5 Spatial Access Methods
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- Aubrey Clarke
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1 5 Spatial Access Methods 5.1 Quadtree 5.2 R-tree 5.3 K-d tree 5.4 BSP tree 3 27 A 17 5 G E B J K 5.5 Grid file Summary D C F H Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 419
2 5 Spatial Access Methods Speeding up queries by well-directed access on relevant tuples Query types Point query Range query Nearest neighbour query Problems Maintenance of the topological structure Density of objects varies strongly Dynamic reorganisation Representation of objects: points as well as extended objects Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 420
3 5 Spatial Access Methods For one-dimensional data: B-Tree Upper (U) and lower (L) bound for number of links usually : 2*L = U nsertion All insertions happen at the leaf nodes key value data pointer tree node node pointers Nodes are split during insertion as soon as they contain more than U-2 keys The median is chosen from among the leaf's elements and the new element The splitting may go all the way up to the root Deletion Rebalancing necessary if node contains less than L keys Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 421
4 5 Spatial Access Methods Very efficient data structure for disk storage O(log n) for all operations Even better Guaranteed maximum node-accesses to locate a key is Balanced binary tree guarantees only Accessing a node is expensive on disks huge improvement No degenerated cases Self-balancing rarely necessary as most updates affect just one node Wasted space decreased due to guaranteed minimal fill factor Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 422
5 5 Spatial Access Methods B + -Tree Different nodes for leaf nodes and internal nodes Data pointers only in leaf nodes All leafs are linked to each other in-order Advantages mproved traversal performance ncreased search efficiency ncreased memory efficiency Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 423
6 5 Spatial Access Methods Why not use B-Tree or B + -Tree for indexing twodimensional data? Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 424
7 5 Spatial Access Methods One-dimensional orderings Mapping multidimensional data to one dimension Try to preserve object proximity Use a uniform grid to partition the space Assign a unique number to each cell in a way that Adjacent cells get similar numbers t can be defined by a recursion of any depth Each cell can be refined separately without disturbing the order of the other numbers Space filling curve Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 425
8 5 Spatial Access Methods Examples for space filling curves: Z-order Gray code Hilbert s curve Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 426
9 5 Spatial Access Methods Example of point-shaped objects: weather stations Attributes Position Owner Year of construction Analog/digital D A E H F C K J B G Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 427
10 5 Spatial Access Methods Sorting points by Z-order Recursive decomposition of space in quadrants until every quadrant contains at most one point A C D F E H K B G J Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 428
11 5 Spatial Access Methods Sorting points by Z-order Recursive decomposition of space in quadrants until every quadrant contains at most one point A C D F E H K B G J Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 428
12 5 Spatial Access Methods Sorting points by Z-order Recursive decomposition of space in quadrants until every quadrant contains at most one point A C D F E H K B G J Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 428
13 5 Spatial Access Methods Sorting points by Z-order Recursive decomposition of space in quadrants until every quadrant contains at most one point A C 01 D F E H 11 K B 00 G 10 J Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 428
14 5 Spatial Access Methods Sorting points by Z-order Recursive decomposition of space in quadrants until every quadrant contains at most one point A 0101 C 0100 D 0111 E F 0110 H K B 00 G J Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 428
15 5 Spatial Access Methods Sorting points by Z-order Recursive decomposition of space in quadrants until every quadrant contains at most one point A 0101 C 0100 D 0111 E F 0110 H K 1110 B 00 G J Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 428
16 5 Spatial Access Methods Sorting points by Z-order Recursive decomposition of space in quadrants until every quadrant contains at most one point A 0101 C 0100 D 0111 E F 0110 H K 1110 B 00 G 1000 J 1011 Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 428
