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Carnegie Mellon Carnegie Mellon Univ. Dept. of Computer Science 15-415 - Database Applications C. Faloutsos Spatial Access Methods - z-ordering.

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Presentation on theme: "Carnegie Mellon Carnegie Mellon Univ. Dept. of Computer Science 15-415 - Database Applications C. Faloutsos Spatial Access Methods - z-ordering."— Presentation transcript:

1 Carnegie Mellon Carnegie Mellon Univ. Dept. of Computer Science 15-415 - Database Applications C. Faloutsos Spatial Access Methods - z-ordering

2 Carnegie Mellon 15-415 - C. Faloutsos2 General Overview Relational model – SQL; db design Indexing; Q-opt;Transaction processing Advanced topics –Distributed Databases –RAID –Authorization / Stat. DB –Spatial Access Methods (SAMs) –Multimedia Indexing

3 Carnegie Mellon 15-415 - C. Faloutsos3 SAMs - Detailed outline spatial access methods –problem dfn –z-ordering –R-trees

4 Carnegie Mellon 15-415 - C. Faloutsos4 Spatial Access Methods - problem Given a collection of geometric objects (points, lines, polygons,...) organize them on disk, to answer spatial queries (like??)

5 Carnegie Mellon 15-415 - C. Faloutsos5 Spatial Access Methods - problem Given a collection of geometric objects (points, lines, polygons,...) organize them on disk, to answer –point queries –range queries –k-nn queries –spatial joins (‘all pairs’ queries)

6 Carnegie Mellon 15-415 - C. Faloutsos6 Spatial Access Methods - problem Given a collection of geometric objects (points, lines, polygons,...) organize them on disk, to answer –point queries –range queries –k-nn queries –spatial joins (‘all pairs’ queries)

7 Carnegie Mellon 15-415 - C. Faloutsos7 Spatial Access Methods - problem Given a collection of geometric objects (points, lines, polygons,...) organize them on disk, to answer –point queries –range queries –k-nn queries –spatial joins (‘all pairs’ queries)

8 Carnegie Mellon 15-415 - C. Faloutsos8 Spatial Access Methods - problem Given a collection of geometric objects (points, lines, polygons,...) organize them on disk, to answer –point queries –range queries –k-nn queries –spatial joins (‘all pairs’ queries)

9 Carnegie Mellon 15-415 - C. Faloutsos9 Spatial Access Methods - problem Given a collection of geometric objects (points, lines, polygons,...) organize them on disk, to answer –point queries –range queries –k-nn queries –spatial joins (‘all pairs’ within ε)

10 Carnegie Mellon 15-415 - C. Faloutsos10 SAMs - motivation Q: applications?

11 Carnegie Mellon 15-415 - C. Faloutsos11 SAMs - motivation salary age traditional DB GIS

12 Carnegie Mellon 15-415 - C. Faloutsos12 SAMs - motivation salary age traditional DB GIS

13 Carnegie Mellon 15-415 - C. Faloutsos13 SAMs - motivation CAD/CAM find elements too close to each other

14 Carnegie Mellon 15-415 - C. Faloutsos14 SAMs - motivation CAD/CAM

15 Carnegie Mellon 15-415 - C. Faloutsos15 day 1365 day 1365 S1 Sn F(S1) F(Sn) SAMs - motivation eg, avg eg,. std

16 Carnegie Mellon 15-415 - C. Faloutsos16 SAMs - Detailed outline spatial access methods –problem dfn –z-ordering –R-trees

17 Carnegie Mellon 15-415 - C. Faloutsos17 SAMs: solutions z-ordering R-trees (grid files) Q: how would you organize, e.g., n-dim points, on disk? (C points per disk page)

18 Carnegie Mellon 15-415 - C. Faloutsos18 z-ordering Q: how would you organize, e.g., n-dim points, on disk? (C points per disk page) Hint: reduce the problem to 1-d points(!!) Q1: why? A: Q2: how?

19 Carnegie Mellon 15-415 - C. Faloutsos19 z-ordering Q: how would you organize, e.g., n-dim points, on disk? (C points per disk page) Hint: reduce the problem to 1-d points (!!) Q1: why? A: B-trees! Q2: how?

20 Carnegie Mellon 15-415 - C. Faloutsos20 z-ordering Q2: how? A: assume finite granularity; z-ordering = bit- shuffling = N-trees = Morton keys = geo- coding =...

