Ch28 Guttman

Spatial Data Management: 

Spatial Data Management Chapter 28

Types of Spatial Data: 

Types of Spatial Data Point Data Points in a multidimensional space E.g., Raster data such as satellite imagery, where each pixel stores a measured value E.g., Feature vectors extracted from text Region Data Objects have spatial extent with location and boundary DB typically uses geometric approximations constructed using line segments, polygons, etc., called vector data.

Types of Spatial Queries: 

Types of Spatial Queries Spatial Range Queries Find all cities within 50 miles of Madison Query has associated region (location, boundary) Answer includes ovelapping or contained data regions Nearest-Neighbor Queries Find the 10 cities nearest to Madison Results must be ordered by proximity Spatial Join Queries Find all cities near a lake Expensive, join condition involves regions and proximity

Applications of Spatial Data: 

Applications of Spatial Data Geographic Information Systems (GIS) E.g., ESRI’s ArcInfo; OpenGIS Consortium Geospatial information All classes of spatial queries and data are common Computer-Aided Design/Manufacturing Store spatial objects such as surface of airplane fuselage Range queries and spatial join queries are common Multimedia Databases Images, video, text, etc. stored and retrieved by content First converted to feature vector form; high dimensionality Nearest-neighbor queries are the most common

Single-Dimensional Indexes: 

Single-Dimensional Indexes B+ trees are fundamentally single-dimensional indexes. When we create a composite search key B+ tree, e.g., an index on <age, sal>, we effectively linearize the 2-dimensional space since we sort entries first by age and then by sal. Consider entries: <11, 80>, <12, 10> <12, 20>, <13, 75> 11 12 13 70 60 50 40 30 20 10 80 B+ tree order

Multidimensional Indexes: 

Multidimensional Indexes A multidimensional index clusters entries so as to exploit “nearness” in multidimensional space. Keeping track of entries and maintaining a balanced index structure presents a challenge! Consider entries: <11, 80>, <12, 10> <12, 20>, <13, 75>

Motivation for Multidimensional Indexes: 

Motivation for Multidimensional Indexes Spatial queries (GIS, CAD). Find all hotels within a radius of 5 miles from the conference venue. Find the city with population 500,000 or more that is nearest to Kalamazoo, MI. Find all cities that lie on the Nile in Egypt. Find all parts that touch the fuselage (in a plane design). Similarity queries (content-based retrieval). Given a face, find the five most similar faces. Multidimensional range queries. 50 < age < 55 AND 80K < sal < 90K

What’s the difficulty?: 

What’s the difficulty? An index based on spatial location needed. One-dimensional indexes don’t support multidimensional searching efficiently. (Why?) Hash indexes only support point queries; want to support range queries as well. Must support inserts and deletes gracefully. Ideally, want to support non-point data as well (e.g., lines, shapes). The R-tree meets these requirements, and variants are widely used today.

The R-Tree: 

The R-Tree The R-tree is a tree-structured index that remains balanced on inserts and deletes. Each key stored in a leaf entry is intuitively a box, or collection of intervals, with one interval per dimension. Example in 2-D:

R-Tree Properties: 

R-Tree Properties Leaf entry = < n-dimensional box, rid > This is Alternative (2), with key value being a box. Box is the tightest bounding box for a data object. Non-leaf entry = < n-dim box, ptr to child node > Box covers all boxes in child node (in fact, subtree). All leaves at same distance from root. Nodes can be kept 50% full (except root). Can choose a parameter m that is <= 50%, and ensure that every node is at least m% full.

Example of an R-Tree: 

Example of an R-Tree R8 R9 R10 R11 R12 R17 R18 R19 R13 R14 R15 R16 R1 R2 R3 R4 R5 R6 R7 Leaf entry Index entry Spatial object approximated by bounding box R8

Example R-Tree (Contd.): 

Example R-Tree (Contd.) R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16 R17 R18 R19

Search for Objects Overlapping Box Q: 

Search for Objects Overlapping Box Q Start at root. 1. If current node is non-leaf, for each entry <E, ptr>, if box E overlaps Q, search subtree identified by ptr. 2. If current node is leaf, for each entry <E, rid>, if E overlaps Q, rid identifies an object that might overlap Q. Note: May have to search several subtrees at each node! (In contrast, a B-tree equality search goes to just one leaf.)

