Range Searching. 2010/11/22 Yang Lei

Transcription

1 Range Searching 2010/11/22 Yang Lei

2 Outline What is range searching? 1D range searching (Property 26.1) 2D range searching Algorithm 1: Projection (worst cases: L-shape) Algorithm 2: Grid Method (Predetermined) (Property 26.2) How to divide grids? Too coarse, too fine? (size? maxg?) Worst case: Clustered data. Algorithm 3: Two-Dimensional Trees (Adpative) Property 26.3, property 26.4 Multidimensional Range Searching (3D, 4D,.,kD)

3 Definition Finding all records in a database that satisfy specified range restrictions on a specified set of attributes (dimensions) is called range searching. Example: (1) On Amazon.com : search all watches with price $100<P<$200; (1D) (2) In UH: search for all undergraduate students graduated between year 2000 to 2009 with GPA between 3.0 to 4.0(3D) (3) Select a group of files by draging a rectangle [P1(x1,y1), P2(x2,y2)] using mouse on your computer. (2D)

4 P1(x1,y1) and P2(x2,y2) Define a Rect. (x1<x<x2)&&(y1<y<y2)

5 One Dimensional Range Searching Method1: 1.Sorting all points for preprocessing, 2.Then do binary search on the endpoints of the interval to return all points fall between. Method2: 1.Build a binary search tree, 2.Do recursive traversal of the tree. Property 26.1 It can be done with O(N*logN) steps for preprocessing, and O(R+logN) for range searching, where R is the number of points actually falling in the range.

6 Note: Exceptions The algorithm we considered here are not expected for queries a large number of points, say R is small. If R is as large as N, use other algorithm. Batching all requests within the memory capacity and do a sequential search through an external database (big&slow), return all requested records. Example: we receive 1K users searching requests per second User 1 requests: $50<watch<$100; User 2 requests: $100<watch<$200; User 3 requests: $50< watch <$450; User 999 requests: $xxx<watch<$yyyy; Go through database only once!

7 Two-Dimensional Range Searching Algorithm 1: Projection If compare sequentially, N=16, there are 2N=16*2=32 comparisons.

8 Algorithm 1: Projection Project points to the narrower side of Rect. Compare X coordinates, 16 comparisons; then compare Y coordinates for {D, E, F, C, H}, 5 comparisons. Total=21 comparisons. Saving 35%.

9 Worst Case of Projection L-shape distribution. Project points to X, compare X coordinates, 16 comparisons; then compare Y coordinates for {LGPEOADHNC}, 10 comparisons; total 26 comparisons. Saving 18%.

10 Algorithm 2: Grid Method Draw artificial grids. Each grid is (size by size large). Total maxg*maxg grids. (i, j)

11 Algorithm 2: Grid Method If there are N points and we want M points per grid square, then we need N/M grid squares, then: The width of a grid square size is sqrt(n/m). The number of a grid square maxg is (int)(n/m), where MAX is the maximum points coordinates. There are maxg by maxg (N/M) grid squares, the point coordinates range between [0, size*maxg]. These estimates are not accurate for small values, but are useful for most situations.

12 A2 Grid Method: Initialize data Preprocessing: artificial grids and short lists Input the N points (array) Determine the grid size size and number of grids maxg *maxg Initialize the grid array and the corresponding lists For each point p i (i=0,1, N-1), divide its coordinates by maxg to get the grid indexed by ( floor(p i.x/maxg), floor(p i.y/maxg)) Insert point p i into the list for the grid it belongs to Loop until all points are processed Complexity: O(N)

13 A2 Grid Method: Searching Range searching Read in the Range R(x1,x2,y1,y2) Determine the grids intersecting the range by indexing (i, j): i from R.x1/maxG to R.x2/maxG && j from R.y1/maxG to R.y2/maxG For each grid intersecting the range Traverse its linked list representing the points contained in that grid Call InsideRect() for every point in the list to examine the insideness Loop until all the intersecting grids (and the corresponding lists) are processed, output the points within the range

14 Grid method is a predetermined approach. Property 26.2 The grid method for range searching is linear in the number of points in the range, on the average, and linear in the total number of points in the worst case. The grid method for range searching is linear in the number of points in the range, on the average, and linear in the total number of points N in the worst case. This depends on choosing parameters so that expected number of points in each grid is constant. The grid method works well if points are well distributed over the assumed range. The worst case is clustered data. All data in one grid, other grids are all empty.

15 Two-Dimensional Trees

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17 Two-Dimensional Trees 2D dimensional trees are dynamic, adaptable data structures that are similar with binary tree, but divide up a geometric space in a manner convenient for range search. The idea is to build binary search trees with points in the nodes, using the y & x coordinates of the points as keys in a strict alternating sequence. Space division is adaptive to the distribution of the points, so that range searching can be done efficiently.

18 Algorithm 3: 2D Trees Inserting points into 2D trees is same as in normal binary search tree; but at root we use y coordinate. For smaller y than the root, go left; For larger y than the root, go right. At next level use x coordinate, then y at next level; altering until an external node is encountered.

19 Algorithm 3: 2D Trees Dividing up the plane in a simple way: All points below the point at the root point go in left sub-tree; All those above the root point go in right sub-tree; All points above the root point and to the left of the point in the right sub-tree go in the left sub-tree of the right sub-tree of the root.

20 A3 2D Trees: Preprocessing Preprocessing: building of the 2D-Tree Input the N points (array) and initialize the empty tree Pick one (randomly) as the root of the empty tree For each point p i (i=1,2, N-1), insert it into the tree by Going down the tree by comparing its keys (two coordinates) to the root nodes and other nodes in the tree Alternating the x and y coordinates as the comparing keys while going down the tree level by level For every comparison, if smaller, go to left sub-tree; if bigger, go to right sub-tree. Repeat until reach a external node Insert point p i into the current position Loop until all points are inserted

21 A3 2D Trees: Range searching Range searching Read in the Range R(x1,x2,y1,y2) Travel through the 2D-Tree by comparing the key of the node point to the corresponding interval (x1,x2) or (y1,y2): Go down to the left sub-tree, if x2 <= p.x ( or y2 <= p.y ) Go down to the right sub-tree, if x1 >= p.x ( or y1 >= p.y ) Check both sub-trees, if x1 < p.x < x2 ( or y1 < p.y <y2 ) Check the other coordinate of those node points which are cut through the interval or on the edge lines Repeat until reach external nodes, output the points within R

22 A3: 2D Tree Algorithm Analysis The complexity of preprocessing is O(2N*lnN). The complexity of range searching is O(R+logN), where R is number of points to be found, and N is number of total points.