17 5 Spatial Access Methods Sorting points by Z-order Recursive decomposition of space in quadrants until every quadrant contains at most one point A 0101 C D 0111 E F 0110 H K 1110 B, C, A, F, D, E, G, J, H, K, B 00 G 1000 J 1011 Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 428
18 5 Spatial Access Methods A more realistic example (150 points) Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 429
19 5 Spatial Access Methods A more realistic example (200 points) Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 429
20 5 Spatial Access Methods A more realistic example (250 points) Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 429
21 5 Spatial Access Methods Efficient calculation of the Z-value by bitinterleaving The Z-value of a quadrant consists of alternating bits from the binary representations of its x and y coordinate values, 11 i.e.: x-coordinate y-coordinate Z-value Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 430
22 5 Spatial Access Methods Example: range query Calculate Z-values of the query window: Min=16, Max=30 Search for Min Check all points whose Z-values Max A C Min B D Max E F G H J K B C A F D E G J H K Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 431
23 5 Spatial Access Methods Problems Only suitable for points False positives, the solutions supplied by the index have to be verified Adequacy depends on the position of the query window Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 432
24 5 Spatial Access Methods ndexes can be classified according to Data types, which can be indexed Points (Point Access Method) or extended objects (Spatial Access Method) Principle of space decomposition: structure dependant on the insertion order data-driven (tree) yes space-driven (trie) no indexing dead space no yes guaranteed storage utilization yes no Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 433
25 5.1 Quadtree Main principle: Successive decomposing of space into quadrants Search tree whose nodes have four children each Decomposition may be space- or data-driven Can be used for points as well as for extended objects Not balanced, can degenerate nw sw ne se Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 434
26 5.1 Quadtree Point quadtree (data-driven) ndexing of points Data pointers in leaf and internal nodes nsertion Search ends at Nil node, replace this node with the new node containing the element Deletion of internal nodes difficult Remove the subtree whose root is the node to be deleted Re-insert nodes of the subtree tree node key value data pointer node pointer Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 435
27 5.1 Quadtree Example: insertion F A C D E F H K B G J Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 436
28 5.1 Quadtree Example: insertion E F A C D E F H K B G J Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 436
29 5.1 Quadtree Example: insertion E F A C D E F H K J D A B G Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 436
30 5.1 Quadtree Example: insertion E C F A C D E F H K J D A B G Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 436
31 5.1 Quadtree Example: insertion E C F A C D E F H K J D A B B G Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 436
32 5.1 Quadtree Example: insertion E C J 5 15 F A C D E F H K J D A B G K H B G Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 436
33 5.1 Quadtree Range query: Min(x, y), Max(x,y) 4 comparisons per node skip N skip W skip E skip S Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 437
34 5.1 Quadtree Range query: Min(5, 16), Max(20,19) 4 comparisons per node x < Min.x skip w D F y < Min.y skip s x > Max.x skip e y > Max.y skip n A C Min E F Max H K E C J J B G D A B G K H Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 438
35 5.1 Quadtree Range query: Min(5, 16), Max(20,19) 4 comparisons per node x < Min.x skip w D F y < Min.y skip s x > Max.x skip e y > Max.y skip n A C Min E F Max H K E C J J B G D A B G K H Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 438
36 5.1 Quadtree Range query: Min(5, 16), Max(20,19) 4 comparisons per node x < Min.x skip w D F y < Min.y skip s x > Max.x skip e y > Max.y skip n A C Min E F Max H K E C J J B G D A B G K H Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 438
37 5.1 Quadtree Range query: Min(5, 16), Max(20,19) 4 comparisons per node x < Min.x skip w D F y < Min.y skip s x > Max.x skip e y > Max.y skip n A C Min E F Max H K E C J J B G D A B G K H Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 438
38 5.1 Quadtree Range query: Min(5, 16), Max(20,19) 4 comparisons per node x < Min.x skip w D F y < Min.y skip s x > Max.x skip e y > Max.y skip n A C Min E F Max H K E C J J B G D A B G K H Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 438
39 5.1 Quadtree Range query: Min(5, 16), Max(20,19) 4 comparisons per node x < Min.x skip w D F y < Min.y skip s x > Max.x skip e y > Max.y skip n A C Min E F Max H K E C J J B G D A B G K H Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 438
