1 / 41

R-TREES

R-TREES. CMPE 422:Database systems Ahmet Menteş. TOPICS COVERED. Introduction Spatial Data and Queries R-Tree Index Algorithms and Complexities Performance R-Tree Variations Conclusion. INTRODUCTION. First Proposed By Antonin Guttman, 1984 from UC Berkeley

gary
Télécharger la présentation

R-TREES

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. R-TREES CMPE 422:Database systems Ahmet Menteş

  2. TOPICS COVERED • Introduction • Spatial Data and Queries • R-Tree Index • Algorithmsand Complexities • Performance • R-Tree Variations • Conclusion

  3. INTRODUCTION • First Proposed By Antonin Guttman, 1984 from UC Berkeley -“A Dynamic Index Structure For Spatial Searching” • R-Tree structures are used for effectively storing and indexing spatial data. Usage areas are wide • -Computer Aided Design-CAD • -Video Recognition-VR • -Image Processing

  4. INTRODUCTION • R-Trees have correspondance with B-Trees but used for keeping spatial data. -Based on B-trees • B-Trees cannot store new types of data • Wanted to store geometrical data and multi-dimensional data • Region-trees (R-trees)

  5. SPATIAL DATA • Any Type of Geometry • Point • City • Line • Trail • Polygon • Border • A Collection of Geometries • Ski Resort Trails • Any Coordinate System • Meters • Pixels • WGS84 (GPS)

  6. SPATIAL QUERY TYPES • Standard Insert, Search and Delete Queries • Spatial Range Queries • Find all cities within 30 kmof İstanbul • Nearest Neighbor Queries • Find the closest restourants to my address • Spatial Join Queries • Find all neighborhoods that are within 5 km of a university • Geometry Set Operations • Equal(), Disjoint(), Intersect(), Touch(), Cross(), Within(), Contains(), Overlap(), Distance(), Intersection(), Union(), Difference() and more...

  7. R-TREE • R-tree is a depth-balanced tree –Each node corresponds to a disk page –Leaf node: an array of leaf entries A leaf entry: (mbb,oid) –Non-leaf node: an array of node entries A node entry: (dr, nodeid)

  8. R-TREE

  9. R-TREE INDEX STRUCTURE • Data objects in the map are represented by the Minimum Bounding Rectangles (MBRs)

  10. R-TREE INDEX STRUCTURE Formal Description • Structure consisting of (regular) nodes containing tuples <I, child-pointer> • At the lowest level: leaf-nodes with tuples <I, tuple-identifier> • I = (I0, I1, … In) corresponds to MBR • n = Number of Dimensions in the Geometry • Each I is a set of the form [a,b] describing the range of the rectangle along the dimension • a or b can be equal to infinity • Tuple-identifier points to a record

  11. R-TREE INDEX STRUCTURE • Important Parameters for R-Tree index • M is the maximum number of entries in one node • Parameter m ≤ M/2 specifies the minimum number of entries in a node • Every Leaf Node Contains Between m and M index records unless it is root. • For each index record, <I, tuple-identifier> in a leaf node is the smallest rectangle that spatially contains the n-dimensional data object. • Every non-leaf node has between m and M children unless it is the root. • For each entry <I, child-pointer> in a non-leaf node, I is the smallest rectangle that spatially contains the rectangles in the child nodes. • The root node has at least two children unless it is a leaf. • All leaves appear on the same level. • Height of an R-Tree  logmN-1

  12. ALGORITHMS OVERVIEW • For all queries, it is possible to check if a point is within a rectangle in linear time (O(n)). • Query Types To Be Reviewed • Search • Splitting • Insertion • Deletion • Nearest Neighbor

  13. SEARCH • Similar to B-tree search • Quite easy & straight forward(Traverse the whole tree starting at theroot node) • No guarantee on good worst-case performance!(Possible overlapping of rectangles of entries within a single node!)

  14. SEARCH • For all entries in a non-leaf node: Check if overlapping If yes: check node pointed to by this entry. • If node is a leaf-node: Check all entries if overlapping the search object If yes: entry is a qualifying record!

  15. SEARCH

  16. SEARCH COMPLEXITY • Worst Case • Linear Search • Every MBR overlaps the search area • Best Case • No more than one overlap at each level • Complexity becomes O(logMn) • Again, dependent on geometries

  17. SPLITTING • Splitting of nodes necessary afterunderflow oroverflow (as a result of a delete or insert operation) • Ultimate goal: Minimize the resulting node’s MBRs • Secondary aim: Speed of algorithm • 3 Implementations to split an R-Tree: Exhaustive, Quadratic-cost, Linear-cost

  18. SPLITTING • Minimal resulting MBRs: • Exhaustive Approach: • Simply try all possible split-ups but this kind of approach is very costly in terms of complexity issue.

  19. SPLITTING • Quadratic-cost Algorithm: • Pick the 2 out of M entries that would consume the most space if put together. Put one in each group • For all remaining entries: Pick the one that would makethe biggest difference in area when put to one of the two groups. Add it to the one with the least difference • Finished when all entries are put in either group! • Quadratic in M, linear in the number of dimensions.

