Reading. K-D Trees and Quad Trees. Geometric Data Structures. k-d Trees. Range Queries. Nearest Neighbor Search. Chapter 12.6

Transcription

1 Rn Cptr 12.6 K-D Trs n Qu Trs CSE 326 Dt Struturs Ltur 9 2/2/05 K-D Trs n Qu Trs - Ltur 9 2 Gomtr Dt Struturs Ornzton o ponts, lns, plns, to support str prossn Appltons Astropsl smulton voluton o ls Grps omputn ojt ntrstons Dt omprsson Ponts r rprsnttvs o 22 loks n n m Nrst nor sr 2/2/05 K-D Trs n Qu Trs - Ltur 9 3 k- Trs Jon Bntl, 1975, wl n unrrut Tr us to stor sptl t. Nrst nor sr. Rn qurs. Fst look-up k- tr r urnt lo 2 n pt wr n s t numr o ponts n t st. Trtonll, k- trs stor ponts n - mnsonl sp w r quvlnt to vtors n -mnsonl sp. 2/2/05 K-D Trs n Qu Trs - Ltur 9 4 Rn Qurs Nrst Nor Sr qur Rtnulr qur Crulr qur Nrst nor s. 2/2/05 K-D Trs n Qu Trs - Ltur 9 5 2/2/05 K-D Trs n Qu Trs - Ltur 9 6 1

2 k- Tr Construton k- Tr Construton (1) I tr s just on pont, orm l wt tt pont. Otrws, v t ponts n l ln prpnulr to on o t s. Rursvl onstrut k- trs or t two sts o ponts. Dvson strts v ponts prpnulr to t s wt wst spr. v n roun-ron son (ook os t ts w) v prpnulr to t wst spr. 2/2/05 K-D Trs n Qu Trs - Ltur 9 7 2/2/05 K-D Trs n Qu Trs - Ltur 9 8 k- Tr Construton (2) k- Tr Construton (3) 2/2/05 K-D Trs n Qu Trs - Ltur 9 9 2/2/05 K-D Trs n Qu Trs - Ltur 9 10 k- Tr Construton (4) k- Tr Construton (5) 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur

3 k- Tr Construton (6) k- Tr Construton (7) 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur 9 14 k- Tr Construton (8) k- Tr Construton (9) 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur 9 16 k- Tr Construton (10) k- Tr Construton (11) 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur

4 k- Tr Construton (12) k- Tr Construton (13) 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur 9 20 k- Tr Construton (14) k- Tr Construton (15) 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur 9 22 k- Tr Construton (16) k- Tr Construton (17) 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur

5 k- Tr Construton (18) k- tr ll 2- Tr Domposton 2 1 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur 9 26 k- Tr Splttn k- Tr Splttn sort ponts n mnson m spr s t m o - n -. In t slt mnson t ml pont n t lst splts t t. To ul t sort lsts or t otr mnsons sn t sort lst n pont to on o two sort lsts. sort ponts n mnson ntor or st sn sort ponts n mnson n to orrt st 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur 9 28 k- Tr Construton Complt 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 v nto two susts n on n O(n) tm. W v t rurrn T(n,) < 2T(n/2,) + O(n) Construtn t k- tr n on n O(n lo n) n n stor No Strutur or k- Trs A no s 5 ls s (splttn s) vlu (splttn vlu) lt (lt sutr) rt (rt sutr) pont (ols pont lt n rt lrn r null) 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur

6 Rtnulr Rn Qur Rtnulr Rn Qur (1) Rursvl sr vr ll tt ntrsts t rtnl. 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur 9 32 Rtnulr Rn Qur (2) Rtnulr Rn Qur (3) 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur 9 34 Rtnulr Rn Qur (4) Rtnulr Rn Qur (5) 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur

7 Rtnulr Rn Qur (6) Rtnulr Rn Qur (7) 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur 9 38 Rtnulr Rn Qur (8) Rtnulr Rn Qur prnt_rn(low,, low, :ntr, root: no pontr) { Cs { root = null: rturn; root.lt = null n root.rt = null : low < root.pont. n root.pont. < n low < root.pont. n root.pont. < tn prnt(root); (root.s = n low < root.vlu ) or (root.s = n low < root.vlu ) : prnt_rn(low,, low,, root.lt); (root.s = n low > root.vlu ) or (root.s = n low > root.vlu ) : prnt_rn(low,, low,, root.rt); }} 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur 9 40 Anlss o Rtnulr Rn Qur Tm or rtnulr rn qur s O(k + lo n) wr k s t numr o ks oun. Anlss: Lt T(n,R) t tm or k- tr o sz 1 n rtnl qur R. T(1,R) < T(n,R) < T(n/2,R 1 ) + T(n/2,R 2 ) + wr R s t unon o R 1 n R 2 Sow nuton on n tt T(n,R) < (k + lo n) or som. k- Tr Nrst Nor Sr Sr rursvl to n t pont n t sm ll s t qur. On t rturn sr sutr wr losr pont tn t on ou lr know out mt oun. 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur

8 k- Tr NNS (1) k- Tr NNS (2) qur pont qur pont 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur 9 44 k- Tr NNS (3) k- Tr NNS (4) qur pont qur pont w 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur 9 46 k- Tr NNS (5) k- Tr NNS (6) qur pont w qur pont w 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur

9 k- Tr NNS (7) k- Tr NNS (10) qur pont w qur pont w 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur 9 50 k- Tr NNS (11) k- Tr NNS (12) qur pont w qur pont w 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur 9 52 k- Tr NNS (13) k- Tr NNS (14) qur pont w qur pont w 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur

10 qur pont w k- Tr NNS (15) 2/2/05 K-D Trs n Qu Trs - Ltur 9 55 Mn s NNS(q,root,null,nnt) Nrst Nor Sr NNS(q: pont, n: no, p: pont, w: stn) : pont { n.lt = null tn {l s} stn(q,n.pont) < w tn rturn n.pont ls rturn p; ls w = nnt tn q(n.s) < n.vlu tn p := NNS(q,n.lt,p,w); w := stn(p,q); q(n.s) + w > n.vlu tn p := NNS(q, n.rt, p, w); ls p := NNS(q,n.rt,p,w); w := stn(p,q); q(n.s) - w < n.vlu tn p := NNS(q, n.lt, p, w); ls //w s nt// q(n.s) - w < n.vlu tn p := NNS(q, n.lt, p, w); w := stn(p,q); q(n.s) + w > n.vlu tn p := NNS(q, n.rt, p, w); rturn p } 2/2/05 K-D Trs n Qu Trs - Ltur 9 56 T Contonl Worst-Cs or Nrst Nor Sr n.s = q(n.s) + w > n.vlu qur pont Hl o t ponts vst or qur Worst s O(n) q w But: on vr (n n prt) nrst nor qurs r O(lo N) Currnt nrst pont n.vlu q(n.s) + w 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur 9 58 Nots on k- NNS Hs n sown to run n O(lo n) vr tm pr sr n rsonl mol. (Assum onstnt) Stor or t k- tr s O(n). Prprossn tm s O(n lo n) ssumn s onstnt. Sp Prttonn Qu Trs 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur

11 Sp Prttonn Qu Trs Sp Prttonn Qu Trs 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur 9 62 A B Cs Nots on Qu Trs Numr o nos s O(n(1+ lo( /n))) wr n s t numr o ponts n s t rto o t wt (or t) o t k sp n t smllst stn twn two ponts Ht o t tr s O(lo n + lo ) 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur 9 64 K-D vs Qu k-d Trs Dnst ln trs Ht o t tr s O(lo n) wt t nsrton Goo o or mnson Supports nsrt, n, nrst nor, rn qurs Qu Trs Sp prttonn tr M not ln Not oo o or mnson Supports nsrt, lt, n, nrst nor, rn qurs Gomtr Dt Struturs Gomtr t struturs r ommon. T k- tr s on o t smplst. Nrst nor sr Rn qurs Otr t struturs us or 3- rps mols Psl smultons 2/2/05 K-D Trs n Qu Trs - Ltur /2/05 K-D Trs n Qu Trs - Ltur