40 5.1 Quadtree Range query: Min(5, 16), Max(20,19) 4 comparisons per node x < Min.x skip w D F y < Min.y skip s x > Max.x skip e y > Max.y skip n A C Min E F Max H K E C J J B G D A B G K H Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 438
41 5.1 Quadtree Range query: Min(5, 16), Max(20,19) 4 comparisons per node x < Min.x skip w D F y < Min.y skip s x > Max.x skip e y > Max.y skip n A C Min E F Max H K E C J J B G D A B G K H Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 438
42 5.1 Quadtree Range query: Min(5, 16), Max(20,19) 4 comparisons per node x < Min.x skip w D F y < Min.y skip s x > Max.x skip e y > Max.y skip n A C Min E F Max H K E C J J B G D A B G K H Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 438
43 5.1 Quadtree Range query: Min(5, 16), Max(20,19) 4 comparisons per node x < Min.x skip w D F y < Min.y skip s x > Max.x skip e y > Max.y skip n A C Min E F Max H K E C J J B G D A B G K H Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 438
44 5.1 Quadtree Range query: Min(5, 16), Max(20,19) 4 comparisons per node x < Min.x skip w D F y < Min.y skip s x > Max.x skip e y > Max.y skip n A C Min E F Max H K E C J J B G D A B G K H Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 438
45 5.1 Quadtree Unfavorable insertion order 2 19 A 15 6 G E B 23 9 J K A C Min D E F Max H K 8 23 D 5 15 C B G J F H Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 439
46 5.1 Quadtree Example: locations of 400 weather stations in Germany Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 440
47 5.1 Quadtree unfavorable insertion order optimal insertion order visualization tool: [Bu10] Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 441
48 5.1 Quadtree unfavorable insertion order (bucket size = 20) optimal insertion order visualization tool: [Bu10] Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 442
49 5.1 Quadtree Disadvantages High storage usage 4 Nil nodes per leaf at least, additional ones may be in internal nodes Deletion expensive K comparisons for point queries, 2k for range queries per node Variants: Pseudo-Quadtree Using arbitrary points for the partitioning process Data pointers only in leaf nodes 9 20 D E A F 6 10 C B K G J H Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 443
50 5.1 Quadtree Point quadtree (space-driven) Recursive subdivision of the space into quadrants until every quadrant contains at most the given number of points Extent of space has to be known B A C D E F H K A C F G J H K D E B G J Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 444
51 5.1 Quadtree nternal nodes contain only node pointer Data pointer only in leaf nodes Structure depends on the location of points nsertion Search ends at Nil node: insert new leaf node Search ends at leaf node: replace leaf with a new internal node, insert both points as new leaves Deletion Only leaves are deleted f an internal node has only one child after deletion it is replaced by its child Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 445
52 5.1 Quadtree Region quadtree (space-driven) ndexing of surfaces Decomposition of space until the quadrants are empty or completely covered by an object X X Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 446
53 5.1 Quadtree Region quadtree (space-driven) ndexing of surfaces Decomposition of space until the quadrants are empty or completely covered by an object X X Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 446
54 5.1 Quadtree Region quadtree (space-driven) ndexing of surfaces Decomposition of space until the quadrants are empty or completely covered by an object X X Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 446
55 5.1 Quadtree Region quadtree (space-driven) ndexing of surfaces Decomposition of space until the quadrants are empty or completely covered by an object X X Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 446
56 5.1 Quadtree Region quadtree (space-driven) ndexing of surfaces Decomposition of space until the quadrants are empty or completely covered by an object X X Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 446
57 5.1 Quadtree Region quadtree (space-driven) ndexing of surfaces Decomposition of space until the quadrants are empty or completely covered by an object X X Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 446
58 5.1 Quadtree Region quadtree (space-driven) ndexing of surfaces Decomposition of space until the quadrants are empty or completely covered by an object X / X Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 446
59 5.1 Quadtree Region quadtree (space-driven) ndexing of surfaces Decomposition of space until the quadrants are empty or completely covered by an object X / X Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 446