  20. SPLITTING • Linear-cost Implementation: • Basically the same as quadratic-cost algorithm • Differences: First pair is picked by finding the two rectangles with the greatest normalizedseparation in anydimension. Remaining pairs are selected randomly. • Linear in M as well as in the number ofdimensions

  21. INSERTION • Similar to corresponding B-tree operation • Basically consist of 3 parts: 1. CHOOSELEAF (Find place to insert new object) 2. INSERT (Insert the new object) 3. ADJUSTTREE (Adjusting preceding nodes)

  22. INSERTION Description: • CHOOSELEAF: • Start with root and run through all nodes then; • Find the one which would have to be leastenlarged to include given object! • INSERT: • Check if room for another entry Insert new entry directly OR after callingSPLITNODE (in case of no room)

  23. INSERTION • ADJUSTTREE: • Ascend from node with new entry: • Adjust all MBRs! • In case of node split: • Add new entry to parent node(If no room in parent node, invokeSPLITNODE again) • Propagate upwards till root is reached!

  24. INSERTION

  25. INSERTION

  26. INSERTION

  27. INSERTION COMPLEXITY • IMPORTANT: Make sure nodes are split so they cover the smallest possible area. • Minimize search time • Important variables • N = Number of entries in each node • T = Tree height • Worst Case • 2 * N * T • O(n) GOOD SPLIT! BAD!

  28. DELETION • NOT similar to B-tree DELETE (Treatment of underflows) • B-tree: Merge under-full node with “neighbor” • R-tree: Delete under-full node and re-insert • Why? Re-use of INSERT routine Incrementally refines spatial structure

  29. DELETION Description • FINDLEAF: • Locate the leaf that contains object to be deleted • DELETE: • Delete entry from node • CONDENSETREE: • If node with the removed entry has too few entries, reallocate them • Recursively check the parent nodes until reaching the root • Update all MBR and remove all nodes that underfull • Reinsert all entries removed from the removed nodes according to the INSERT algorithm.

  30. DELETION

  31. DELETION COMPLEXITY • Deletion variables • N = number of entries in each node • T = tree height • Complexity • 2 * N * T

  32. NEAREST NEIGHBOR • Two Options • Branch-and-bound search • Best first search • Branch-and-bound • Find two distances to each object • Minimum distance from the search point to any side of the other object’s MBR • Minmax distance • Least of the furthest distance in every dimension • Lowest upper bound on the distance from the point to an object • Best First Search • Calculates minmax distance for all objects • R-Tree sorted by minmax distance • Removes nodes from sorted tree • If node has no children it is the nearest neighbor.

  33. NEAREST NEIGHBOR • Branch-and-Bound • Takes longer because it searches all nodes that have not been pruned • Best First Search • Investigates only the closest nodes • Large priority queue data structure in memory • Can cause thrashing • Run-time complexity subject to geometries • How many overlap and how large

  34. PERFORMANCE Why do Benchmarking? • Proof practicality of the structure • Find suitable values for m and M • Test various node-splitting algorithms Testing environment: • C-implementation running under UNIX • Layout of a RISC-II chip with 1057 rectangles • Different page sizes (128 – 2048 bytes) • Various values for M (6 – 102)

  35. PERFORMANCE • Linear INSERT fastest • Linear INSERT quite insensitive to M and m • Quadratic INSERT depends on M as well as m • DELETE extremely sensitive to m

  36. PERFORMANCE • Same page hit / miss performance for linear and quadratic split • Slight advantage for exhaustive version • Space usage strongly depending on m

  37. PERFORMANCE • Quadratic-cost splitting with jumps at INSERT operations for increasing amount of data • Linear INSERT extremely constant • R-tree structure very effective in directing search to small subtrees

  38. R-TREE VARIATIONS R+-Tree: • R+ trees differ from R trees in that: • Nodes are not guaranteed to be at least half filled • The entries of any internal node do not overlap • An object ID may be stored in more than one leaf node. • Advantages • Because nodes are not overlapped with each other, point query performance benefits since all spatial regions are covered by at most one node. • A single path is followed and fewer nodesare visited than with the R-tree. • Disadvantages • Since rectangles are duplicated, an R+ tree can be larger than an R tree built on same data set. • Construction and maintenance of R+ trees is more complex than the construction and maintenance of R trees and other variants of the R tree. R-Tree MBRs R+-Tree MBRs

  39. R-TREE VARIATIONS • R*-Tree • Do not split nodes on insert • Take entries from the overfull node and reinsert them into the tree • Changes MBRs • Saves time and possibly rebalances the tree

  40. CONCLUSION Reasons for R-trees: • Importance of being able to store and effectively search spatial data. • Strong limitations with most spatialstructures (e.g.point data only, not dynamic, performance with paged memory, …) Basic Idea: • Structure similar to B-tree. • Represent objects by their MBR. • Allow overlapping.

  41. CONCLUSION Algorithms: • Search and insert similar to B-tree operations • Delete performing re-insert instead of merge for under-full nodes (incrementally refines spatial structure) • Node splitting: Benchmarks show that linear version performs quiteas well as quadratic-cost implementation. Modifications: • Structure is easy to adapt for special applications and their needs. • Performance advantages are usually paid for with price of astructure that gets harder to maintain.

More Related