21 Carnegie Mellon 15-415 - C. Faloutsos21 z-ordering Q2: how? A: assume finite granularity (e.g., 2 32 x2 32 ; 4x4 here) Q2.1: how to map n-d cells to 1-d cells?

22 Carnegie Mellon 15-415 - C. Faloutsos22 z-ordering Q2.1: how to map n-d cells to 1-d cells?

23 Carnegie Mellon 15-415 - C. Faloutsos23 z-ordering Q2.1: how to map n-d cells to 1-d cells? A: row-wise Q: is it good?

24 Carnegie Mellon 15-415 - C. Faloutsos24 z-ordering Q: is it good? A: great for ‘x’ axis; bad for ‘y’ axis

25 Carnegie Mellon 15-415 - C. Faloutsos25 z-ordering Q: How about the ‘snake’ curve?

26 Carnegie Mellon 15-415 - C. Faloutsos26 z-ordering Q: How about the ‘snake’ curve? A: still problems: 2^32

27 Carnegie Mellon 15-415 - C. Faloutsos27 z-ordering Q: Why are those curves ‘bad’? A: no distance preservation (~ clustering) Q: solution? 2^32

28 Carnegie Mellon 15-415 - C. Faloutsos28 z-ordering Q: solution? (w/ good clustering, and easy to compute, for 2-d and n-d?)

29 Carnegie Mellon 15-415 - C. Faloutsos29 z-ordering Q: solution? (w/ good clustering, and easy to compute, for 2-d and n-d?) A: z-ordering/bit-shuffling/linear-quadtrees ‘looks’ better: few long jumps; scoops out the whole quadrant before leaving it a.k.a. space filling curves

30 Carnegie Mellon 15-415 - C. Faloutsos30 z-ordering z-ordering/bit-shuffling/linear-quadtrees Q: How to generate this curve (z = f(x,y) )? A: 3 (equivalent) answers!

31 Carnegie Mellon 15-415 - C. Faloutsos31 z-ordering z-ordering/bit-shuffling/linear-quadtrees Q: How to generate this curve (z = f(x,y))? A1: ‘z’ (or ‘N’) shapes, RECURSIVELY order-1 order-2... order (n+1)

32 Carnegie Mellon 15-415 - C. Faloutsos32 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)

33 Carnegie Mellon 15-415 - C. Faloutsos33 z-ordering z-ordering/bit-shuffling/linear-quadtrees Q: How to generate this curve (z = f(x,y) )? A: 3 (equivalent) answers! Method #2?

34 Carnegie Mellon 15-415 - C. Faloutsos34 z-ordering bit-shuffling 00 01 10 11 10 01 00 x y x 0 y 1 z =( 0 1 0 1 ) 2 = 5

35 Carnegie Mellon 15-415 - C. Faloutsos35 z-ordering bit-shuffling 00 01 10 11 10 01 00 x y x 0 y 1 z =( 0 1 0 1 ) 2 = 5 How about the reverse: (x,y) = g(z) ?

36 Carnegie Mellon 15-415 - C. Faloutsos36 z-ordering bit-shuffling 00 01 10 11 10 01 00 x y x 0 y 1 z =( 0 1 0 1 ) 2 = 5 How about n-d spaces?

37 Carnegie Mellon 15-415 - C. Faloutsos37 z-ordering z-ordering/bit-shuffling/linear-quadtrees Q: How to generate this curve (z = f(x,y) )? A: 3 (equivalent) answers! Method #3?

38 Carnegie Mellon 15-415 - C. Faloutsos38 z-ordering linear-quadtrees : assign N->1, S->0 e.t.c. 0 1 0 1 00... 01... 10... 11... W E NSNS

39 Carnegie Mellon 15-415 - C. Faloutsos39 z-ordering... and repeat recursively. Eg.: z blue-cell = WN;WN = (0101) 2 = 5 0 1 0 1 00... 01... 10... 11... W E NSNS 00 11

40 Carnegie Mellon 15-415 - C. Faloutsos40 z-ordering Drill: z-value of magenta cell, with the three methods? 0 1 0 1 W E NSNS

41 Carnegie Mellon 15-415 - C. Faloutsos41 z-ordering Drill: z-value of magenta cell, with the three methods? 0 1 0 1 W E NSNS method#1: 14 method#2: shuffle(11;10)= (1110) 2 = 14

42 Carnegie Mellon 15-415 - C. Faloutsos42 z-ordering Drill: z-value of magenta cell, with the three methods? 0 1 0 1 W E NSNS method#1: 14 method#2: shuffle(11;10)= (1110) 2 = 14 method#3: EN;ES =... = 14

