Indices in a DBMS.

Indices in a DBMS

Organization of Records in Files
Heap – a record can be placed anywhere in the file where there is space Sequential – store records in sequential order, based on the value of the search key of each record Hashing – a hash function computed on some attribute of each record; the result specifies in which block of the file the record should be placed Records of each relation may be stored in a separate file. In a clustering file organization records of several different relations can be stored in the same file Motivation: store related records on the same block to minimize I/O

Index Classification Primary vs. Secondary Clustered vs Unclustered
primary – the index on the primary key unique – an index on a candidate key secondary – not primary Clustered vs Unclustered clustered – key order corresponds with record order E.g. B-tree separate from record file Index-organized table  B-tree leaves store records (no file)‏ unclustered – index contains record-IDs in random order

B+Tree n=4 Root 100 120 150 180 30 3 5 11 120 130 180 200 30 35 100 101 110 150 156 179

Sample non-leaf 57 81 95 to keys to keys to keys to keys
<  k< k< 95

Sample leaf node: with key 57 with key 81 To record with key 85 57 81
From non-leaf node to next leaf in sequence 57 81 95 with key 57 with key 81 To record with key 85

Non-root nodes have to be at least half-full
Use at least Non-leaf: n/2 children Leaf: (n-1)/2 pointers to data

n=4 Full node min. node Non-leaf Leaf 120 150 180 30 3 5 11 30 35

(a) simple case (b) leaf overflow (c) non-leaf overflow (d) new root
Insert into B+tree (a) simple case space available in leaf (b) leaf overflow (c) non-leaf overflow (d) new root

(simple case) Insert key = 32
100 30 3 5 11 30 31 32

(leaf overflow) Insert key = 7
100 30 7 3 5 11 30 31 3 5 7

(internal overflow) Insert key = 160
100 160 120 150 180 180 150 156 179 180 200 160 179

(new root) insert 45 n=4 30 new root 10 20 30 40 1 2 3 10 12 20 25 30
32 40 40 45

n=4 insert: 1, 2, 10, 20, 3, 12, 30, 32, 25, 40, 45

(a) Simple case - no example (b) Coalesce with neighbor (sibling)‏
Deletion from B+tree (a) Simple case - no example (b) Coalesce with neighbor (sibling)‏ (c) Re-distribute keys (d) Cases (b) or (c) at non-leaf

(b) Coalesce with sibling
Delete 50 n=5 10 40 100 40 10 20 30 40 50

(c) Redistribute keys Delete 50 n=4 10 40 100 35 20 30 35 40 50

n=4 new root (d) Non-leaf coalesce Delete 37 25 25 20 30 25 26 10 14
22 30 37 30

B+tree deletions in practice
Often, coalescing is not implemented Too hard and not worth it!

n is number of keys / node
Interesting problem: For B+tree, how large should n be? … n is number of keys / node

Assumptions You have the right to set the disk page size for the disk where a B-tree will reside. Compute the optimum page size n assuming that The items are 4 bytes long and the pointers are also 4 bytes long. Time to read a node from disk is n Time to process a block in memory is unimportant B+tree is full (I.e., every page has the maximum number of items and pointers

 FIND nopt by f’(n) = 0 Disk bandwidth increases?
What happens to nopt as Disk bandwidth increases? Access time stays behind? CPU get faster?

f(n) = time to find a record = logn(T) * (10 + 0.0002n)

1994 (book)  2004 (now)‏ N=500  n=4000

1994 Table 1M records 10ms access time 4MB/s bandwidth n~ 4KB / 8KB pages Be conservative to limit RAM consumption

2004 Table 10M records 6ms access time 40MB/s bandwidth n~ 8KB / 32KB pages relative benefit decreases so don’t overdo it

Answer should be nopt = “few thousand”
 FIND nopt by f’(n) = 0 Answer should be nopt = “few thousand” What happens to nopt as  block sizes are increasing.. Disk bandwidth increases? Access time stays behind? CPU get faster?

