database design and implementation 07.multidim

8/8/2019 Database design and Implementation 07.multidim

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Database Systems Implementation, Bongki Moon 1

Multidimensional Access Methodsfor Point and Spatial Data

Ramakrishnan&Gehrke: Chap. 28.1-28.3

Various articles


Single-Dimensional Indexes

B+ trees are fundamentally single-dimensionalindexes.

When we create a composite search key B+ tree,e.g., an index on , we effectively linearizethe 2-dimensional space since we sort entries firstby age and then by sal.

Consider entries:, ,

11 12 13

70605040302010

80

B+ treeorder


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Multidimensional Queries

Multidimensional equality queries (e.g.) age = 50 AND sal = 70K

Multidimensional range queries 50 < age < 55 AND 80K < sal < 90K

Similarity queries (NN; content-based retrieval) Given a face, find the five most similar faces.

Spatial queries (GIS, CAD)

Find all hotels within 5 miles from the conference venue. Find the city with population 500,000 or more that isnearest to Kalamazoo, MI.

Find all cities that lie on the Nile in Egypt. Find all parts that touch the fuselage (in a plane design).


Desired Properties

A multidimensional index clusters entries so as toexploit nearness in multidimensional space.

Symmetric: efficient access w.r.t. all indexed

attributes.

Consider entries:, ,

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Spatialclusters


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Whats the difficulty?

An index based on spatial location needed.

Must support range queries as well aspoint queries.

Must support inserts and deletes gracefully.

Need to support non-point data as well.

(e.g.) lines, shapes, polygons, etc.

Numerous index structures have been proposed. Points: Grid file, k-d-B tree, hB-tree, etc.

Spatial: R-tree and its variants (R+-, R*-, P-trees).


K-d Tree [Bentley 1975]

Structurally, a binary tree. Different (alternate) discriminatorat each level. Not necessarily balanced. Efficient exact-matching, range queries.

But, main memory data structure.


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Point Quad-tree [Finkel&Bentley 1974]

Essentially, multidimensional (binary) search tree. Fanout is 2d for a d-dimensional space.

Like k-d trees, decomposes the space by hyperplanes.

Searching is similar to binary search tree.

At each level, all d key values are compared.

To insert a new data point, first perform a search

then, insert the new data as a new leaf node,

its corresponding partition is divided into 2d-waysubspaces.

Not necessarily balanced.


Point quad-tree: example


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Grid Files [Nievergelt 1984]

Dynamic version of multi-attribute hashing.

Superimpose a d-dimensional orthogonal grid.

The subspace represented by each grid cell(bucket) may be of different shape and size.

Structure

Scales (for x and y dimensions): stay in memory.

Directory: mapping grid buckets to data blocks.

Each bucket corresponds to a disk block. But adjacentbuckets can share the same disk block.


Grid Files (contd)

For any exact-match query, only 2 disk accesses. One for directory, and the other for a data block.

Symmetric structure w.r.t. the index attributesallows efficient range queries w.r.t. all attributes.

Adapts to non-uniform data distributions. More cuts in scales in the dense data regions.

Typically cuts occur on pre-defined points in scales.

If an insertion causes overflow in a data block, If the block is associated with several buckets, split.

O/W, introduce a new split on a scale; Very expensive.

How about deletion?


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Grid File: example


Grid File: disadvantages

Split and Merge are quite complicated.

If attributes are correlated, large directory and

many empty buckets; A variant: Rotated grid file. The most serious problem with grid files is the

size of directory.

Directory growsfaster than lineareven for uniformlydistributed data set: O(n1+(d-1)/(dB+1)) [Regnier 1984].

The Curse of Dimensionality!!


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k-d-B tree [Robinson 1981]

Combines (adaptive) k-d tree and B-tree.

Dynamic, multi-way, balanced, fixed-size page.

Unlike B-tree, 50% utilization is not guaranteed.

Node structures

Internal node: a collection of .

Leaf node: .

Space decomposition

Space decomposed in a manner similar to k-d tree. Subspaces at the same level are mutually disjoint (no

overlap).

Union of subspaces constitutes the complete space of theirparent.


k-d-B tree: example


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k-d-B tree: Insertion

If a leaf node overflows, it is split; about half theentries are shifted to the new leaf node.

Various heuristics for optimal split.

Like B-tree, the split may be propagated up the tree.

However, unlike B-tree, split of an internal node mayaffect the space decomposition of its child nodes.

The hyperplane used for the new split will split all thedescendant nodes as well. Cascading splits!!

