Module 4: Index Structures Lecture 16: Voronoi Diagrams and Tries. The Lecture Contains: Voronoi diagrams. Tries. Index structures
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1 The Lecture Contains: Voronoi diagrams Tries Delaunay triangulation Algorithms Extensions Index structures 1-dimensional index structures Memory-based index structures Disk-based index structures Classification of index structures file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture16/16_1.htm[6/14/2012 3:39:09 PM]
2 Voronoi diagrams Partition of a plane with N points (sites) into convex polygons such that Each polygon contains one and only one site Every point in the polygon is closest to the polygon's site than any other site Also known as Voronoi tessellation or Dirichlet tessellation file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture16/16_2.htm[6/14/2012 3:39:09 PM]
3 Voronoi diagrams Partition of a plane with N points (sites) into convex polygons such that Each polygon contains one and only one site Every point in the polygon is closest to the polygon's site than any other site Also known as Voronoi tessellation or Dirichlet tessellation Polygons are called Voronoi regions or Voronoi cells or Thiessen polygons or Dirichlet domains Vertices of the resulting graph Voronoi vertices Edges are Voronoi edges file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture16/16_3.htm[6/14/2012 3:39:10 PM]
4 Delaunay triangulation Dual of Voronoi diagram Triangulate such that circumcircles of every triangle is empty Voronoi vertices are circumcircles of Delaunay triangles file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture16/16_4.htm[6/14/2012 3:39:10 PM]
5 Algorithms Used for 1-NN searches Fortune's algorithm to construct Voronoi diagrams Plane sweep Time to construct is Optimal Requires space to store file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture16/16_5.htm[6/14/2012 3:39:10 PM]
6 Extensions or Mahalanobis distance may be used instead of No guarantee that such a structure exists Weighted may be used file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture16/16_6.htm[6/14/2012 3:39:10 PM]
7 Extensions or Mahalanobis distance may be used instead of No guarantee that such a structure exists Weighted may be used In dimensions, requires space file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture16/16_7.htm[6/14/2012 3:39:10 PM]
8 Extensions or Mahalanobis distance may be used instead of No guarantee that such a structure exists Weighted may be used In dimensions, requires space Voronoi diagram of order- Partition according to closest sites Useful for -NN searches Ordinary Voronoi is order-1 Order-( ) is called farthest point Voronoi diagram In two dimensions, requires space and time file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture16/16_8.htm[6/14/2012 3:39:10 PM]
9 Extensions or Mahalanobis distance may be used instead of No guarantee that such a structure exists file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture16/16_9.htm[6/14/2012 3:39:10 PM]
10 Extensions or Mahalanobis distance may be used instead of No guarantee that such a structure exists Weighted may be used file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture16/16_10.htm[6/14/2012 3:39:10 PM]
11 Extensions or Mahalanobis distance may be used instead of No guarantee that such a structure exists Weighted may be used In dimensions, requires space file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture16/16_11.htm[6/14/2012 3:39:11 PM]
12 Extensions or Mahalanobis distance may be used instead of No guarantee that such a structure exists Weighted may be used In dimensions, requires space Voronoi diagram of order- Partition according to closest sites Useful for -NN searches Ordinary Voronoi is order-1 Order-( ) is called farthest point Voronoi diagram In two dimensions,requires space and time file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture16/16_12.htm[6/14/2012 3:39:11 PM]
13 Tries Comes from the word retrieval Mostly used for strings Structure Root represents null string Each edge defines the next character Each node stores a string or a preffix of a string Strings with same preffix share the path Advantages over binary search trees Search time is where is the length of the query Size is generally less Related structures: preffix tree, radix tree, suffix tree file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture16/16_13.htm[6/14/2012 3:39:11 PM]
14 Index structures Organize objects for effcient searching and retrieval Three main classes 1-d index structures Memory-based index structures Disk-based index structures file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture16/16_14.htm[6/14/2012 3:39:11 PM]
15 1-dimensional index structures Hashing Hash on single-dimensional key Static hashing, Dynamic hashing Some structures can be extended to multiple dimensions B-tree based structures Key is single-dimensional Supports disk-based accesses file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture16/16_15.htm[6/14/2012 3:39:11 PM]
16 Memory-based index structures K-d-tree, Range tree, Quadtree, etc. No page-based organization of nodes. Fanout (i.e., number of children) does not depend on page capacity. Mostly used for low-dimensional data. file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture16/16_16.htm[6/14/2012 3:39:11 PM]
17 Disk-based index structures Grid le, R-tree, K-d-B-tree, X-tree, M-tree. Not all objects t in memory. Disk-page-oriented organization of objects. Aim is to reduce number of random accesses for queries. file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture16/16_17.htm[6/14/2012 3:39:11 PM]
18 Classification of index structures Space-partitioning Fanout independent of dimensionality Problem of dead space indexing No guarantee on space usage K-d-B-tree, K-d-tree, Quadtree, Grid le Data-partitioning Fanout decreases with dimensionality Eliminates dead space Guarantee on space usage R-tree, R*-tree, X-tree file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture16/16_18.htm[6/14/2012 3:39:11 PM]
19 Classification of index structures Space-partitioning Fanout independent of dimensionality Problem of dead space indexing No guarantee on space usage K-d-B-tree, K-d-tree, Quadtree, Grid le Data-partitioning Fanout decreases with dimensionality Eliminates dead space Guarantee on space usage R-tree, R*-tree, X-tree Point access methods (PAM) Only points are indexed Spatial access methods (SAM) Spatial objects can be indexed file:///c /Documents%20and%20Settings/iitkrana1/My%20Documents/Google%20Talk%20Received%20Files/ist_data/lecture16/16_19.htm[6/14/2012 3:39:11 PM]
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