43 Carnegie Mellon 15-415 - C. Faloutsos43 z-ordering - Detailed outline spatial access methods –z-ordering main idea - 3 methods use w/ B-trees; algorithms (range, knn queries...) non-point (eg., region) data analysis; variations –R-trees

44 Carnegie Mellon 15-415 - C. Faloutsos44 z-ordering - usage & algo’s Q1: How to store on disk? A: Q2: How to answer range queries etc

45 Carnegie Mellon 15-415 - C. Faloutsos45 z-ordering - usage & algo’s Q1: How to store on disk? A: treat z-value as primary key; feed to B-tree SF PGH

46 Carnegie Mellon 15-415 - C. Faloutsos46 z-ordering - usage & algo’s MAJOR ADVANTAGES w/ B-tree: already inside commercial systems (no coding/debugging!) concurrency & recovery is ready SF PGH

47 Carnegie Mellon 15-415 - C. Faloutsos47 z-ordering - usage & algo’s Q2: queries? (eg.: find city at (0,3) )? SF PGH

48 Carnegie Mellon 15-415 - C. Faloutsos48 z-ordering - usage & algo’s Q2: queries? (eg.: find city at (0,3) )? A: find z-value; search B-tree SF PGH

49 Carnegie Mellon 15-415 - C. Faloutsos49 z-ordering - usage & algo’s Q2: range queries? SF PGH

50 Carnegie Mellon 15-415 - C. Faloutsos50 z-ordering - usage & algo’s Q2: range queries? A: compute ranges of z-values; use B-tree SF PGH 9,11-15

51 Carnegie Mellon 15-415 - C. Faloutsos51 z-ordering - usage & algo’s Q2’: range queries - how to reduce # of qualifying of ranges? SF PGH 9,11-15

52 Carnegie Mellon 15-415 - C. Faloutsos52 z-ordering - usage & algo’s Q2’: range queries - how to reduce # of qualifying of ranges? A: Augment the query! SF PGH 9,11-15 -> 8-15

53 Carnegie Mellon 15-415 - C. Faloutsos53 z-ordering - usage & algo’s Q2’’: range queries - how to break a query into ranges? 9,11-15

54 Carnegie Mellon 15-415 - C. Faloutsos54 z-ordering - usage & algo’s Q2’’: range queries - how to break a query into ranges? A: recursively, quadtree-style; decompose only non-full quadrants 9,11-15 12-15

55 Carnegie Mellon 15-415 - C. Faloutsos55 z-ordering - usage & algo’s Q2’’: range queries - how to break a query into ranges? A: recursively, quadtree-style; decompose only non-full quadrants 9,11-15 12-15 9, 11

56 Carnegie Mellon 15-415 - C. Faloutsos56 z-ordering - Detailed outline spatial access methods –z-ordering main idea - 3 methods use w/ B-trees; algorithms (range, knn queries...) non-point (eg., region) data analysis; variations –R-trees

57 Carnegie Mellon 15-415 - C. Faloutsos57 z-ordering - usage & algo’s Q3: k-nn queries? (say, 1-nn)? SF PGH

58 Carnegie Mellon 15-415 - C. Faloutsos58 z-ordering - usage & algo’s Q3: k-nn queries? (say, 1-nn)? A: traverse B-tree; find nn wrt z-values and... SF PGH

59 Carnegie Mellon 15-415 - C. Faloutsos59 z-ordering - usage & algo’s... ask a range query. SF PGH 5 3 12 nn wrt z-value

60 Carnegie Mellon 15-415 - C. Faloutsos60 z-ordering - usage & algo’s... ask a range query. SF PGH 5 3 12 nn wrt z-value

61 Carnegie Mellon 15-415 - C. Faloutsos61 z-ordering - usage & algo’s Q4: all-pairs queries? ( all pairs of cities within 10 miles from each other? ) SF PGH (we’ll see ‘spatial joins’ later: find all PA counties that intersect a lake)

62 Carnegie Mellon 15-415 - C. Faloutsos62 z-ordering - Detailed outline spatial access methods –z-ordering main idea - 3 methods use w/ B-trees; algorithms (range, knn queries...) non-point (eg., region) data analysis; variations –R-trees –...