60 5.1 Quadtree Region quadtree (space-driven) ndexing of surfaces Decomposition of space until the quadrants are empty or completely covered by an object X / X X X X Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 446
61 5.1 Quadtree Region quadtree (space-driven) ndexing of surfaces Decomposition of space until the quadrants are empty or completely covered by an object X / X X X X X X Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 446
62 5.2 R-tree The R-tree is the prototype of a multidimensional extension of the B-tree The root has at least two child nodes Every internal node has got between m and M child nodes (multiway tree) M and m M are predisposed For each entry of an internal node the MBR containing all the rectangles of its child nodes is stored Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 447 node pointer MBR data pointer internal node leaf
63 5.2 R-tree Overlapping clusters (the more overlap the less efficient) Objects are stored in leaves, internal nodes for navigation only Height balanced tree Dynamic index structure (supports insert, update, delete) Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 448
64 5.2 R-tree Construction a e f c g j b h i d Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 449
65 5.2 R-tree Construction a e B A f c g j b h D i C d a c f e g j b d h i Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 449
66 5.2 R-tree Construction a e B A f c g j b h D i C d A B C D a c f e g j b d h i Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 449
67 5.2 R-tree Construction a e B A f c g j b h D i C d A B C D a c f e g j b d h i Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 449
68 5.2 R-tree Search Recursively from the root to the leaves One path at a time f the object has not been found in this subtree, the next search path will be traversed The traversal order of paths is arbitrary Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 450
69 5.2 R-tree Good performance is not guaranteed n the worst case all paths have to be searched (because of overlap) Search algorithms try to skip as many irrelevant subtrees as possible (pruning) Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 451
70 5.2 R-tree Search in internal node: Check for each entry if its MBR intersects S For all intersecting entries continue searching in all its child nodes Search in leaf nodes: Check for each entry if its MBR intersects S For all intersecting entries check if the original objects satisfy the search condition, if so they belong to the solution Search algorithms for R-trees are the most efficient if overlap and coverage are minimized Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 452
71 5.2 R-tree Example: a e B A f c S g j b h D i C d A B C D a c f e g j b d h i Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 453
72 5.2 R-tree Example: a e B A f c S g j b h D i C d A B C D a c f e g j b d h i Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 453
73 5.2 R-tree Example: a e B A f c S g j b h D i C d A B C D a c f e g j b d h i Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 453
74 5.2 R-tree Example: a e B A f c S g j b h D i C d A B C D a c f e g j b d h i Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 453
75 5.2 R-tree Example: a e B A f c S g j b h D i C d A B C D a c f e g j b d h i Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 453
76 5.2 R-tree Example: a e B A f c S g j b h D i C d A B C D a c f e g j b d h i Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 453
77 5.2 R-tree Example: a e B A f c S g j b h D i C d A B C D a c f e g j b d h i Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 453
78 5.2 R-tree nsertion Search the best leaf node, according to spatial criteria (ChooseLeaf) f the node contains less than M entries, insert the object there Otherwise overflow handling is necessary and the node is split (SplitNode), so that overlap and coverage are minimized Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 454
79 5.2 R-tree MBR of the parent node has to be enlarged if the new object is not entirely inside (AdjustTree) f the root is splitted create a new root whose children are the splitted nodes of the old root Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 455
80 nsert k either into C or into D f D is chosen the added area is bigger, but there is no overlap Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig R-tree a b c d e g f h i j k A B C D a b c d e g f h i j k A B C D
81 5.2 R-tree Heuristic Always insert a new object into the node resulting in smallest increase in volume f it lies inside a MBR no enlargement is necessary f there is more than one possibility, choose the node with the smallest volume the Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 457