So, impossible to guarantee minimum storage utilization.


hB-tree [Lomet&Salzberg 1989]

A variant of the k-d-B tree.

Unlike k-d-B tree, split may be done by more thanone hyperplane, allowing L-shaped subspaces.

Structure Each node is a k-d tree, which represents a L-shaped or brick

with holes (holey brick).

The leaves of the k-d tree point to the lower-level nodes.

More than one pointer can reference the same node.

Split guarantees 1:2 data distribution in the worstcase.

Splits are localized; avoid cascading splits.


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hB-tree: example

The node u represents an L-shaped subspaceA+B+E+G. G in turn represents an L-shaped.

w u


Summary: Point Data

Data objects are points in a multidimensional space.

Popular queries are Exact-match (point) query,

Range (including partial-match) query, Nearest-neighbor query.

K-d trees and quad-trees are main memory indexes.

Grid file, k-d-B tree, hB-tree are for point data. Grid file: superlinear directory growth.

K-d-B tree: cascading splits.

hB-tree: 33% guaranteed utilization, duplicate references.


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The R-Tree [Guttman 1984]

Like B-tree, R-tree remains balanced on insertsand deletes, and guarantees 50% storageutilization.

Each key is a minimum bounding rectangle(MBR).

Example in 2-D:

X

Y

Root ofR Tree

Leaflevel


Example: 2-D View

R8R9

R10

R11

R12

R17

R18

R19

R13

R14

R15

R16

R1

R2

R3

R4

R5

R6

R7

Leaf entry

Index entry

Spatial objectapproximated bybounding box R8


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Example: Tree Structure

R1 R2

R3 R4 R5 R6 R7

R8 R9 R10 R11 R12 R13 R14 R15 R16 R17 R18 R19


R-Tree Properties

Internal node: a set of pairs

An MBR covers all the MBRs in subtree.

Leaf node: a set of pairs

MBR is the tightest bounding box for a data object.

Nodes can be kept 50% full (except root).

Root node has at least two entries.

Each node except root contains between m andMentries.The 50% utilization can be achieved by properly settingm.

MBRs (even at the same level) are allowed to overlap.


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Intersection (or Range) Query

Start at root.1. If current node is non-leaf, for each

entry , if boxE overlaps Q,search subtree identified by ptr.

2. If current node is leaf, for each entry, if E overlaps Q, rid identifies

an object that might overlap Q.

Note: May have to search severalseveral subtreessubtrees at each node!(In contrast, a B-tree equality search goes to just one leaf.)


Inserting an Entry

Algorithm ChooseLeaf Start at root and go down to best-fit leaf node L.

At each level, choose one among nodes whose MBR needs leastenlargement to cover the MBR of the new object.

Tie-breaking by going to a child node with smallest area.

Algorithm SplitNode If best-fit leaf L is already full, split L into L1 and L2.

Algorithm AdjustTree Adjust the MBR of L (or L1) in its parent while ascending to the root.

If L is split, add an entry for L2 to its parent. (This could cause theparent node to recursively split.)


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Splitting a Node During Insertion

The entries in node L plus the newly inserted entry mustbe distributed between L1 and L2.

Goal is to reduce likelihood of both L1 and L2 beingsearched on subsequent queries.

Two alternative goals to avoid multiple paths from theroot and to prune the search space:

Minimize coverage (area)

Minimize overlap

GOOD SPLIT!

BAD!


Guttmans Split Algorithm

Exhaustive : O(2M) to try all possible groupings.

Quadratic : O(M2*d)

First, find a pair (R,S) that maximizes

area(MBR(R,S)) - area(MBR(R) MBR(S)).

For the rest of entries,1) Choose an entry E such that |R S| is maximal, where x is the

area increase required in GroupR or GroupS to include the entry E.

2) Add E to a group so that coverage will be enlarged by the leastamount. Tie-breaking by adding to a group with smaller area,then with fewer entries.

3) Repeat this until all the entries are assigned to groups.


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Guttmans Split Algorithm

Linear : O(M*d)

First, find a pair (R,S) for each dimension that maximizesdisti(R, S). Then, pick one among the d pairs with themaximum normalized distance. disti(R,S) is the distance between R and S along i-axis.

Normalized distance is disti(R,S) / max_distancei.

Then, add entries to groups so that coverage will beenlarged by the least amount.


Deleting an Entry

Deletion consists of searching for the entry to bedeleted, removing it.

If the node becomes underflow (i.e., contains lessthen the minimum entries), re-insert theremaining entries.


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R*-tree [Beckmann et al. 1990]

Major insight : node splitting is critical for performance.

Basically, a set of optimizing techniques for R-tree.

