1. P ARALLEL O RTHOGONAL R ANGE S EARCHING Project Presentation by Savitha Parur Venkitachalam

2. O RTHOGONAL R ANGE S EARCHING - REVIEW Input : a) Given a set of point P ={p 1, p 2,…..p n } in the d- dimensional space with coordinate axes (x 1, x 2 …x d ) b) A query range Output: Report all the points that lie within the d-range Project focuses on 2D Range searching

3. P REPROCESSING Process the data and store it in a KD tree KD tree is a binary tree in which every node is a K-dimensional point

4. B UILD KD T REE - P SEUDOCODE Input The set of data D The current depth of the tree which will be zero initially Output Root of the KD tree storing the set of data Buildkdtree (D, depth) If (D has only one point) then return this point else if (depth is even) Split D into two subsets through the median of X coordinates i.e. with a vertical line else Split D into two subsets through the median of Y coordinates i.e. with a horizontal line. Create a node with the median value and whose children are defined as Left_child  Buildkdtree(D 1, depth+1) Right_child  Buildkdtree(D 2, depth+1) Return node

5. R ANGE SEARCHING IN KD TREE Input: 1) A node of the KD tree 2) Input Query range Searchkdtree (node N, Range R) { If (N is a leaf) Report the value at N if it falls in the range. Else Consider the left subtree. If (left subtree is fully contained in the R) Report the left subtree Else if (left subtree intersects R) Searchkdtree(leftsubtree, R) Consider the right subtree. If (right subtree is fully contained in the R) Report the right subtree Else if (right subtree intersects R) Searchkdtree(rightsubtree, R) }

6. P ARALLEL A LGORITHM Data Distribution In the real world, databases exists in different processors For simulating the scenario, initially server has all the data Server scatters the data among the processors

7. P ARALLEL A LGORITHM

8. Range searching Server sends the query range to all the processors Processors search their local KD tree for the data in the range Processors report the results to the server Server takes the union of the results and reports it to the user

9. C OMPLEXITY – R ANGE S EARCHING Range searching in KD trees has a complexity of O ( dn 1 − 1/ d + k ), k is the total number of points included in the search, n is the total number of points and d is the dimensions Parallel range search will have a complexity of O( d(n/p) 1 − 1 d + k ) + communication time

10. R ESULTS - P ERFORMANCE #pTimespeed up 1 0.9957121 2 0.5439111.830652 5 0.2322664.286947 10 0.1567166.353608 20 0.1197198.317076 25 0.1093589.105068

11. R ESULTS Time taken for varying load and varying #processors Range - size/2 Sizes125102025 10000.0004990.0005280.0008960.0013390.002795 0.002270 10000 0.0067720.0038030.0020920.0022320.002539 0.003019 1000000.0859910.0480470.0252250.02070.0190850.018542 10000000.9957120.5439110.2322660.15670.1197190.109358

12. T ASKS PENDING Research on the scope of multi threading with in each processor Read Input data from a file to measure performance accurately Check for any redundant data structures or unwanted copying of data that may affect performance Writing the output to a file Testing for accuracy on a data with predictable ranges More testing with varying ranges

13. R EFERENCES http://2011.cccg.ca/PDFschedule/papers/paper106.pdf http://www.cs.arizona.edu/classes/cs437/fall12/Lecture5.prn.pdf http://www.cs.wustl.edu/~pless/506/l11w.html Parallel Computational Geometry – Aggarwal. A; O'Dunlaing. C; Yap.C Parallel Processing and Applied Mathematics: 5th International Conference, PPAM 2003, Czestochowa, Poland, September 7-10, 2003. Revised Paper Computational Geometry - Algorithms and Applications - Mark de Berg, TU Eindhoven, Otfried Cheong, KAIST,Marc van Kreveld, Mark Overmars

14. Q UESTIONS