn t som pont n splttn t t orzontll trou t, s sn r: 1 < 1 I w ontnu ts onstruton to omplton, our rsultn k-tr wll look lk ts: Hr, t ol no s t root, nos on lvl own r r, nos two lvls p r rn, n nos tr lvls p r lu

69 To v ou ttr sns or t omtr ntuton n ts k-tr, lt's tr trou wt ppns wn w tr lookn up wtr vn pont s n t k-tr. In prtulr, lt's s wt ppns s w tr to look up t no n t ottom-rt ornr o t k-tr. W n t t root o t k-tr n onsr wtr our no's oornt s lss tn or rtr tn t root no's oornt. Ts s quvlnt to splttn t pln vrtll t t root no, tn skn w l-sp our no s n. Our no ppns to n t rt l-sp, n so w n nor ll o t nos n t lt l-sp n rursvl plor t rt. Ts s sown rpll low, wr t r-out ron orrspons to prts o t pln w wll nvr look n: Now, w k wtr our no s ov or low t r no, w s t root o t tr n ts l-sp. Our no s low t, so w n sr t top l-sp n look n t ottom. Ts s sown r: Nt, w k wtr w'r to t lt or t rt o t rn no tt's t root o ts ron o sp. W'r to t rt, so w sr t slvr o l-sp to t lt o tt no n on - tnu on: - 6 -

70 At ts pont, w v r t no w'r lookn or, n t sr lortm trmnts. Nrst-Nor Lookup n k-trs Now tt ou v ttr omtr ntuton or t k-tr, w n tlk out t most ntrstn oprton on t k-tr: nrst-nor lookup. Ts qur works s ollows: vn k-tr n pont n sp (ll t tst pont), w pont n t k-tr s losst to t tst pont? (T pont n t t st losst to t tst pont s ll ts nrst nor). Bor w suss t tul lortm or on nrst-nor lookup, w'll suss t ntuton n t lortm. Suppos tt w v uss o wt w tnk t nrst nor to t tst pont s. For mpl, suppos tt t tst pont s nt t str n tt w tnk t nrst nor s t pont onnt to t str t s ln: Gvn our uss o wt t nrst nor s, w n mk rul osrvton. I tr s pont n ts t st tt s losr to t tst pont tt our urrnt uss, t must l n t rl ntr t t tst pont tt psss trou t urrnt uss. Ts rl s sown r: Altou n ts mpl ts ron s rl, n tr mnsons t woul spr, n n nrl w ll t t nt prspr. T rson tt ts osrvton s so mportnt s tt t lts us prun w prts o t tr mt ol t tru nrst nor. In prtulr, not tt ts rl s ntrl to t rt o t splttn prpln runnn vrtll trou t root o t tr. Consquntl, n pont to t lt o t root o t tr nnot possl n t nt prspr, n onsquntl n't n ttr tn our urrnt uss. In otr wors, on w v uss out wr t nrst nor s, w n strt lmntn prts o t tr wr t tul nswr nnot. Ts nrl tnqu o srn lr sp n prunn optons s on prtl rsults s ll rn-n-oun

71 From t ptur t's lr tt t rl o possl nrst nors os not ross t ml splttn prpln, ut ow n w trmn ts mtmtll? In nrl, vn rl n ln (or, mor nrll, prspr n prpln), t's t trk to trmn wtr tt rl ntrsts t ln. Fortuntl, tou, t t tt w'v osn ll o t splttn prplns to s-ln rtl smpls ts tsk. Blow s n rtrr ln n two rls, on o w rosss t ln n on o w os not: = 0 r 2 ( 2, 2 ) r 1 ( 1, 1 ) Now, onsr t stn rom t ntrs o ts rls to t ln = 0. Ts s smpl t solut vlu o t rn twn t rls' oornts n 0, s sn r: = r ( 2, 2 ) r 1 ( 1, 1 ) Not tt t stn 1 0 rom t ntr o t lu rl to t ln s rtr tn t rus o t rl, n so t rl os not ross t ln. On t otr n, t stn rom t ntr o t r rl to t ln s lss tn t rus o t rl, n so som prt o tt rl os ross t ln. Ts vs nrl rtron or trmnn wtr nt prspr rosss prtulr splttn prpln. In prtulr, vn k-tr no oln pont ( 0, 1, 2,..., k ) n prspr o rus r ntr t ( 0, 1, 2,..., k ), t no prttons ponts s on tr t omponnt, tn t prspr rosss t no's splttn pln onl < r. To rp: Gvn uss out w no s t nrst nor, w n onstrut nt prspr ntr t t tst pont n runnn trou t uss pont. T nrst nor to t tst pont must l ns ts prspr. I ts prspr s ull to on s o splttn prpln, tn ll ponts on t otr s o t splttn prpln nnot ontn n t spr n tus nnot t nrst nor