Primary or Auxiliary Structure
Primary index Leaf blocks in sequence  clustered index Main storage structure for a database table E.g. B+-tree organized file / hash structured files Typically an index on an unique key But not necessarily Normally, you can have only one clustered index! Secondary index Also called unclustered index A separate file from where the table is stored Refers with (block/offset) pointers to records in the table file You can define many as you want (to maintain)‏

Clustered vs. Unclustered Index
Primary B-Tree index Primary index Leaf blocks in sequence  clustered index Main storage structure for a database table E.g. B+-tree organized file / hash structured files Typically an index on an unique key But not necessarily Normally, you can have only one clustered index! Secondary index Also called unclustered index A separate file from where the table is stored Refers with (block/offset) pointers to records in the table file You can define many as you want (to maintain)‏ low high 1 access only (rest is ‘just’ bandwidth)‏

Clustered vs. Unclustered Index
Primary B-Tree index Primary index Leaf blocks in sequence  clustered index Main storage structure for a database table E.g. B+-tree organized file / hash structured files Typically an index on an unique key But not necessarily Normally, you can have only one clustered index! Secondary index Also called unclustered index A separate file from where the table is stored Refers with (block/offset) pointers to records in the table file You can define many as you want (to maintain)‏ low high 1 access only (rest is ‘just’ bandwidth)‏ Pay N times access cost Secondary B-tree index

Are Unclustered Indices a Good Idea?
Secondary indices depend on random I/O can do asynchronous I/O (multiple I/Os at-a-time)‏ degenerates into full table scans

Block size for sequential reads?

When do random I/Os make sense?

Are Unclustered Indices a Good Idea?
Secondary indices depend on random I/O can do asynchronous I/O (multiple I/Os at-a-time)‏ degenerates into full table scans Is not using an index at all better? I.e. read the entire table sequentially without any index Use redundant clustered orderings Materialized views C-STORE (Stonebraker et al, VLDB 2005), MonetDB/X100 Database Cracking (Kersten, CIDR )‏

Geographic Data Raster data consist of bit maps or pixel maps, in two or more dimensions. Example 2-D raster image: satellite image of cloud cover, where each pixel stores the cloud visibility in a particular area. Additional dimensions might include the temperature at different altitudes at different regions, or measurements taken at different points in time. Design databases generally do not store raster data.

Geographic Data (Cont.)
Vector data are constructed from basic geometric objects: points, line segments, triangles, and other polygons in two dimensions, and cylinders, speheres, cuboids, and other polyhedrons in three dimensions. Vector format often used to represent map data. Roads can be considered as two-dimensional and represented by lines and curves. Some features, such as rivers, may be represented either as complex curves or as complex polygons, depending on whether their width is relevant. Features such as regions and lakes can be depicted as polygons.

Applications of Geographic Data
Examples of geographic data map data for vehicle navigation distribution network information for power, telephones, water supply, and sewage Vehicle navigation systems store information about roads and services for the use of drivers: Spatial data: e.g, road/restaurant/gas-station coordinates Non-spatial data: e.g., one-way streets, speed limits, traffic congestion Global Positioning System (GPS) unit - utilizes information broadcast from GPS satellites to find the current location of user with an accuracy of tens of meters. increasingly used in vehicle navigation systems as well as utility maintenance applications.

Spatial Access Develop a data structure to speedup spatial data access
Nearness queries request objects that lie near a specified location. Nearest neighbor queries, given a point or an object, find the nearest object that satisfies given conditions. Region queries deal with spatial regions. e.g., ask for objects that lie partially or fully inside a specified region. Queries that compute intersections or unions of regions. Spatial join of two spatial relations with the location playing the role of join attribute.

Indexing of Spatial Data
k-d tree - early structure used for indexing in multiple dimensions. Each level of a k-d tree partitions the space into two. choose one dimension for partitioning at the root level of the tree. choose another dimensions for partitioning in nodes at the next level and so on, cycling through the dimensions. In each node, approximately half of the points stored in the sub-tree fall on one side and half on the other. Partitioning stops when a node has less than a given maximum number of points. The k-d-B tree extends the k-d tree to allow multiple child nodes for each internal node; well-suited for secondary storage.

Division of Space by a k-d Tree
Each line in the figure (other than the outside box) corresponds to a node in the k-d tree the maximum number of points in a leaf node has been set to 1. The numbering of the lines in the figure indicates the level of the tree at which the corresponding node appears.

Division of Space by Quadtrees
Each node of a quadtree is associated with a rectangular region of space; the top node is associated with the entire target space. Each non-leaf nodes divides its region into four equal sized quadrants correspondingly each such node has four child nodes corresponding to the four quadrants and so on Leaf nodes have between zero and some fixed maximum number of points (set to 1 in example).