63 Carnegie Mellon 15-415 - C. Faloutsos63 z-ordering - regions Q: z-value for a region? z B = ?? z C = ?? A B C

64 Carnegie Mellon 15-415 - C. Faloutsos64 z-ordering - regions Q: z-value for a region? A: 1 or more z-values; by quadtree decomposition z B = ?? z C = ?? A B C

65 Carnegie Mellon 15-415 - C. Faloutsos65 z-ordering - regions Q: z-value for a region? 0 1 0 1 00... 01... 10... 11... W E NSNS 00 11 A B C z B = 11** z C = ?? “don’t care”

66 Carnegie Mellon 15-415 - C. Faloutsos66 z-ordering - regions Q: z-value for a region? 0 1 0 1 00... 01... 10... 11... W E NSNS 00 11 A B C z B = 11** z C = {0010; 1000} “don’t care”

67 Carnegie Mellon 15-415 - C. Faloutsos67 z-ordering - regions Q: How to store in B-tree? Q: How to search (range etc queries) A B C

68 Carnegie Mellon 15-415 - C. Faloutsos68 z-ordering - regions Q: How to store in B-tree? A: sort (*<0<1) Q: How to search (range etc queries) A B C

69 Carnegie Mellon 15-415 - C. Faloutsos69 z-ordering - regions Q: How to search (range etc queries) - eg ‘red’ range query A B C

70 Carnegie Mellon 15-415 - C. Faloutsos70 z-ordering - regions Q: How to search (range etc queries) - eg ‘red’ range query A: break query in z-values; check B-tree A B C

71 Carnegie Mellon 15-415 - C. Faloutsos71 z-ordering - regions Almost identical to range queries for point data, except for the “don’t cares” - i.e., A B C 1100 ?? 11**

72 Carnegie Mellon 15-415 - C. Faloutsos72 z-ordering - regions Almost identical to range queries for point data, except for the “don’t cares” - i.e., z1= 1100 ?? 11** = z2 Specifically: does z1 contain/avoid/intersect z2? Q: what is the criterion to decide?

73 Carnegie Mellon 15-415 - C. Faloutsos73 z-ordering - regions z1= 1100 ?? 11** = z2 Specifically: does z1 contain/avoid/intersect z2? Q: what is the criterion to decide? A: Prefix property: let r1, r2 be the corresponding regions, and let r1 be the smallest (=> z1 has fewest ‘*’s). Then:

74 Carnegie Mellon 15-415 - C. Faloutsos74 z-ordering - regions r2 will either contain completely, or avoid completely r1. it will contain r1, if z2 is the prefix of z1 A B C 1100 ?? 11** region of z1: completely contained in region of z2

75 Carnegie Mellon 15-415 - C. Faloutsos75 z-ordering - regions Drill (True/False). Given: z1= 011001** z2= 01****** z3= 0100**** T/F r2 contains r1 T/F r3 contains r1 T/F r3 contains r2

76 Carnegie Mellon 15-415 - C. Faloutsos76 z-ordering - regions Drill (True/False). Given: z1= 011001** z2= 01****** z3= 0100**** T/F r2 contains r1 - TRUE (prefix property) T/F r3 contains r1 - FALSE (disjoint) T/F r3 contains r2 - FALSE (r2 contains r3)

77 Carnegie Mellon 15-415 - C. Faloutsos77 z-ordering - regions Drill (True/False). Given: z1= 011001** z2= 01****** z3= 0100**** z2

78 Carnegie Mellon 15-415 - C. Faloutsos78 z-ordering - regions Drill (True/False). Given: z1= 011001** z2= 01****** z3= 0100**** z3 z2 T/F r2 contains r1 - TRUE (prefix property) T/F r3 contains r1 - FALSE (disjoint) T/F r3 contains r2 - FALSE (r2 contains r3)

79 Carnegie Mellon 15-415 - C. Faloutsos79 z-ordering - regions Spatial joins: find (quickly) all counties intersecting lakes

80 Carnegie Mellon 15-415 - C. Faloutsos80 z-ordering - regions Spatial joins: find (quickly) all counties intersecting lakes Naive algorithm: O( N * M) Something faster?

81 Carnegie Mellon 15-415 - C. Faloutsos81 z-ordering - regions Spatial joins: find (quickly) all counties intersecting lakes

82 Carnegie Mellon 15-415 - C. Faloutsos82 z-ordering - regions Spatial joins: find (quickly) all counties intersecting lakes Solution: merge the lists of (sorted) z-values, looking for the prefix property footnote#1: ‘*’ needs careful treatment footnote#2: need dup. elimination

83 Carnegie Mellon 15-415 - C. Faloutsos83 z-ordering - Detailed outline spatial access methods –z-ordering main idea - 3 methods use w/ B-trees; algorithms (range, knn queries...) non-point (eg., region) data analysis; variations –R-trees

84 Carnegie Mellon 15-415 - C. Faloutsos84 z-ordering - variations Q: is z-ordering the best we can do?

85 Carnegie Mellon 15-415 - C. Faloutsos85 z-ordering - variations Q: is z-ordering the best we can do? A: probably not - occasional long ‘jumps’ Q: then?

