Multidimensional Data Structures

Transkripsi

1 Multidimensional Data Structures 2002, Arry Akhmad Arman Laboratory for Signal & Systems Electrical Engineering Department Bandung Institute of Technology [email protected]

2 N-dimensional Data Most media data requires the ability to reason about both time and space. Such data is typically referred to as n-dimensional data, reflection the fact that data has associated attributes drawn from n-dimensional space. Example : o Typical two dimensional coordinates (x, y) o Three dimensional coordinates (x, y, z) o Space time (x, y, z, t) Most techniques to store n-dimensional data do by using hierarchical decomposition of space that are typically represented by various kind of trees : k-d trees, point quadtrees, MX quadtrees, and R-trees , ARRY AKHMAD ARMAN - Electrical Engineering Dept. of ITB 2

3 K-d Trees : Introduction The k-d trees is used to store k-dimensional point data such as that shown in figure below. For k=2, k-d tree is 2-d tree For k-3, k-d tree is 3-d tree , ARRY AKHMAD ARMAN - Electrical Engineering Dept. of ITB 3

4 K-d Trees : Node Structure nodetype = record INFO : infotype; XVAL : real; YVAL : real; LLINK : ^infotype; RLINK : ^infotype; end; INFO LLINK XVAL YVAL RLINK , ARRY AKHMAD ARMAN - Electrical Engineering Dept. of ITB 4

5 K-d Trees : Rules (1) Suppose T is a pointer to the root of a 2-d tree. If N is a node in this tree, then the level of node N is define as Level(N) = 0 if N is the root of the tree Level(N) = level(p)+1 if N s parent is P , ARRY AKHMAD ARMAN - Electrical Engineering Dept. of ITB 5

6 K-d Trees : Rules (2) If N is a node in the tree such that level(n) is even, then o every node M in th subtree rooted at N.LLINK has the propeeerty that M.XVAL < M.XVAL, o every node P in th subtree rooted at N.RLINK has the propeeerty that P.XVAL >= P.XVAL, If N is a node in the tree such that level(n) is odd, then o every node M in th subtree rooted at N.LLINK has the propeeerty that M.YVAL < M.YVAL, o every node P in th subtree rooted at N.RLINK has the propeeerty that P.YVAL < P.YVAL, , ARRY AKHMAD ARMAN - Electrical Engineering Dept. of ITB 6

7 K-d Tree : Insertion City (XVAL, YVAL) Banja Luka (19, 45) Derventa (40, 50) Teslic (38, 38) Tuzla (54, 40) Sinj (4,4) , ARRY AKHMAD ARMAN - Electrical Engineering Dept. of ITB 7

8 K-d Tree : Insertion (2) , ARRY AKHMAD ARMAN - Electrical Engineering Dept. of ITB 8

9 K-d Tree : Characteristics Tree structure dipengaruhi oleh urutan pemasukan node Titik yang sesungguhnya direpresentasikan dengan setiap node pada tree Setiap node selalu berisi informasi (tidak ada node perantara yang kosong) Setiap node informasi mungkin terletak diujung tree, diawal (root) atau ditengah. Hal ini akan menyulitkan proses deletion jika node yang akan dihapus tidak terletak di ujung , ARRY AKHMAD ARMAN - Electrical Engineering Dept. of ITB 9

10 Point Quad Tree : Node Structure nodetype = record INFO : infotype; XVAL : real; YVAL : real; NW : ^infotype; SW : ^infotype; NE : ^infotype; SE : ^infotype; end; INFO XVAL YVAL NW SW NE SE , ARRY AKHMAD ARMAN - Electrical Engineering Dept. of ITB 10

11 Point Quad Tree : Rules (1) , ARRY AKHMAD ARMAN - Electrical Engineering Dept. of ITB 11

12 Point Quad Tree : Rules (2) , ARRY AKHMAD ARMAN - Electrical Engineering Dept. of ITB 12

13 Point Quad Tree : Insertion City (XVAL, YVAL) Banja Luka (19, 45) Derventa (40, 50) Teslic (38, 38) Tuzla (54, 40) Sinj (4,4) , ARRY AKHMAD ARMAN - Electrical Engineering Dept. of ITB 13

14 Point Quad Tree : Insertion (2) , ARRY AKHMAD ARMAN - Electrical Engineering Dept. of ITB 14

15 Point Quad Tree : Characteristics Tree structure dipengaruhi oleh urutan pemasukan node Titik yang sesungguhnya direpresentasikan dengan setiap node pada tree Setiap node selalu berisi informasi (tidak ada node perantara yang kosong) Setiap node informasi mungkin terletak diujung tree, diawal (root) atau ditengah. Hal ini akan menyulitkan proses deletion jika node yang akan dihapus tidak terletak di ujung. Kedalaman tree lebih pendek dibandingkan k-d tree , ARRY AKHMAD ARMAN - Electrical Engineering Dept. of ITB 15

16 MX Quad Tree : Node Structure nodetype = record INFO : infotype; XVAL : real; YVAL : real; NW : ^infotype; SW : ^infotype; NE : ^infotype; SE : ^infotype; end; INFO XVAL YVAL NW SW NE SE , ARRY AKHMAD ARMAN - Electrical Engineering Dept. of ITB 16

17 MX Quad Tree : Rules (1) , ARRY AKHMAD ARMAN - Electrical Engineering Dept. of ITB 17