Improving Search Using Constraints: 

Improving Search Using Constraints It is convenient to store boxes in the R-tree as approximations of arbitrary regions, because boxes can be represented compactly. But why not use convex polygons to approximate query regions more accurately? Will reduce overlap with nodes in tree, and reduce the number of nodes fetched by avoiding some branches altogether. Cost of overlap test is higher than bounding box intersection, but it is a main-memory cost, and can actually be done quite efficiently. Generally a win.

Insert Entry <B, ptr>: 

Insert Entry <B, ptr> Start at root and go down to “best-fit” leaf L. Go to child whose box needs least enlargement to cover B; resolve ties by going to smallest area child. If best-fit leaf L has space, insert entry and stop. Otherwise, split L into L1 and L2. Adjust entry for L in its parent so that the box now covers (only) L1. Add an entry (in the parent node of L) for L2. (This could cause the parent node to recursively split.)

Splitting a Node During Insertion: 

Splitting a Node During Insertion The entries in node L plus the newly inserted entry must be distributed between L1 and L2. Goal is to reduce likelihood of both L1 and L2 being searched on subsequent queries. Idea: Redistribute so as to minimize area of L1 plus area of L2. Exhaustive algorithm is too slow; quadratic and linear heuristics are described in the paper. GOOD SPLIT! BAD!

Splitting Algorithm: 

Splitting Algorithm If the node to split has M entries, we have 2M possible groupings to consider for the split. Alternative Quadratic-Cost Algorithm: Pick two “seed” entries s.t. the most area is “wasted” if they’re in the same group (build covering rectangle and subtract areas of each entry; we want to maximize this quantity) Assign remaining entries one-by-one to one of the two new nodes. The next entry inserted is the one with the greatest “preference” (difference of area-expansion costs) for one of these nodes. Linear Cost Algorithm: Pick a pair of seeds that are “farthest apart” along a dimension. Pick next entry to insert in any order.

R-Tree Variants: 

R-Tree Variants The R* tree uses the concept of forced reinserts to reduce overlap in tree nodes. When a node overflows, instead of splitting: Remove some (say, 30% of the) entries and reinsert them into the tree. Could result in all reinserted entries fitting on some existing pages, avoiding a split. R* trees also use a different heuristic, minimizing box perimeters rather than box areas during insertion. Another variant, the R+ tree, avoids overlap by inserting an object into multiple leaves if necessary. Searches now take a single path to a leaf, at cost of redundancy.

GiST: 

GiST The Generalized Search Tree (GiST) abstracts the “tree” nature of a class of indexes including B+ trees and R-tree variants. Striking similarities in insert/delete/search and even concurrency control algorithms make it possible to provide “templates” for these algorithms that can be customized to obtain the many different tree index structures. B+ trees are so important (and simple enough to allow further specialization) that they are implemented specially in all DBMSs. GiST provides an alternative for implementing other tree indexes in an ORDBS.

Indexing High-Dimensional Data: 

Indexing High-Dimensional Data Typically, high-dimensional datasets are collections of points, not regions. E.g., Feature vectors in multimedia applications. Very sparse Nearest neighbor queries are common. R-tree becomes worse than sequential scan for most datasets with more than a dozen dimensions. As dimensionality increases contrast (ratio of distances between nearest and farthest points) usually decreases; “nearest neighbor” is not meaningful. In any given data set, advisable to empirically test contrast.

Summary: 

Summary Spatial data management has many applications, including GIS, CAD/CAM, multimedia indexing. Point and region data Overlap/containment and nearest-neighbor queries Many approaches to indexing spatial data R-tree approach is widely used in GIS systems Other approaches include Grid Files, Quad trees, and techniques based on “space-filling” curves. For high-dimensional datasets, unless data has good “contrast”, nearest-neighbor may not be well-separated

Comments on R-Trees: 

Comments on R-Trees Deletion consists of searching for the entry to be deleted, removing it, and if the node becomes under-full, deleting the node and then re-inserting the remaining entries. Overall, works quite well for 2 and 3 D datasets. Several variants (notably, R+ and R* trees) have been proposed; widely used. Can improve search performance by using a convex polygon to approximate query shape (instead of a bounding box) and testing for polygon-box intersection.