72 To trmn wtr t nt prspr rosss splttn prpln tt omprs oornt, w k wtr < r. Ts osrvtons, tkn totr, sust t ollown lortm or nn t nrst nor to tst pont: Lt t tst pont ( 0, 1,..., k ). Mntn lol st stmt o t nrst nor, ll 'uss.' Mntn lol vlu o t stn to tt nor, ll 'stdst' St 'uss' to NULL. St 'stdst' to nnt. Strtn t t root, ut t ollown prour: urr == NULL rturn /* I t urrnt loton s ttr tn t st known loton, * upt t st known loton. */ stn(urr, uss) < stdst stdst = stn(urr, uss) uss = urr /* Rursvl sr t l o t tr tt ontns t tst pont. */ < urr rursvl sr t lt sutr on t nt s ls rursvl sr t rt sutr on t nt s /* I t nt prspr rosss ts splttn pln, look on t * otr s o t pln mnn t otr sutr. */ urr < stdst rursvl sr t otr sutr on t nt s Intutvl, ts prour works wlkn own to t l o t k-tr s w wr srn t tr or t tst pont. As w strt unwnn t rurson n wlkn k up t tr, w k wtr no s ttr tn t st stmt w v so r. I so, w upt our st stmt to t urrnt no. Fnll, w k wtr t nt prspr s on our urrnt uss oul ross t splttn prpln o t urrnt no. I t osn't, tn w n lmnt ll ponts on t otr s o t splttn prpln rom onsrton n wlk k up to t nt no n t tr. Otrws, w must look n tt s o t tr to s tr r n losr ponts. Ts lortm n sown to run n O(lo n) tm on ln k-tr wt n t ponts prov tt tos ponts r rnoml strut. In t worst s, tou, t ntr tr mt v to sr. Howvr, n low-mnsonl sps, su s t Crtsn pln or tr-mnsonl sp, ts s rrl t s

73 k-nrst Nor Srs n Boun Prort Quus In ts susson, w'v onl onsr t prolm o nn t snl nrst nor to tst pont. A mor ntrstn quston s, vn tst pont n som numr k, to n t k-nrst-nors o tt pont. Ts sr s otn rrr to s k-nn sr. It turns out tt t prvous lortm n sl pt to o k-nn sr nst o 1-NN sr. T lortm s lmost ntl, pt tt nst o mntnn just t st pont, w mntn lst o t k st ponts w'v sn so r. Bor srn t lortm, w'll ntrou spl t strutur ll oun prort quu (or BPQ or sort). A oun prort quu s smlr to rulr prort quu, pt tt tr s uppr oun on t numr o lmnts tt n stor n t BPQ. Wnvr nw lmnt s to t quu, t quu s t pt, t lmnt wt t st prort vlu s jt rom t quu. For mpl, suppos tt w v BPQ wt mmum sz v tt ols t ollown lmnts: Vlu A B C D E Prort Suppos tt w wnt to nsrt t lmnt F wt prort 0.4 nto ts oun prort quu. Bus ts BPQ s mmum sz v, ts wll nsrt t lmnt F, ut tn vt t lowst-prort lmnt (E), ln t ollown BPQ: Vlu A B F C D Prort Now suppos tt w ws to nsrt t lmnt G wt prort 4.0 nto ts BPQ. Bus G's prort vlu s rtr tn t mmum-prort lmnt n t BPQ, upon nsrtn G t wll mmtl vt. In otr wors, nsrtn n lmnt nto BPQ wt prort rtr tn t mmum-prort lmnt o t BPQ s no t. Gvn ss to BPQ, w n prorm k-nn sr n k-tr s ollows:

74 Lt t tst pont P = ( 0, 1,..., k ). Mntn BPQ o t nt nrst nors, ll 'pq' St t mmum sz o 'pq' to k Strtn t t root, ut t ollown prour: urr == NULL rturn /* A t urrnt pont to t BPQ. Not tt ts s no-op t * pont s not s oo s t ponts w'v sn so r. */ nquu urr nto pq wt prort stn(urr, P) /* Rursvl sr t l o t tr tt ontns t tst pont. */ < urr rursvl sr t lt sutr on t nt s ls rursvl sr t rt sutr on t nt s /* I t nt prspr rosss ts splttn pln, look on t * otr s o t pln mnn t otr sutr. */ : pq sn't ull -or- urr s lss tn t prort o t m-prort lm o pq tn rursvl sr t otr sutr on t nt s Tr r two mnor ns to ts lortm tt rntt t rom t ntl 1-NN sr lortm. Frst, wn trmnn wtr to look on t oppost s o t splttn pln, w us s t rus o t nt prspr t stn rom t tst pont to t mmum-prort pont n t BPQ. T rtonl n ts s tt wn nn t k nrst nors, our nt prspr or t k nrst ponts ns to nompss ll k o tos nors, not just t losst. T otr mn n s tt wn w onsr wtr to look on t oppost s o t splttn pln, our son tks nto ount wtr t BPQ ontns t lst k ponts. Ts s trml mportnt! I w prun out prts o t tr or w v m t lst k usss, w mt ntll trow out on o t losst ponts. Consr t ollown stup: Suppos tt w ws to prorm 2-NN lookup or t tst pont nt t str. W rursvl k t lt sutr o t splttn pln, n n t pont nt n lu s nt nrst nor. Sn w vn't oun two nrst nors t, w stll n to look on t otr s o t splttn pln or mor nors, vn tou t nt prspr os not ross t splttn prpln

75 T Assnmnt Your ssnmnt s to mplmnt lss rprsntn k-tr, w w'll ll KDTr, tt llows lnts to ul k-trs, qur k-trs or mmrsp, n ut k-nn lookups on tm. In t ours o on so, ou'll n prn wt lss mplmntton, onst-orrtnss, tmplts, op untons, oprtor ovrlon, n pton-nln. Atonll, ou'll t to prn rstn t powr o k-nn lookups sn ppltons tt ul o o our KDTr lss. T mount o o tt ou tull n to wrt s not too rt on t orr o two unr lns tou t wll rqur ou to v sol unrstnn o t lnu turs w'v plor ovr t pst wks. To mk t sr to omplt t ssnmnt, I'v rokn t prorm own nto srs o v smllr stps. I vs ompltn t ssnmnt n ts orr, ut ou'r r to mplmnt KDTr s ou s t. Stp Zro: St up t Projt Unlk Evl Hnmn, ts ssnmnt s r mount o strtr o, mostl or t smpl ppltons. St t up lk ou woul n otr QT projt. Stp On: Implmnt Bs Funtonlt Now tt ou'v ottn t projt st up, t's tm to strt mplmntn KDTr. T KDTr mplmntton ou'll wrtn s tull slt vrnt on t k-tr strutur sr rlr n ts nout tt ssots ulr t wt pont. In sns, our KDTr wll t lk n mp rom ponts n sp to vlus. For mpl, ou oul us KDTr to mp rom lontu/lttu prs to ts, rom omtr t to ss pronoss, or rom ms to lls on tos ms. Blow s prtl spton o t KDTr lss, ltn t untons ou'll n to wrt to t s untonlt workn. Bs (nomplt) KDTr ntr tmplt <sz_t N, tpnm ElmTp> lss KDTr { pul: KDTr(); ~KDTr(); sz_t mnson() onst; sz_t sz() onst; ool mpt() onst; vo nsrt(onst Pont<N>& pt, onst ElmTp& vlu); }; ool ontns(onst Pont<N>& pt) onst; ElmTp& oprtor[] (onst Pont<N>& pt); ElmTp& t(onst Pont<N>& pt); onst ElmTp& t(onst Pont<N>& pt) onst; You m v not tt KDTr s n unusul tmplt sntur:

76 tmplt <sz_t N, tpnm ElmTp> lss KDTr You not msr tt t KDTr mplmntton s prmtrz ovr sz_t s wll s tp. W v not suss ntr tmplt rumnts or, ut t v just lk rulr tp tmplt rumnts. I ou wnt to rt KDTr tt mps rom ponts n tr-mnsonl sp to strns, ou oul lr t s KDTr<3, strn> mkdtr; T ks n t KDTr r ojts o tp Pont<N>, wr N s t mnson o t KDTr. Tt s, KDTr<3, strn> uss Pont<3>s s ks, KDTr<2, tt> woul us Pont<2>s s ks, t. I'v prov ull-workn mplmntton o Pont n t strtr o; t vs lk -sz STL vtor<oul>. For mpl: Pont<3> pt; pt[0] = 137.0; pt[1] = 42.0; pt[2] = ; I vs lookn ovr t Pont. r l to s wt otr untonlt sts. You'r r to tn ts lss owvr ou l, ut ou souln't n to o so or ts ssnmnt. Gvn ts tl out t KDTr n Pont tps, ou soul n t ssnmnt mplmntn t ollown mmr untons on KDTr: KDTr(); ~KDTr(); sz_t mnson() onst; sz_t sz() onst; ool mpt() onst; vo nsrt (onst Pont<N>& pt, onst ElmTp& vlu); ool ontns(onst Pont<N>& pt) onst; ElmTp& oprtor[] (onst Pont<N>& pt); Construts nw, mpt KDTr. Dstros t KDTr n llots ll ts rsours. Rturns t mnson o t ponts stor n t KDTr. (Ts s t vlu o t tmplt prmtr N). Rturns t numr o lmnts stor n t KDTr n wtr or not t s mpt, rsptvl. Insrts t sp pont nto t KDTr wt ssot vlu vlu. I t pont lr sts n t KDTr, t ol vlu s ovrwrttn. Rturns wtr t sp Pont s ontn n t KDTr. Rturns rrn to t vlu ssot wt t pont pt. I t pont os not st n t KDTr, t s wt t ult vlu o ElmTp s ts vlu, n rrn to ts nw vlu s rturn. Ts s t sm vor s t STL mp's oprtor[]. Not tt ts unton os not v onst ovrlo us t unton m mutt t tr

77 ElmTp& t(onst Pont<N>& pt); onst ElmTp& t(onst Pont<N>& pt) onst; Rturns rrn to t vlu ssot wt t pont pt, t sts. I t pont s not n t tr, tn ts unton trows n out_o_rn pton. Ts unton s onst-ovrlo, sn t os not n t tr. Not tt t lst our untons (ontns, oprtor[], n t two vrsons o t) ll o som sr o t KDTr lookn or prtulr vlu, rn onl n tr vor wn t pont s not ontn n t tr. ontns rturns ls, oprtor[] s nw lmnt, n t trows n out_o_rn pton. Rtr tn wrtn t o to trvrs t tr our tms n ustomzn t vor wn n lmnt sn't oun, I stronl sust wrtn lpr unton tt srs t tr or prtulr pont, tn rturns pontr to t no ontnn t. You n tn mplmnt ts untons on top o ts ommon suroutn. As n mpl, r's smpl mplmntton o ontns tt ssums t stn o lpr unton nno: tmplt <sz_t N, tpnm ElmTp> ool KDTr<N, ElmTp>::ontns(onst Pont<N>& pt) onst { rturn nno(pt)!= NULL; } To k wtr ou v our o workn, ou n run t rst st o tsts rom t projt tst-rnss. I ts rport n rrors, sur to orrt tm or movn on. You m lso wnt to tsts o our own. Stp Two: Implmnt Nrst-Nor Lookup Now tt ou v t s untonlt r, t's tm to mplmnt k-nn srs. Your nt tsk s to mplmnt t knnvlu unton, w looks lk ts: Etn (stll nomplt) KDTr ntr tmplt <sz_t N, tpnm ElmTp> lss KDTr { pul: KDTr(); ~KDTr(); sz_t mnson() onst; sz_t sz() onst; ool mpt() onst; vo nsrt(onst Pont<N>& pt, onst ElmTp& vlu); ool ontns(onst Pont<N>& pt) onst; ElmTp& oprtor[] (onst Pont<N>& pt); ElmTp& t(onst Pont<N>& pt); onst ElmTp& t(onst Pont<N>& pt) onst; }; ElmTp knnvlu(onst Pont<N>& pt) onst; Ts unton tks n pont n sp n numr o nors. It soul tn o k-nn sr n t k-tr usn pt s t tst pont. Atr on so, t wll v oun ollton o t k nr