Quadtrees (Cont.) Region quadtrees store array (raster) information.
A node is a leaf node if all the array values in the region that it covers are the same. Otherwise, it is subdivided further into four children of equal area, and is therefore an internal node. Each node corresponds to a sub-array of values. The sub-arrays corresponding to leaves either contain just a single array element, or have multiple array elements, all of which have the same value. Extensions of k-d trees and quadtrees have been proposed to index line segments and polygons Require splitting segments/polygons into pieces at partitioning boundaries Same segment/polygon may be represented at several leaf nodes

R-Trees R-trees are a N-dimensional extension of B+-trees, useful for indexing sets of rectangles and other polygons. Supported in “many” modern database systems, along with variants like R+ -trees and R*-trees. Basic idea: generalize the notion of a one-dimensional interval associated with each B+ -tree node to an N-dimensional interval, that is, an N-dimensional rectangle. Will consider only the two-dimensional case (N = 2) generalization for N > 2 is straightforward, although R-trees work well only for relatively small N

R Trees (Cont.) A rectangular bounding box is associated with each tree node. Bounding box of a leaf node is a minimum sized rectangle that contains all the rectangles/polygons associated with the leaf node. The bounding box associated with a non-leaf node contains the bounding box associated with all its children. Bounding box of a node serves as its key in its parent node (if any) Bounding boxes of children of a node are allowed to overlap A polygon is stored only in one node, and the bounding box of the node must contain the polygon The storage efficiency or R-trees is better than that of k-d trees or quadtrees since a polygon is stored only once

Example R-Tree A set of rectangles (solid line) and the bounding boxes (dashed line) of the nodes of an R-tree for the rectangles. The R-tree is shown on the right.

Search in R-Trees To find data items (rectangles/polygons) intersecting (overlaps) a given query point/region, do the following, starting from the root node: If the node is a leaf node, output the data items whose keys intersect the given query point/region. Else, for each child of the current node whose bounding box overlaps the query point/region, recursively search the child Can be very inefficient in worst case since multiple paths may need to be searched but works acceptably in practice. Simple extensions of search procedure to handle predicates contained-in and contains

Insertion in R-Trees To insert a data item:
Find a leaf to store it, and add it to the leaf To find leaf, follow a child (if any) whose bounding box contains bounding box of data item, else child whose overlap with data item bounding box is maximum Handle overflows by splits (as in B+ -trees) Split procedure is different though (see below) Adjust bounding boxes starting from the leaf upwards Split procedure: Goal: divide entries of an overfull node into two sets such that the bounding boxes have minimum total area This is a heuristic. Alternatives like minimum overlap are possible Finding the “best” split is expensive, use heuristics instead See next slide

Splitting an R-Tree Node
Quadratic split divides the entries in a node into two new nodes as follows Find pair of entries with “maximum separation” that is, the pair such that the bounding box of the two would has the maximum wasted space (area of bounding box – sum of areas of two entries) Place these entries in two new nodes Repeatedly find the entry with “maximum preference” for one of the two new nodes, and assign the entry to that node Preference of an entry to a node is the increase in area of bounding box if the entry is added to the other node Stop when half the entries have been added to one node Then assign remaining entries to the other node Cheaper linear split heuristic works in time linear in number of entries, Cheaper but generates slightly worse splits.

Deleting in R-Trees Deletion of an entry in an R-tree done much like a B+-tree deletion. In case of underful node, borrow entries from a sibling if possible, else merging sibling nodes Alternative approach removes all entries from the underfull node, deletes the node, then reinserts all entries

Indices in database systems focus on:
Find a trusted fortune teller Indices in database systems focus on: All tuples are equally important for fast retrieval There are ample resources to maintain indices MonetDB cracks the database into pieces based on actual query load M.Kersten 2008

Cracking algorithms Physical reorganization happens per column based on selection predicates. Split a piece of a column in two new pieces A<10 A<10 A>=10 M.Kersten 2008

Cracking algorithms Physical reorganization happens per column
Split a piece of a column in two new pieces Split a piece of a column in three new pieces A<5 A<10 A<10 5<A<10 5<A<10 A>=10 A>=10 M.Kersten 2008

Cracking example 17 3 8 6 2 12 13 4 15 select A>5 and A<10
M.Kersten 2008

Cracking example select A>5 and A<10 17 17 3 3 8 8 6 6 2 2 15 15 13 13 4 4 12 12 M.Kersten 2008

Cracking example >=10 >=10 17 17 3 3 8 8 6 6 2 2 15 15 13 13 4 4
select A>5 and A<10 >=10 17 17 3 3 8 8 6 6 2 2 15 15 13 13 4 4 12 12 >=10 M.Kersten 2008