82 5.2 R-tree Overflow f a new entry should be inserted into a full node than M+1 entries have to be distributed to two nodes During subsequent searches it should not be necessary to search both subtrees The smaller the MBRs the less overlap Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 458
83 5.2 R-tree Calculate the sum of the area of both rectangles and minimize the dead space The volume of both MBRs should be as small as possible Decision how to split is not trivial There are many possibilities to distribute the objects to two MBRs: x 2 n 1 1 Compare all possible distributions takes exponential time (practically impossible) Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 459
84 5.2 R-tree Example: possible distributions for 5 objects Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 460
85 5.2 R-tree visualization tool: [Sp10] Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 461
86 5.2 R-tree Example: possible distributions for 6 objects visualization tool: [Sp10] Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 462
87 5.2 R-tree Overflow handling with a quadratic complexity For each pair of objects calculate the area of their MBR and choose the pair with the largest MBR As these objects should not belong to the same node they are used as seed points of two new nodes Calculate for all other objects the difference in increase in volume in respect of both MBRs nsert the object with the largest difference into the adequate MBR and update it Repeat this process for the remaining objects Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 463
88 5.2 R-tree Determining the seeds with a linear complexity For each dimension: Find the rectangle with the highest low edge and the rectangle with the lowest high edge Calculate the distance between them and normalise it by dividing it by the width of the MBR of the overflowing node The final seeds are the two objects having the greatest normalized separation This linear cost algorithm is usually sufficient as the quality of the splits i.e. the minimality of overlaps is similar to that of the quadratic cost algorithm Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 464
89 5.2 R-tree Example Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 465
90 5.2 R-tree Example Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 465
91 5.2 R-tree Example Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 465
92 5.2 R-tree Example Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 465
93 5.2 R-tree Example Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 465
94 5.2 R-tree Example Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 465
95 5.2 Example Overflow Handling m A B C D a e B A f c g j a c f b d k e g j h i b h D i C d k Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 466
96 5.2 Example Overflow Handling m A B C D e g j a c f b d k e g j h i Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 466
97 5.2 Example Overflow Handling X-direction: Choose e and j 9 m d x = 2/11 0,18 Y-direction: e 3,5 Choose m and j d y = 3,5/9 0,43 g 2 j As d y > d x choose m and j 11 Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 467
98 5.2 Example Overflow Handling X-direction: Choose e and j 9 m d x = 2/11 0,18 Y-direction: e 3,5 Choose m and j d y = 3,5/9 0,43 g 2 j As d y > d x choose m and j 11 Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 467
99 5.2 Example Overflow Handling nsert g to m: 24 9 m to j: 41,5 Difference: 17,5 e 3,5 nsert e to m: 14 g 2 j to j: Difference: 38 Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 468
100 5.2 Example Overflow Handling nsert g to m: 24 9 m to j: 41,5 Difference: 17,5 e 3,5 nsert e to m: 14 g 2 j to j: Difference: 38 Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 468
101 5.2 Example Overflow Handling nsert g to m: 24 9 m to j: 41,5 Difference: 17,5 e 3,5 nsert e to m: 14 g 2 j to j: Difference: 38 Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 468
102 5.2 Example Overflow Handling Result a E e m A f c g j B b h D i A B E C D C d k j a c f e m g b d k h i Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 469
103 5.2 R-tree Deletion Search for the leaf containing the entry (FindLeaf) Delete the entry from the leaf (DeleteRecord) Condense the tree if the node contains less than m entries, i.e. the node is deleted completely and the other entries are reinserted (CondenseTree) f only one child of the root is left this child becomes the new root Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 470
104 5.2 R-tree Updates f an object is updated its MBR might change n that case the corresponding entry has to be deleted, updated and reinserted Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 471
105 5.2 R-tree Updates f an object is updated its MBR might change n that case the corresponding entry has to be deleted, updated and reinserted Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 471