78 st ponts n sp, lon wt t ElmTp vlus ssot wt tm. T rturn vlu o ts unton soul t most-rquntl-ourrn vlu ssot wt t k-nrst-nors o t tst pont. In t vnt o t, ou n rturn n o t strns tt t or most rqunt. For mpl, vn t ollown ollton o ponts (ll wt strns) n t nt tst pont: Y Y N N N Y N N N Y Y Y N I w 3-NN lookup, t knnvlu unton soul rturn "Y". T lortm or on k-nn lookup ssums t stn o oun prort quu, n to mk ts ssnmnt sr to omplt I'v prov ou BounPQuu lss w os just tt. You m wnt to look ovr ts ntr or strtn work on ts prt o t ssnmnt. You mt wonrn w ts unton rturns t most ommon ll o t nr ponts rtr tn t ponts tmslvs. Ts s mostl us t smpl ppltons unl wt ts projt ll us t k-nn sr n t mnnr port ts unton n I n't l lk nlssl upltn o. T tst rnss ontns two untons w tst ts unton. Enl tm n onrm tt our o works or movn on to t nt ston. Stp Four: Implmnt Cop Funtons As wrttn, t KDTr lss s strutor ut no op onstrutor or ssnmnt oprtor. Ts mns tt C++ wll prov t lss ult vrsons o ts untons, w wll us rs - s. To prvnt ts, ou wll n to mplmnt op onstrutor n ssnmnt oprtor or t KDTr lss. Ts rsults n t nl ntr o t KDTr lss: Complt KDTr ntr tmplt <sz_t N, tpnm ElmTp> lss KDTr { pul: KDTr(); ~KDTr(); KDTr(onst KDTr& otr); KDTr& oprtor= (onst KDTr& otr); sz_t mnson() onst; sz_t sz() onst; ool mpt() onst; vo nsrt(onst Pont<N>& pt, onst ElmTp& vlu); ool ontns(onst Pont<N>& pt) onst;