Cracking example >=10 17 17 3 3 8 8 6 6 2 2 15 15 13 13 4 4 12 12
select A>5 and A<10 >=10 17 17 3 3 8 8 6 6 2 2 15 15 13 13 4 4 12 12 M.Kersten 2008

Cracking example >=10 <=5 17 17 3 3 8 8 6 6 2 2 15 15 13 13 4 4
select A>5 and A<10 >=10 17 17 3 3 8 8 6 6 2 2 15 15 13 13 4 4 <=5 12 12 M.Kersten 2008

select A>5 and A<10 17 4 3 3 8 8 6 6 2 2 15 15 13 13 >=10 4 17 12 12 M.Kersten 2008

select A>5 and A<10 17 4 3 3 8 8 6 6 2 2 15 15 >=10 13 13 4 17 12 12 M.Kersten 2008

Cracking example <=5 17 4 3 3 8 8 6 6 2 2 15 15 13 13 4 17 12 12
select A>5 and A<10 17 4 3 3 8 8 6 6 2 2 <=5 15 15 13 13 4 17 12 12 M.Kersten 2008

Cracking example <=5 <=5 17 4 3 3 8 8 6 6 2 2 15 15 13 13 4 17
select A>5 and A<10 17 4 3 3 <=5 8 8 6 6 2 2 <=5 15 15 13 13 4 17 12 12 M.Kersten 2008

Cracking example >5 and <10 <=5 17 4 3 3 8 8 6 6 2 2 15 15 13
select A>5 and A<10 17 4 3 3 >5 and <10 8 8 6 6 2 2 <=5 15 15 13 13 4 17 12 12 M.Kersten 2008

Cracking example >5 and <10 17 4 3 3 8 2 6 6 2 8 15 15 13 13 4
select A>5 and A<10 17 4 3 3 8 2 6 6 >5 and <10 2 8 15 15 13 13 4 17 12 12 M.Kersten 2008

Cracking example select A>5 and A<10 17 4 3 3 <= 5 8 2 6 6 > 5 2 8 15 15 13 13 >= 10 4 17 12 12 M.Kersten 2008

Improve data access for future queries
Cracking example Improve data access for future queries select A>5 and A<10 17 4 3 3 <= 5 8 2 6 6 > 5 2 8 15 15 13 13 >= 10 4 17 12 12 M.Kersten 2008

Cracking example Improve data access for future queries select A>5 and A<10 select A>3 and A<14 17 4 3 3 <= 5 8 2 6 6 > 5 2 8 15 15 13 13 >= 10 4 17 12 12 M.Kersten 2008

Cracking example Improve data access for future queries select A>5 and A<10 select A>3 and A<14 17 4 4 3 3 <= 5 3 <= 5 8 2 2 6 6 6 > 5 > 5 2 8 8 15 15 15 13 13 13 >= 10 >= 10 4 17 17 12 12 12 M.Kersten 2008

racking example Improve data access for future queries select A>5 and A<10 select A>3 and A<14 17 4 4 >3 and <14 3 3 <= 5 3 <= 5 8 2 2 <=3 6 6 6 > 5 > 5 2 8 8 15 15 15 13 13 13 >= 10 >= 10 4 17 17 12 12 12 M.Kersten 2008

Cracking example Improve data access for future queries select A>5 and A<10 select A>3 and A<14 17 4 4 >3 and <14 3 3 <= 5 2 3 <= 5 8 2 <=3 6 6 6 > 5 > 5 2 8 8 15 15 15 13 13 13 >= 10 >= 10 4 17 17 12 12 12 M.Kersten 2008

Cracking example Improve data access for future queries select A>5 and A<10 select A>3 and A<14 17 4 >3 and <14 3 3 <= 5 2 3 <= 5 8 2 4 <=3 6 6 6 > 5 > 5 2 8 8 15 15 15 13 13 13 >= 10 >= 10 4 17 17 12 12 12 M.Kersten 2008

Cracking example Improve data access for future queries select A>5 and A<10 select A>3 and A<14 17 4 2 >3 and <14 3 3 <= 5 3 <= 5 8 2 4 <=3 6 6 6 > 5 > 5 2 8 8 15 15 15 13 13 13 >= 10 >= 10 4 17 17 12 12 12 M.Kersten 2008

Cracking example Improve data access for future queries select A>5 and A<10 select A>3 and A<14 17 4 2 3 3 <= 5 3 <= 5 <=3 8 2 4 6 6 6 > 5 > 5 2 8 8 15 15 15 13 13 13 >= 10 >= 10 4 17 17 12 12 12 M.Kersten 2008