106 5.2 R-tree Updates f an object is updated its MBR might change n that case the corresponding entry has to be deleted, updated and reinserted Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 471
107 5.2 R-tree Updates f an object is updated its MBR might change n that case the corresponding entry has to be deleted, updated and reinserted Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 471
108 5.2 R-tree R + tree Overlap of adjacent MBRs is forbidden Several leaves may contain an entry for one object Search is more efficient, but its scalability is similar to R-trees b C a A f d c E e k h g j D i B A B E C D a e e g j c f b d h i k Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 472
109 5.2 R-tree Differences between R + trees and R-trees nsertion: object may be inserted in more than one leaf nsertion: propagation of splits down the tree as overlaps are not allowed Deletion: there is no minimum number of child nodes anymore a e e g j c f b d h i k The biggest advantage of R + -trees is the improved search efficiency as only a single path has to be searched for a point query A disadvantage is that nodes are often under-filled due to many splits A B E C D Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 473
110 5.3 K-d tree K-dimensional, binary search tree On every level the space is decomposed perpendicular to one dimension one comparison ndexing of points Variant for extended objects: Bintree Not balanced key value A K-d tree for n points Needs O(n) space May be constructed in O(n log n) time data pointer discriminant: dimension used for splitting node pointer tree node Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 474
111 5.3 K-d tree Example: data driven, alternating splits in x- and y-dimension B C 8 17 D y J F x y K A C D E F H K 4 7 x 9 31 x x x J A 3 27 y E y G 17 5 y H y B G Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 475
112 5.3 K-d tree Split at data points or at arbitrary points Choice of split dimension Alternating or dimension with the largest extent Choice of split position Median or average Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 476
113 5.3 K-d tree Determination of the split plane Calculation of the median of n elements necessary Complex algorithm to determine the median in O(n) or: Sort all points in two lists one according to the x- and one to the y-coordinate After a split these lists are splited respectively traversal in linear time possible Determining the median in O(1) and creation of sublists in O(n) time Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 477
114 5.3 K-d tree Bintree Like the region quadtree, but the space is not divided into quadrants but into half-spaces 20 x y 20 y 4 x 8 x 23 x 28 x 24 X y 8 y 24 y 10 y 24 y 2 x 2 x 8 x 4 x x X 12 X 8 y / 16 y 8 y X 4 23 Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 478
115 5.3 K-d tree Deletion Similar to binary search tree: the node to delete is replaced by its immediate predecessor or successor Problem: Several different orderings of the elements F (one for each dimension) Solution: Search the node which is the immediate predecessor or successor with respect to the discriminant of the node to delete A B x C y Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig D 9 31 E y x y J x G x y y K x H y
116 5.3 K-d tree Deletion Similar to binary search tree: the node to delete is replaced by its immediate predecessor or successor Problem: Several different orderings of the elements F (one for each dimension) Solution: Search the node which is the immediate predecessor or successor with respect to the discriminant of the node to delete A B x C y Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig D 9 31 E y x y J x G x y y K x H y
117 5.3 K-d tree Deletion Similar to binary search tree: the node to delete is replaced by its immediate predecessor or successor Problem: Several different orderings of the elements F (one for each dimension) Solution: Search the node which is the immediate predecessor or successor with respect to the discriminant of the node to delete B 4 7 x A Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig D 9 31 E y x y J x G x y y K x H y
118 5.4 BSP tree Binary Space Partitioning (BSP) Recursive bipartitioning of the n-dimensional space by (arbitrary) (n-1)-dimensional hyperplanes (in 2d: lines) generalized k-d tree Usually used to index surfaces or lines Class variables of a BSP tree node class Node{ Hyperplane ph; Node leftchild; Node rightchild; Set oset; } //split plane (instead of the discriminant) //left child, is in front of the split plane //right child, is behind the split plane //set of objects in the plane Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 480