Algorithm ChooseSubtree

If the current node is a leaf, done.

If the current node is a parent of leaf nodes, use minimum overlapenlargement for selecting a leaf node, with tie-breaking by least areaenlargement then smallest area. Complexity is O(M2). An O(MlogM)approximate algorithm was suggested.

If the current node is a parent of internal nodes, use minimum area

enlargement for selecting an internal node, with tie-breaking bysmallest area. [Same as Guttmans ChooseLeaf algorithm]

Split algorithm that considers area, margin and overlap. Based on plane-sweeping paradigm; O(d*MlogM).


R*-tree : Deferred Splitting

Algorithm Reinsert: a new idea Deferred Splitting

Dynamic reorganization byforced-reinsert; at the firsttime overflow at any level, reinsert p out of M+1

entries. The p entries whose centers are farthest from the

center of the enclosing MBR are chosen for reinsertion.

Significant performance improvement for point datasets.


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R*-tree : Node Splitting

Node splitting based on Plane-Sweeping:

(1) Sort MBRs by xlow coordinates (for each dimension).

(2) Generate (M-2m+2) 2-way partitions:k=1: m M-m+1k=2: m+1 M-m..k=M-2m+2: M-m+1 m

(3) For each dimension i, compute Si = margink, where margink is thesum of perimeters of two MBRs (by each 2-way partitioning).

(4) Choose a dimensionj, such that Sj is minimal.

(5) For the dimensionj, select one out of (M-2m+2) 2-way partitionssuch that overlap is minimized. Tie-break by the minimum area.


R*-tree : Performance Evaluation


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Summary on R-Trees

Overall, works quite well for low dimensional (d < 10)data sets; widely used.

Other variants: P-tree [Jagadish 1990]: use a convex polygon for MBR.

SR-tree [Katayama 1997]: use a spherical bounding region.

X-tree [Berchtold 1996]: proposed for high dimensional data sets.

Parallel R-tree [Kamel&Faloutsos 1992]: for multi-disk systems.

Hilbert R-tree [Kamel&Faloutsos 1994]: use the Hilbert space-filling curve for deferred splitting.

Packed R-tree [Rous. 1985; Kamel 1993]: to bulk-load static data.

Still, one of the hottest research tools!!!


Space-filling Curves: background

Historical background: [Peano 1890]: showed there exists a continuous and

surjective mapping, f: [0,1] [0,2]2.

[Hilbert 1891]: demonstrated its existence by ageometric construction.


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Space-filling Curves forMultidimensional Indexing

Main difficulty in multidimensional indexing: No total order that preserves spatial proximity.

If total order found, then one-dimensional access methods can beused. UB-Tree (Universal B-Tree): z-ordering with a variant of B+-Tree

Linear Mappingby a Space-Filling Curve First, partition the m-D space with a grid.

Map the coordinates of each grid cell into a unique number by anSFC.

A spatial object is represented by a list of grid cells intersected.

The way of mapping determines how closely adjacent cells (ordata in the cells) are stored in an index, or a file on disk. [dataclustering]


Z-Curve by Bit-Interleaving

Popular curves are: Z-curve (aka, Morton curve), Hilbert, Gray-coded.

Z-curve (aka Morton curve) Easy to implement, adopted by commercial DBMS. A single common prefix can be used to represent a set

of grid cells. [If z1 is the prefix of z2, then the region of z1encloses that of z2.]

Clustering not as good as Hilbert curve.


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Z-curve for spatial objects: example

Nine regionsapproximate thepolygon: {011100,011110,110100,01

001*,0110**,1100**,000111,001***,10010*}

01001*=010010+010011

0110**=011000+011001+011010+011011


Data Clustering by SFC

SFC preserves data locality while mapping.

Clusteris a set of consecutive grid points.

A 2x2 range query below retrieves 4 grid points in

two clusters, which may amount to two disk seeks.


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Clustering Properties of SFC

Comparative studies show SFCs have different clusteringproperties. [Jagadish 1990] For a 2x2 range query, the avg number of clusters is

2.65 (z-curve), 2.5 (Gray-coded), 2.0 (Hilbert curve).

[Rong 1991] For a 2-d rectangular query, the avg number of clustersby z-curve is (perimeter + 2*one-side)/3.

[Moon 1996] For a d-dimensional query region of any rectilinear

polyhedron, the avg number of clusters by Hilbert curve is Sq/2d,where Sq is the hyper-surface area of the query.

For a 2-d rectangular query, the avg number of clusters byHilbert curve isperimeter/4.

database design and implementation 07.multidim

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