79 }; ElmTp& oprtor[] (onst Pont<N>& pt); ElmTp& t(onst Pont<N>& pt); onst ElmTp& t(onst Pont<N>& pt) onst; ElmTp knnvlu(onst Pont<N>& pt) onst; You r r to mplmnt ts untons s ou s t, ut I stronl nour ou to r ovr Cptr 11 n t ours rr or on so. It s surprsnl s to t ts untons wron, n ou wll wnt to nsur tt ou unrstn wt to wt out or or ou strt on tm up. T tstn rnss ontns two tsts tt rs t op untons, on kn t s untonlt n on lusvl kn ss. Mk sur tt our mplmntton psss t tsts or movn on. Stp Four: Run Smpl Appltons Conrtultons! You'v just omplt our KDTr. Tk som tm to pl roun wt t smpl ppltons tt v n unl wt t strtr o. Tr r tr ppltons ou n k out, o w s sr r: Mp Lookup. Ts prorm prsnts mp o t worl n lts ou lk on vrous lotons. It tn uss 1-NN lookup to trmn w ountr t slt loton s n, lon wt t stt/provn wtn tt ountr t loton s n. T prorm uss ol US ovrnmnt t rom t Ntonl Gosptl-Intlln An n US Golol Surv. I ou' lk to rtrv t rw t ls on w t t or ts prorm s s, k out t ollown lnks: tp://tp.n.ml/pu2/ns_t/onms ms_t_ zp ttp://onms.uss.ov/os/sttz/ntonlfl_ zp ttp://rt-no.n.ml/ns/tml/geopolitical_codes.ls Color Nmn. Rnll Munro, utor o t wom k, rn surv n w prtpnts wr sown rnom olor n sk to nm tt olor. T rsults o t olor surv wr tn rls to t nrl pul on s l (ttp://l.k.om). T t st ontns tr mllon prs o olors (no s RGB trplts) n t rsponnts' nms or tos olors. T Color Nmn pplton pulls up t sstm olor oosr lo, lts lnts oos olors, tn rports t 3-NN nm o tt olor s on ru sust o tt t. I ou' lk t rw t ls I us to ul t t st, ou n n t onln t t lnk low. B wrn t t s not n ltr n som o t olor nms r rtnl NSFW. ttp://k.om/olor/olorsurv.tr.z Dt Clsston. Erlr n t qurtr w m r or nto mn lrnn wrtn prptron lssr tt oul ronz nwrttn ts. An ltrntv mns or prormn ts lsston uss t k-nn lortm. T Dt Clssr pplton prsnts ou nvs on w ou n rw t twn 0 n 9, tn uss k-nn to uss wt t t ou wrot ws. T prorm s vr oo ur, tou t

80 os mk t osonl mstk. T rw t or ts prorm ws otn rom t MNIST ts t ttp://nn.lun.om//mnst/ Wn runnn ts smpl prorms, I sust ompln tm wt optmzton turn on n un turn o. Lon n prossn mts o t tks tm, n t ovr rom un nstrumntton n mk t prorms tk vr lon tm to lo. Evn wt optmzton turn on, t prorms n stll tk wl to lo t Color Nmn prorm tks n spll lon tm to lo sn t s to ul k-tr out o two mllon t ponts. Also, wr tt ts prorms wll us lot o RAM! Av, Tps, n Trks Hr r w sp pontrs tt mt mk our l lot sr s ou o trou ts ssnmnt: Don't stt to sk qustons! Ts ssnmnt uss mn o t C++ tnqus w'v sn ovr t pst w wks. I ou'r vn troul ttn our o to ompl, or n't rmmr wt kwor ou'r suppos to usn somwr, ml t st lst ( 106lst@s.stnor.u), m, or o to t LIR n I n tr to pont ou n t rt rton. Ts ssnmnt s not s r s t m sm. Ts nout s rl ns, ut t tul mount o o ou n to wrt s not tt rt. You r onl rsponsl or mplmntn w untons, som o w n mplmnt n snl ln o o. I ou tk t tm to tnk trou ow ll t untons r rlt to on notr, ou n sv oursl mu on ort mplmntn t untons n trms o otr. Wt out or tpnm wrnss wn mplmntn untons. Your mplmntton o KDTr wll rqur t us o nst tp to rprsnt nos n t tr. I ou wrt n prvt lpr untons tt rturn ojts o ts tp, ou wll n to us t tpnm kwor wn mplmntn tos untons. For mpl, suppos tt ou n lpr strut ll No n tn n unton tt rturns No*, s sown r: prvt: strut No { /*... */ }; No* nno(onst Pont<N>& pt); T mplmntton o ts unton woul tn v ts sntur: tmplt <sz_t N, tpnm ElmTp> tpnm KDTr<N, ElmTp>::No* KDTr<N, ElmTp>::nNo(onst Pont<N>& pt); Tt's rl moutul, n unortuntl t's t onl w to ommunt to t omplr wt ou'r trn to mplmnt. Mk sur ou unrstn t us o tpnm, lon wt w t tmplt rumnts r uplt n two pls