86 Carnegie Mellon 15-415 - C. Faloutsos86 z-ordering - variations Q: is z-ordering the best we can do? A: probably not - occasional long ‘jumps’ Q: then? A1: Gray codes

87 Carnegie Mellon 15-415 - C. Faloutsos87 z-ordering - variations A2: Hilbert curve! (a.k.a. Hilbert-Peano curve)

88 Carnegie Mellon 15-415 - C. Faloutsos88 z-ordering - variations ‘Looks’ better (never long jumps). How to derive it?

89 Carnegie Mellon 15-415 - C. Faloutsos89 z-ordering - variations ‘Looks’ better (never long jumps). How to derive it? order-1 order-2... order (n+1)

90 Carnegie Mellon 15-415 - C. Faloutsos90 z-ordering - variations Q: function for the Hilbert curve ( h = f(x,y) )? A: bit-shuffling, followed by post-processing, to account for rotations. Linear on # bits. See textbook, for pointers to code/algorithms (eg., [Jagadish, 90])

91 Carnegie Mellon 15-415 - C. Faloutsos91 z-ordering - variations Q: how about Hilbert curve in 3-d? n-d? A: Exists (and is not unique!). Eg., 3-d, order- 1 Hilbert curves (Hamiltonian paths on cube) #1 #2

92 Carnegie Mellon 15-415 - C. Faloutsos92 z-ordering - Detailed outline spatial access methods –z-ordering main idea - 3 methods use w/ B-trees; algorithms (range, knn queries...) non-point (eg., region) data analysis; variations –R-trees –...

93 Carnegie Mellon 15-415 - C. Faloutsos93 z-ordering - analysis Q: How many pieces (‘quad-tree blocks’) per region? A: proportional to perimeter (surface etc)

94 Carnegie Mellon 15-415 - C. Faloutsos94 z-ordering - analysis (How long is the coastline, say, of England? Paradox: The answer changes with the yard- stick -> fractals...)

95 Carnegie Mellon 15-415 - C. Faloutsos95 z-ordering - analysis Q: Should we decompose a region to full detail (and store in B-tree)?

96 Carnegie Mellon 15-415 - C. Faloutsos96 z-ordering - analysis Q: Should we decompose a region to full detail (and store in B-tree)? A: NO! approximation with 1-3 pieces/z- values is best [Orenstein90]

97 Carnegie Mellon 15-415 - C. Faloutsos97 z-ordering - analysis Q: how to measure the ‘goodness’ of a curve?

98 Carnegie Mellon 15-415 - C. Faloutsos98 z-ordering - analysis Q: how to measure the ‘goodness’ of a curve? A: e.g., avg. # of runs, for range queries 4 runs3 runs (#runs ~ #disk accesses on B-tree)

99 Carnegie Mellon 15-415 - C. Faloutsos99 z-ordering - analysis Q: So, is Hilbert really better? A: 27% fewer runs, for 2-d (similar for 3-d) Q: are there formulas for #runs, #of quadtree blocks etc? A: Yes ([Jagadish; Moon+ etc] see textbook)

100 Carnegie Mellon 15-415 - C. Faloutsos100 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)

101 Carnegie Mellon 15-415 - C. Faloutsos101 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)

102 Carnegie Mellon 15-415 - C. Faloutsos102 z-ordering - fun observations Observation #2: Hilbert (like) curve for video encoding [Y. Matias+, CRYPTO ‘87]: Given a frame, visit its pixels in randomized hilbert order; compress; and transmit

103 Carnegie Mellon 15-415 - C. Faloutsos103 z-ordering - fun observations In general, Hilbert curve is great for preserving distances, clustering, vector quantization etc

104 Carnegie Mellon 15-415 - C. Faloutsos104 SAMs - Detailed outline spatial access methods –problem dfn –z-ordering –R-trees

105 Carnegie Mellon 15-415 - C. Faloutsos105 Conclusions z-ordering is a great idea (n-d points -> 1-d points; feed to B-trees) used by TIGER system and (most probably) by other GIS products works great with low-dim points

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