K-D Trees and Quad Trees. James Fogarty Autumn 2007 Lecture 12

Transcription

1 K-D Trs n Qu Trs Jms Fort Autumn 2007 Ltur 12

2 Rn Qurs Tnk o rn qur. Gv m ll ustomrs Gv m ll ounts wort $5m to $15m Cn on n tm. Wt w wnt ot: Gv m ll ustomrs wt ounts wort twn $5m n $15m. 2

3 Gomtr Dt Struturs Ornzton o ponts, lns, plns, t n support o str prossn Appltons Mp normton Grps - omputn ojt ntrstons Dt omprsson - nrst nor sr Dson Trs - mn lrnn 3

4 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. 4

5 Rn Qurs Rtnulr qur Crulr qur 5

6 Nrst Nor Sr qur Nrst nor s. 6

7 k- Tr Construton 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) 7

8 k- Tr Construton (1) v prpnulr to t wst spr. 8

9 k- Tr Construton (2) 9

10 k- Tr Construton (3) 10

11 k- Tr Construton (4) 11

12 k- Tr Construton (5) 12

13 k- Tr Construton (6) 13

14 k- Tr Construton (7) 14

15 k- Tr Construton (8) 15

16 k- Tr Construton (9) 16

17 k- Tr Construton (10) 17

18 k- Tr Construton (11) 18

19 k- Tr Construton (12) 19

20 k- Tr Construton (13) 20

21 k- Tr Construton (14) 21

22 k- Tr Construton (15) 22

23 k- Tr Construton (16) 23

24 k- Tr Construton (17) 24

25 k- Tr Construton (18) k- tr ll 25

26 2- Tr Domposton

27 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. 27

28 k- Tr Splttn sort ponts n mnson ntor or st sn sort ponts n mnson n to orrt st 28

29 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 29

30 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) 30

31 Rtnulr Rn Qur Rursvl sr vr ll tt ntrsts t rtnl. 31

32 Rtnulr Rn Qur (1) 32

33 Rtnulr Rn Qur (2) 33

34 Rtnulr Rn Qur (3) 34

35 Rtnulr Rn Qur (4) 35

36 Rtnulr Rn Qur (5) 36

37 Rtnulr Rn Qur (6) 37

38 Rtnulr Rn Qur (7) 38

39 Rtnulr Rn Qur (8) 39

40 Rtnulr Rn Qur prnt_rn(low,, low, :ntr, root: no pontr) { Cs { root = null: rturn; root.lt = null: low < root.pont. n root.pont. < n low < root.pont. n root.pont. < tn prnt(root); ls (root.s = n low < root.vlu ) or (root.s = n low < root.vlu ) tn prnt_rn(low,, low,, root.lt); (root.s = n low > root.vlu ) or (root.s = n low > root.vlu ) tn prnt_rn(low,, low,, root.rt); }} 40

41 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. 41

42 k- Tr NNS (1) qur pont 42

43 k- Tr NNS (2) qur pont 43

44 k- Tr NNS (3) qur pont 44

45 k- Tr NNS (4) qur pont w 45

46 k- Tr NNS (5) qur pont w 46

47 k- Tr NNS (6) qur pont w 47

48 k- Tr NNS (7) qur pont w 48

49 k- Tr NNS (10) qur pont w 49

50 k- Tr NNS (11) qur pont w 50

51 k- Tr NNS (12) qur pont w 51

52 k- Tr NNS (13) qur pont w 52

53 k- Tr NNS (14) qur pont w 53

54 k- Tr NNS (15) qur pont w 54

55 Nots on k- NNS Hs n sown to run n O(lo n) vr tm pr sr n rsonl mol. Stor or t k- tr s O(n). Prprossn tm s O(n lo n) ssumn s onstnt. 55

56 Worst-Cs or Nrst Nor Sr qur pont Hl o t ponts vst or qur Worst s O(n) But: on vr (n n prt) nrst nor qurs r O(lo N) 56

57 Sp Prttonn Qu Trs 57

58 Sp Prttonn Qu Trs 58

59 Sp Prttonn Qu Trs 59

60 A B Cs 60

61 Nots on Qu Trs Numr o nos s O(n(1+ lo(δ/n))) wr n s t numr o ponts n Δ 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 Δ) 61

62 k-d Trs K-D vs Qu 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 62

63 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 63