119 5.4 BSP tree Pseudocode for the construction of a BSP tree Node buildbsptree(o) { // o is set of objects if o.isempty() return null; Node N = new Node(); N.ph = choosepartitioninghyperplane(o); Set fset = new Set(); //to save the objects in front of ph Set bset = new Set(); //to save the objects behind ph for each Object z in o do{ if z is coincident to ph, add z to N.oSet; if z is in front of ph, add z to fset; if z is behind ph, add z to bset; if z spans ph, split z and add pieces to fset and bset; }N.leftChild = buildbsptree(fset); N.rightChild = buildbsptree(bset); return N; } Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 481
120 5.4 BSP tree Determining the position of an object z relative to a split plane g 2-dimensional split line: ax + by + c = 0 Calculate ax + by + c for all points of z f all values < 0 z is in front of g f all values > 0 z is behind g f all values = 0 z lies on g Otherwise z intersects the split line and has to be splitted 4 2 y x split line: 4x 3y +3 = 0 nsert the points of O1 (3,3) (5,2) = =17 Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 482 O2 split line O1
121 5.4 BSP tree Goals for the choice of split planes Balanced tree Avoid splitting of objects Split planes orthogonal to the axes Heuristic to estimate the costs of a split plane Difference between the number of objects in front of and behind the split plane 15 * number of splitted objects 5 if split plane is not orthogonal to one of the axes Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 483
122 5.4 BSP tree A split plane which do not divide any objects is called Free Partition f a part of a boundary of an object which already intersects a BSP region is chosen as a split plane, it is a free partition in respect of that region Under the assumption that there are no intersections between objects object region split line Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 484
123 5.4 BSP tree Application in computer graphics E.g. visibility computation, ray tracing Surfaces on the same side as the observer May cover surfaces on the other side Cannot be covered by surfaces on the other side Determination of the drawing order Back-to-front: background objects are drawn first and are painted over by objects in the foreground 1 wanted: ordering of the objects with respect to their relative distance from the observer Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 485
124 5.4 BSP tree For each node, determine on which side of it the observer lies norder traversal defines the order of the objects 1. Subtree on the opposite side from the observer 2. Node 3. Subtree on the same side as the observer norder traversal on a binary search tree 1. Left subtree 2. Node 3. Right subtree Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 486
125 5.4 BSP tree Example Construction of a BSP tree by a sequence of insertion operations (without application of the heuristic) 3-dimensional objects (upright walls) 0 0 Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 487
126 5.4 BSP tree Example Construction of a BSP tree by a sequence of insertion operations (without application of the heuristic) 3-dimensional objects (upright walls) Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 487
127 5.4 BSP tree Example Construction of a BSP tree by a sequence of insertion operations (without application of the heuristic) 3-dimensional objects (upright walls) Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 487
128 5.4 BSP tree Example Construction of a BSP tree by a sequence of insertion operations (without application of the heuristic) 3-dimensional objects (upright walls) Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 487
129 5.4 BSP tree Example Construction of a BSP tree by a sequence of insertion operations (without application of the heuristic) 3-dimensional objects (upright walls) b 4f 3 2 4b 2 1 4f Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 487
130 5.4 BSP tree Example Construction of a BSP tree by a sequence of insertion operations (without application of the heuristic) 3-dimensional objects (upright walls) b 4f 3 5b 2 5f 4b 2 1 5f 4f 5b Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 487
131 5.4 BSP tree Example Construction of a BSP tree by a sequence of insertion operations (without application of the heuristic) 3-dimensional objects (upright walls) b 4f 3 5b 2 5f 4b 2 1 5f 4f 5b 6f 6b Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 487
132 5.4 BSP tree Example Construction of a BSP tree by a sequence of insertion operations (without application of the heuristic) 3-dimensional objects (upright walls) b 4f 3 5b 2 5f 4b 2 1 5f 6ff 4f 6fb 6b 5b 6ff 6fb 6b Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 487
133 5.4 BSP tree Drawing order: 3, 4b, 5b, 0, 2, 6ff, 4f, 6fb, 1, 5f, 6b 6fb 6ff f 4b 3 Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 488
