The R-Tree. Yufei Tao. ITEE University of Queensland. INFS4205/7205, Uni of Queensland

Transcription

1 Yu To ITEE Unvrsty o Qunsln

2 W wll stuy nw strutur ll t R-tr, w n tout o s mult-mnsonl xtnson o t B-tr. T R-tr supports ntly vrty o qurs (s w wll n out ltr n t ours), n s mplmnt n numrous ts systms. Our susson n ts ltur wll ous on ortoonl rn rportn.

3 2D Ortoonl Rn Rportn (Wnow Qury) Lt S st o ponts n R 2. Gvn n xs-prlll rtnl q, rn qury rturns ll t ponts o S tt r ovr y q, nmly, S q. T nton n xtn to ny mnsonlty n strtorwr mnnr. Exmpl T rsult s {,, } or t s rtnl q. l k

4 Appltons Fn ll rsturnts n t Mnttn r. Fn ll prossors wos s r n [20, 40] n tr nnul slrs r n [200k, 300k]....

5 R-Tr E l no s twn 0.4B n B t ponts, wr B 3 s prmtr. T only xpton ppls wn t l s t root, n w s t s llow to v twn 1 n B ponts. All t l nos r t t sm lvl. E ntrnl no s twn 0.4B n B l nos, xpt wn t no s t root, n w s t ns to v t lst 2 l nos. In prt, or sk-rsnt R-tr, t vlu o B pns on t lok sz o t sk so tt no s stor n lok.

6 R-Tr For ny no u, not y S u t st o ponts n t sutr o u. Consr now u to n ntrnl no wt l nos v 1,..., v ( B). For v ( ), u stors t mnmum ounn rtnl (MBR) o S v, not s MBR(v ). T ov s n MBR on 7 ponts.

7 Exmpl Assum B = l 6 u u u 3 8 k 3 l k u 4 u 5 u 6 u 7 u 8

8 Answrn Rn Qury Lt q t sr ron o rn qury. Blow w v t psuo-o o t qury lortm, w s nvok s rn-qury(root, q), wr root s t root o t tr. Alortm rn-qury(u, r) 1. u s l tn 2. rport ll ponts stor t u tt r ovr y r 3. ls 4. or l v o u o 5. MBR(v) ntrsts r tn 6. rn-qury(v, r)

9 Exmpl Nos u 1, u 2, u 3, u 5, u 6 r ss to nswr t qury wt t s sr ron l 6 u u u 3 8 k 3 l k u 4 u 5 u 6 u 7 u 8

10 R-Tr Construton Cn B Artrry Hv you wonr wy t l nos r rt n ts wy? For xmpl, s t solutly nssry to roup n l nto l no? l k T R-tr nton s no orml onstrnt wtsovr on t roupn o t nto nos (unlk B-trs), ut som R-trs v poorr prormn tn otrs; s t nxt sl.

11 R-Tr Construton Cn B Artrry Is ts oo R-tr? 5 6 l k u 1 2 u 2 u l k u 4 u 5 u 6 u 7 u 8 3 Implton?

12 R-Tr Construton: A Common Prnpl In nrl, t onstruton lortm o t R-tr ms t mnmzn t prmtr sum o ll t MBRs. For xmpl, t lt tr s smllr prmtr sum tn t rt on. l k 5 6 l k

13 R-Tr Construton: A Common Prnpl Wy not mnmz t r? A rtnl wt smllr prmtr usully s smllr r, ut not t v vrs. Ltr n t ours, w wll s n nlyss tt ormlly vlts ts ntuton. T ov two rtnls v t sm r.

14 Insrton Lt p t pont n nsrt. T psuo-o low soul s nvok s nsrt(root, p), wr root s t root o t tr. Alortm nsrt(u, p) 1. u s l no tn 2. p to u 3. u ovrlows tn /* nmly, u s B + 1 ponts */ 4. nl-ovrlow(u) 5. ls 6. v oos-sutr(u, p) /* w sutr unr u soul w nsrt p nto? */ 7. nsrt(v, p)

15 Coos-Sutr W MBR woul you nsrt p nto? p Alortm oos-sutr(u, p) 1. rturn t l wos MBR rqurs t mnmum nrs n prmtr to ovr p. rk ts y vorn t smllst MBR.

16 Ovrlow Hnln Alortm nl-ovrlow(u) 1. splt(u) nto u n u 2. u s t root tn 3. rt nw root wt u n u s ts l nos 4. ls 5. w t prnt o u 6. upt MBR(u) n w 7. u s l o w 8. w ovrlows tn 9. nl-ovrlow(w)

17 Splttn L Essntlly w r ln wt t ollown prolm: Lt S st o B + 1 ponts. Dv S nto two sont sts S 1 n S 2 to mnmz t prmtr sum o MBR(S 1 ) n MBR(S 2 ), sut to t onton tt S 1 0.4B n S 2 0.4B. Exmpl T lt splt s ttr: k k S 1 = {,,,, } S 1 = {,,,, } S 2 = {,,,,, k} S 2 = {,,,,, k}

18 Splttn L No Lt m = S. In 2D sp, t l-splt prolm n solv n O(m 5 ) tm, notn tt MBR s trmn y 4 ponts. Ts, owvr, s too xpnsv. In prt, ursts r us to lrt t pross, ut tr s no urnt tt w n n t st splt typl trn qulty or ny. T nxt sl xplns ow.

