Z-ordering. Reduce n-dim to 1-dim points snake-curve z-ordering bit-shuffling Linear quad trees queries Hilbert curve

Transcription

1 Z-ordering Reduce n-dim to 1-dim points snake-curve z-ordering bit-shuffling Linear quad trees queries Hilbert curve Space filling curve L-Systems Fractals Hausdorff dimension 1

2 Motivation How would you represent n-dim points by a tree? 1. Use R-trees? 2. Reduce the problem to 1-dim points?...use B-trees! Ideas Reduce the problem to 1-dim points How? (PCA?) Other Ideas? Assume finite granularity (e.g., 2 32 x2 32 ; 4x4 here) How to map n-d cells to 1-d cells? 2

3 Mapping... How to map n-d cells to 1-d cells? Idea: Row-wise Is it good? Row-wise ordering Is row-wise ordering a good idea? Great for x axis; bad for y axis 3

4 Snake-curve How about the snake curve? Snake-curve Problem: Bad distance preservation! 4

5 z-ordering looks better: few long jumps; scoops out the whole quadrant before leaving it it is a space filling curve Space filling curve Example In the nature: Kidney arterial system 5

6 z-ordering How to generate this curve (z = f(x,y))? z (or N ) shapes, RECURSIVELY order-1 order-2... order (n+1) z-ordering Notice: self similar (we ll see about fractals, soon) method is hard to use: z =? f(x,y) order-1 order-2... order (n+1) 6

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8 Y X

9 bit-shuffing y x x 0 0 y 1 1 z =( ) 2 = 5 bit-shuffing y x x 1 1 y 1 0 z =( ) 2 = 14 9

10 Y z =( ) 2 = X Linear quad trees A quad tree is a tree whose nodes either are leaves or have 4 children The children are ordered 0, 1, 2, 3 (00, 01, 10, 11) W E 1 0 W,E,N,S: West,East,North,South N S

11 Quad-trees... and repeat recursively z gray-cell = WN;WN = (0101) 2 = 5 1 W E N S 0 1 z-ordering z-value of magenta cell, with the three methods W E Method 1: 14 (count) N S Method 2: shuffle(11;10)= (1110) 2 = 14 Method 3: EN;ES =... = 14 11

12 z-ordering... ask a range query Queries Find the z-values that contained in the query and then the ranges Q A Q B Q A range [4, 7] Q B ranges [3,4] and [9,10] 12

13 z-ordering - variations Is z-ordering the best we can do? Probably not - occasional long jumps Gray codes? z-ordering - variations Hilbert curve! 13

14 z-ordering - variations Looks better (never long jumps). How to derive it? Hilbert Curve Hilbert Space filling curve in 2 dimensions 1 st order 2 nd order 3 rd order 14

15 z-ordering - variations Looks better (never long jumps). How to derive it? order-1 order-2... order (n+1) 15

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19 z-ordering - variations How about Hilbert curve in 3-d? n-d? Exists (and is not unique!). Eg., 3-d, order-1 Hilbert curves (Hamiltonian paths on cube) #1 #2 Background Information 3-dimensional Hilbert Curve Unit 19

20 Background Information 20

21 z-ordering - analysis How to measure the goodness of a curve? z-ordering - analysis How to measure the goodness of a curve? E.g., avg. # of runs, for range queries 4 runs 3 runs (#runs ~ #accesses on B-tree) 21

22 z-ordering - analysis So, is Hilbert really better? 27% fewer runs, for 2-d (similar for 3-d) Are there formulas for #runs, #of quadtree blocks etc? Yes ([Jagadish; Moon+ etc] see textbook) z-ordering - fun observations Hilbert and z-ordering curves: space filling curves : eventually, they visit every point in n-d space - therefore: order-1 order-2... order (n+1) 22

23 z-ordering - fun observations... they show that the plane has as many points as a line (-> headaches for 1900 s mathematics/topology) (fractals, again!) order-1 order-2... order (n+1) z-ordering - fun observations In general, Hilbert curve is great for preserving distances, clustering, vector quantization etc... For more: H.V. Jagdish. Linear clustering of objects with multi- ple attributes. Proceedings of ACM SIGMOD, pages , May

24 Definition of Space-Filling Curves We can construct a mapping from a 1-D interval to a n-d interval where n is finite, i.e. we can construct mappings as follow: 2 Ι = [0,1] Ω = [0,1] 3 Ι = [0,1] Ω = [0,1] n Ι = [0,1] Ω = [0,1], n = 4,5,... If the curve of this mapping passes through every point of the target space we call this a Space- Filling Curve. Geometric Interpretation First we partition the interval I into n subintervals and the square Ω into n sub squares. Now one can easily establish a continuous mapping between these areas. This idea holds for any finite-dimensional target space / 9 2 / 9 1,1 0,0 1 / 3,0 1,0 24

25 Hilbert s Space-Filling Curve The German mathematician David Hilbert ( ) was the first one to give the so far only algebraically described space-filling curves a geometric interpretation In the 2 dimensional case he splat I into four subintervals and Ω into four sub squares. This splitting is recursively repeated for the emerging subspaces, leading to the below shown curve 0 1 / 4 2 / 4 3 / 4 4 / / 2 / 3 / / 16 1,1 16 1,1 / ,0 0, L-Systems Introduced 1968 by the biologist Aristid Lindenmayer as a Frame work fo studyingthe Development of simple multicellularorganisms 25

26 Deterministic, Context Free L-Systems (DOL) Grammar: a ab b a axiom: b Derivation: b a ab aba abaab abaababa Interpretation of a String String: FFF-FF-F-F+F+FF-F-FFF δ =

27 Hilbert Curve ω : L p 1 : L +RF LFL FR + p 2 : R LF + RFR + FL Replacement: Using additional symbols L, R to controll derivation Three Dimensional Hilbert Curve 27

28 Fractals A fractal is a shape that is recursively constructed or self-similar, that is, a shape that appears similar at all scales of magnification and is therefore often referred to as "infinitely complex Hillbert curve is a fractal We can define a fractal dimension Imagine a line whose length is 9 cm If we try to cover this line with smaller lines whose length is s=3 cm, we will need N=3 copies of our measuring line In this case N=s 28

29 Imagine a square whose every side is 9 cm If we try to cover this square with smaller squares whose side is s=3 cm, we will need N=9 copies of our measuring square N=s 2 N=s D, s=scale factor, N=Number of objects Hausdorff dimension D=log(N)/log(s) Note that for the Hilbert curve that consists of 4(=N) copies of itself Each twice (s=2) as short as the whole, D=log(4)/log(2)=2...surprising as the curve covers the whole of the square whose dimension is 2 29

30 Suprisingly the dimension may be not integer =log(n)/log(s) = log(3)/log(2) = 1.58 Felix Hausdorff (November 8, 1868 January 26, 1942) Felix Hausdorff (November 8, 1868 January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology When the Nazis came to power, Hausdorff, who was Jewish, felt that as a respected university professor he would be spared from persecution However, his abstract mathematics was denounced as "Jewish", useless, and "un-german" He could no longer publish in Germany, Hausdorff continued to be an active research mathematician, publishing in Polish journals Finally, in 1942 when he could no longer avoid being sent to a concentration camp, Hausdorff committed suicide together with his wife and sister-in-law on the 26th of January 30

31 Reduce n-dim to 1-dim points snake-curve z-ordering bit-shuffling Linear quad trees queries Hilbert curve Space filling curve L-Systems Fractals Hausdorff dimension 31