On Op%mality of Clustering by Space Filling Curves

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1 On Op%mality of Clustering by Space Filling Curves Pan Xu Iowa State University Srikanta Tirthapura 1

2 Mul%- dimensional Data Indexing and managing single- dimensional data is a solved problem Mul%- dimensional data is not Computa%onal Geometry (k- d trees, quad trees) Databases (R- trees, Space Filling Curves,etc) Parallel Compu%ng (same as above) 2

3 Space Filling Curve (SFC) Discrete mul%- dimensional universe Path that passes through each point in a mul%- dimensional universe exactly once Recipe for processing mul%- dimensional data Use and SFC to map data to a single dimension Use a single dimensional data structure for indexing (ex: B+ trees) 3

4 Z Space Filling Curve 4

5 Z Space Filling Curve 5

6 Z Space Filling Curve 6

7 Z Space Filling Curve 7

8 Hilbert Space Filling Curve 8

9 Hilbert Space Filling Curve 9

10 Hilbert Space Filling Curve 10

Hilbert Space Filling Curve Source: h^p://www.math.osu.

11 Hilbert Space Filling Curve Source: h^p:// Prof. Zbigniew Fiedorowicz 11

12 Where SFCs? Databases Spa%al databases, Geographic Informa%on Systems Commercial and Open Source Products: Oracle Spa%al, OpenGIS Parallel Compu%ng Par%%oning data among processors Deciding which data to include inside a cache Thousands of Cita%ons 12

13 Performance of an SFC Range Queries Find me all coffee shops between la%tudes x and x, longitudes y and y, and priced between z and z A query is a subset of the universe Rectangular Queries: Intersec%on of halfplanes Rec%linear Queries: Union of mul%ple disjoint rectangles Using SFC, a mul%dimensional range split into many one dimensional ranges Each one dimensional range is evaluated quickly using an index 13

14 Clustering Number 14

15 Clustering Number 1 for the Hilbert Curve 2 for the Z Curve 15

16 Average Clustering Number 16

17 Transla%on of a Query Shape Q = T(q) = all transla%ons of a basic query shape q 17

18 Rota%on With d dimensions, d! possible rota%ons for a single query. 18

19 Ques%ons Given SFC π and query set Q, what is c(q,π)? Lower bounds on c(q,π)? Op%mality of a curve for a class of queries? Seqng: d dimensional universe Total number of cells = n n 1/d n 1/d... 19

20 Prior Work 20

21 Con%nuous SFCs Con%nuous SFC Non- Con%nuous SFC 21

22 Our Results Con%nuous SFCs General SFCs Rectangular Queries General (Rec%linear) Queries Exact Formula Exact Formula Lower Bound, Con%nuous is op%mal Con%nuous need not be op%mal 22

23 Our Results When all rota%ons are considered for rectangular queries, Hilbert SFC is op%mal (and so is any con%nuous SFC) Note: Results for limi%ng n, and for constant query size, which does not increase with n 23

24 General Techniques 24

25 Basic Lemma 25

26 Basic Lemma

27 Basic Lemma

28 Basic Lemma

29 Basic Lemma

30 Average Clustering Number 30

31 Coun%ng Ques%on 31

32 Con%nuous SFC, Arbitrary Query, Transla%ons Only 32

33 Con%nuous SFC 2 Query A Dimension 1 Query B 33

34 Con%nuous SFC, Arbitrary Query Above Formula allows exact computa%on of average clustering number Also extends to Transla%on + Rota%on 34

35 Symmetric SFC, Arbitrary Query 35

36 Arbitrary SFC, Rectangular Query 36

37 Dimension 2 Dimension 1 37

38 Dimension 2 Dimension 1 38

39 Our Results Con%nuous SFCs General SFCs Rectangular Queries General (Rec%linear) Queries Exact Formula Exact Formula Lower Bound, Con%nuous is op%mal Con%nuous need not be op%mal 39

40 Conclusions Fundamental Results about SFCs General Method from first principles Edge- centric perspec%ve, i.e. count the number of queries crossing the edges Answers Explicit Open Ques%ons by Jagadish (1997), Moon et al. (2001) and others 40

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