19 Splttn L No Alortm splt(u) 1. m = t numr o ponts n u 2. sort t ponts o u on x-mnson 3. or = 0.4B to m 0.4B 4. S 1 t st o t rst ponts n t lst 5. S 2 t st o t otr ponts n t lst 6. lult t prmtr sum o MBR(S 1 ) n MBR(S 2 ); ror t ts s t st splt so r 7. Rpt Lns 2-6 wt rspt to y-mnson 8. rturn t st splt oun

20 Exmpl Tr r 3 possl splts lon t x-mnson. Rmmr tt no must v t lst 0.4B = 4 ponts (r B = 10).

21 Tnk: How to mplmnt t lortm n O(n lo n) tm? Fn ountr-xmpl wr t lortm os not v n optml splt. W v suss only t 2D s. How to xtn t lortm to mnsonlty 3?

22 Splttn n Intrnl No Lt S st o B +1 rtnls. Dv S nto two sont sts S 1 n S 2 to mnmz t prmtr sum o MBR(S 1 ) n MBR(S 2 ), sut to t onton tt S 1 0.4B n S 2 0.4B. On n, w wll sttl or n lortm tt s st ut os not lwys rturn n optml splt.

23 Splttn n Intrnl No Alortm splt(u) /* u s n ntrnl no */ 1. m = t numr o ponts n u 2. sort t rtnls n u y tr lt ounrs on t x-mnson 3. or = 0.4B to m 0.4B 4. S 1 t st o t rst rtnls n t lst 5. S 2 t st o t otr rtnls n t lst 6. lult t prmtr sum o MBR(S 1 ) n MBR(S 2 ); ror t ts s t st splt so r 7. Rpt Lns 2-6 wt rspt to t rt ounrs on t x-mnson 8. Rpt Lns 2-7 w.r.t. t y-mnson 9. rturn t st splt oun

24 Exmpl Tr r 3 possl splts w.r.t. t lt ounrs on t x-mnson. Rmmr tt no must v t lst 0.4B = 4 ponts (r B = 10).

25 Insrton Exmpl Assum tt w wnt to nsrt t wt pont m. By pplyn oos-sutr tw, w r t l no u 6 tt soul ommot m. T no ovrlows tr norportn m (rll B = 3) l 3 m k u u 2 u l m k u 4 u 5 u 6 u 7 u 8

26 Insrton Exmpl No u 6 splts, nrtn u 9. An u 9 s l o u 3 uss u 3 to ovrlow l 6 3 m k u u 2 u l m k u 4 u 5 u 6 u 7 u 8 u 9

27 Insrton Exmpl No u 3 splts, nrtn u 10. T nsrton nss tr n u 10 s l o t root l k m 9 8 u u 2 u 3 4 u l m k u 4 u 5 u 6 u 9 u 7 u 8

28 Altou not rqur n ts ours, tr r st lortms to prorm ltons rom n R-tr. T nstrutor wll suss ts n t lss tm prmts.

29 2D Ortoonl Rn Rportn (Wnow Qury) on Rtnls Lt S st o xs-prlll rtnls n R 2. Gvn n xs-prlll rtnl r, rn qury rturns ll t rtnls o S tt r ovr y r, nmly, S r. T nton n xtn to ny mnsonlty n strtorwr mnnr. Exmpl T rsult s {,,,, } or t s rtnl q. l k

30 An R-Tr on Rtnls Sm s n R-tr on ponts wt two ns: L nos stor t rtnls (s oppos to ponts). Splttn l no s prorm usn t ntrnl-splt lortm mnton rlr.

31 An R-Tr on Ln Smnts Motvton Consr orp normton systm (GIS) tt mns ro smnts. T ov ur sows n xmpl wr t lu smnts rprsnt ro, wrs t rn ons rprsnt notr. Suppos tt w wnt to rtrv ll t ro smnts n t s wnow (tnk o t wnow s t usr s rowsr, or nstn). How to pt t R-tr to support ts typ o rtrvl?

32 Fltr Rnmnt A ommon soluton mplmnt n ommrl systms s s on t ltr rnmnt rmwork. In t ltr stp, w qukly rtrv nt st o smnts tt s urnt to ontn t nl rsult. T rnmnt stp tn xmns vry smnt n t nt st to rmov t ls ts. A promnnt ppro o rlz t ltr stp s to rt n MBR on vry smnt, n us n R-tr to n t MBRs tt ntrst t qury. T smnts orrsponn to tos MBRs onsttut t nt st.

33 Fltr Rnmnt T s rtnls r t MBRs on t smnts. Wt r t smnts rtrv y t ltr stp? Ar tr ny ls ts?