134 5.5 Grid file Main idea: k-dimensional, dynamic array as index (on disk) ndex contains (except for management information where necessary) only pointers to data pages (buckets) Data is stored in (big) buckets Two disk accesses for exact match queries: ndependent of the distribution of values and the number of stored objects For points only Data-driven partitioning of space Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 489
135 5.5 Grid file Components: K scales S i (with i from 1 to k) define the grid on the k- dimensional data space D Cell or grid directory (GD): dynamic k-dimensional matrix to map D on the buckets Bucket: storage of the points of one or more cells (bucket region BR) data space cell scales S 1 S 2 d 2 grid directory threshold in data space position in GD d 1 buckets Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 490
136 5.5 Grid file Properties 1:1-relation between cells and elements of GD Element of GD = bucket pointer n:1-relation between cells and buckets Cells have to form a rectangle in order to be assigned to the same bucket Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 491
137 5.5 Grid file Exact match query Search the scales (binary search) for the position of the grid cell that covers the point ( no disk access, if the scales fit into find (10.3,52.2) main memory) Read the grid directory to identify the bucket ( 1st disk access) Read the bucket ( 2nd disk access) S , 54.3 Kiel 10.3, 52.2 Braunschweig 11.5, 51.0 Jena 55 4 S Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 492
138 5.5 Grid file nsertion Search for the point to be inserted and if possible put it in the found bucket n case of bucket overflow Add new bucket, update GD f a grid refinement is necessary, update scales Choice of dimension for the grid refinement e.g. alternating or longest interval Choice of split position e.g. middle or local median Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 493
139 5.5 Grid file Example: insertion with bucket overflow without grid refinement d 2 D A E H S F S ADE HG BCF B C G J Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig d 1
140 5.5 Grid file Example: insertion with bucket overflow without grid refinement d 2 D A E H S F S ADE HG BCF V B C G J Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig d 1
141 5.5 Grid file Example: insertion with bucket overflow without grid refinement d 2 D A E H S F S ADE HG JG BCF H V B C G J Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig d 1
142 5.5 Grid file Example: insertion with bucket overflow and grid refinement D d 2 A E H S 1 S ADE 10 C F K H JG BCF V B G J Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig d 1
143 5.5 Grid file Example: insertion with bucket overflow and grid refinement D d 2 A E H S 1 S ADE H JG BCF V V 10 Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 495 B C F 6 G K J d 1
144 5.5 Grid file Example: insertion with bucket overflow and grid refinement D d 2 A E H S 1 S ADE H JG BCF V V 10 5 Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 495 B C F 6 G K J d 1
145 5.5 Grid file Example: insertion with bucket overflow and grid refinement D d 2 A E H S 1 S ADE H JG BCF V V 10 5 Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 495 B C F 6 G K J d 1
146 5.5 Grid file Example: insertion with bucket overflow and grid refinement D d 2 A E H S 1 S ADE H JG BCF V V 10 5 Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 495 B C F 6 G K J d 1
147 5.5 Grid file Example: insertion with bucket overflow and grid refinement D d 2 A E H S 1 S ADE H JG BCF V V 10 5 Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 495 B C F 6 G K J d 1
148 5.5 Grid file Example: insertion with bucket overflow and grid refinement D S 1 S ADE V 2 H 1 JG K 1 2 BCF V d Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 495 A B C E F 6 G K H J d 1
149 5.5 Grid file Deletion Merging of buckets deally the buckets to be merged were split at some earlier point and none of them was split again buddy system With adjacent cells, complex Not always possible Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 496
150 5.6 Summary One-dimensional orderings mpose a linear ordering on multidimensional points Z-order, Hilbert s curve, Gray code Quadtree Point quadtree: construction, range query Region quadtree R-tree Construction Search nsertion, overflow handling R + -tree Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 497
151 5.6 Summary K-d tree Choice of split plane Deletion Bintree BSP tree Data structure Choice of split plane Rendering Grid file nsertion, overflow handling Deletion y 4 2 O2 split line 2 O1 4 6 x Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 498
152 5.6 Summary collect GS visualize manage analyze objects query evaluation geometry indexes vector approximations Spatial Databases and GS Karl Neumann, Sarah Tauscher fis TU Braunschweig 499
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