81 B rul out onst-orrtnss. I ou rt n prvt mmr untons to ssst n t mplmnttons o t KDTr pul ntr, mk sur tos mmr untons r mrk onst wr pproprt. In prtulr, ontns, t, n knnvlu r onst, so t ll n mmr untons, tos untons must mrk onst s wll. You wll t som rl roous omplr rrors ou tr lln non-onst mmr unton rom onst mmr unton, so wr. Us s nst o s. T <mt> r l ports two smlr-sounn untons to omput solut vlu, s n s. In ts ssnmnt, ou soul not us t s unton. s works on ntrl vlus, so ou pss n oul, t rturn vlu wll norrtl roun to n nt. s s sn to work on lots n ouls, n s mu mor pproprt unton. Rmmr t onst_st/stt_st trk. T KDTr ontns two untons nm t tt r onl n tr onstnss. Rtr tn wrtn two ops o t sm o, ou n us t onst_st/stt_st trk to mplmnt t non-onst vrson n trms o t onst vrson. Look ovr t ltur o or t Vtor lss or mor tls. Etnsons I ou'r ntrst n srpnn our C++ sklls, wnt to o mor vn oprtons on t ktr, or l lk spnn lz Sun on w urousl, w not som tnsons to our KDTr? Blow s lst o possl tnson s, som o w r strtorwr, wl otrs wll rqur snnt tm n ort. I ou n up ompltn n o ts, lt m know n I' l to look ovr wt ou'v wrttn! Bul t k-tr mor ntllntl. Trtonll, k-trs r not ult on lmnt t tm, ut rtr rom omplt t st ll t on. To nsur tt t tr s ln, t lmnts r sort tr rst omponnt, t mn s us s t root o t tr, n t rmnn lmnts r tn rursvl suv nto lrn o t root no. Implmnt nw onstrutor or t KDTr lss to ul up t tr n ts son. A support or otr stn mtrs. Wn on nrst-nor lookup, w us Euln stn s msur o losnss twn two ponts n tr to n pont n t k-tr wt t lst Euln stn to t tst pont. Howvr, t's possl to us ll sorts o otr stn mtrs, su s Mnttn stn or t mmum norm. A support to KDTr to tr out ts nw stn mtrs. How os t vor o t smpl ppltons n? Coos s mor ntllntl. T urrnt k-tr mplmntton ls trou w s t splts on wt lvl o t tr. A mor lvr woul to splt lon t lonst s o t t st wt t ol o sprn t ponts out mor vnl. Upt t KDTr lss to us ts untonlt. A support or rn srs. On ommon oprtons on k-trs s rn sr, wr t nput s rtnl n sp n t output s t st o ponts n t k-tr ontn n tt rtnl. Ts vs mu ttr lortm or t CtFnr prorm

82 tn t on w wrot rlr n t qurtr. Rsr ow to mplmnt ts unton, tn t to KDTr. A support or lmnt rmovl. T KDTr ou'v wrttn n v nw lmnts, ut nnot rmov stn lmnts. Dvlop n lortm to rmov rrr ponts rom k-tr, tn upt our KDTr ntr to support ts. B rtv! Tnk o n lvr uss or k-tr? How out somtn ou oul o to mk t k-tr mor nt? I ou v n s ou' lk to tr out, ll mns o or t n I' lov to s wt ou om up wt. Dlvrls To sumt t ssnmnt, uplo our upt KDTr. l, lon wt n otr ls ou mt v t, to pprlss. I ou'v n tnsons or spl turs I soul wr o, lt our rr know n our ommnts. I woul lso pprt t ou or som k on ts ssnmnt s wll s t lss s wol ws t ntrstn? Too s? Too r? Just rt? Fnll, pt oursl on t k ou'v just omplt t lst ssnmnt o CS106L n r now vtrn C++ prormmr. Conrtultons! Goo luk!

5/1/2018. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees

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