Cracking example Improve data access for future queries select A>5 and A<10 select A>3 and A<14 17 4 2 <=3 3 3 <= 5 3 8 2 4 > 3 6 6 6 > 5 > 5 2 8 8 15 15 15 13 13 13 >= 10 >= 10 4 17 17 12 12 12 M.Kersten 2008

Cracking example Improve data access for future queries select A>5 and A<10 select A>3 and A<14 17 4 2 <=3 3 3 <= 5 3 8 2 4 > 3 6 6 6 > 5 > 5 2 8 8 15 15 15 13 13 13 12 >= 10 >= 10 4 17 17 12 12 M.Kersten 2008

Cracking example Improve data access for future queries select A>5 and A<10 select A>3 and A<14 17 4 2 <=3 3 3 <= 5 3 8 2 4 > 3 6 6 6 > 5 > 5 2 8 8 15 15 12 >=10 13 13 13 >= 10 4 17 17 >= 14 12 12 15 M.Kersten 2008

Cracking example >=10 Improve data access for future queries
The more we crack the more we learn select A>5 and A<10 select A>3 and A<14 17 4 2 <=3 3 3 <= 5 3 8 2 4 >3 6 6 6 > 5 > 5 2 8 8 15 15 12 >=10 13 13 13 >= 10 4 17 17 >= 14 12 12 15 M.Kersten 2008

Design The first time a range query is posed on an attribute A, a cracking DBMS makes a copy of column A, called the cracker column of A A cracker column is continuously physically reorganized based on queries that need to touch attribute such as the result is in a contiguous space For each cracker column, there is a cracker index Cracker Index Cracker Column M.Kersten 2008

Try to avoid useless investments
A simple range query M.Kersten 2008

Try to avoid useless investments
TPC-H query 6 M.Kersten 2008

Cracking works under high volume updates
Try to avoid useless investments Cracking is easy in a column store and is part of the critical execution path Cracking works under high volume updates M.Kersten 2008

Updates Base columns are updated as normally
We need to update the cracker column and the cracker index Efficiently Maintain the self-organization properties Two issues: When How M.Kersten 2008

When to propagate updates in cracking
Follow the workload to maintain self-organization Updates become part of query processing When an update arrives, it is not applied For each cracker column there is a pending insertions column and a pending deletions column Pending updates are applied only when a query needs the specific values M.Kersten 2008

Updates aware select We extended the cracker select operator to apply the needed updates before cracking The select operator: Search the pending insertions column Search the pending deletions column If Steps 1 or 2 find tuples run an update algorithm Search the cracker index Physically reorganize the cracker column Update the cracker index Return a slice of the cracker column M.Kersten 2008

Merging Insert a new tuple with value 9
The new tuple belongs to the blue piece 7 Start position: 1 values: >1 9 2 10 29 Start position: 4 values: >12 25 31 57 Start position: 7 values: >35 42 53 M.Kersten 2008

Merging Insert a new tuple with value 9
The new tuple belongs to the blue piece 7 2 Start position: 1 values: >1 10 9 29 Pieces in the cracker column are ordered 25 Start position: 5 values: >12 31 Tuples inside a piece are not ordered 57 42 Start position: 8 values: >35 Shifting is not a viable solution 53 M.Kersten 2008

Merging by Hopping We need to make enough room to fit the new tuples 9
Insert a new tuple with value 9 7 We need to make enough room to fit the new tuples Start position: 1 values: >1 2 10 29 Start position: 4 values: >12 25 31 42 Start position: 8 values: >35 53 57 M.Kersten 2008

Merge Gradually A query merges only the qualifying values, i.e., only the values that it needs for a correct and complete result Merge Gradually Merge Completely We avoid the large peaks but... Average cost increases significantly M.Kersten 2008

The Ripple Touch only the pieces that are relevant for the current query M.Kersten 2008

The Ripple Touch only the pieces that are relevant for the current query 7 Start position: 1 values: >1 2 10 29 Start position: 4 values: >22 25 31 57 42 Start position: 7 values: >35 53 M.Kersten 2008

The Ripple Touch only the pieces that are relevant for the current query Select 7<= A< 15 7 Start position: 1 values: >1 2 10 29 Start position: 4 values: >22 25 31 57 42 Start position: 7 values: >35 53 M.Kersten 2008

The Ripple Touch only the pieces that are relevant for the current query Select 7<= A< 15 7 Pending insertions Start position: 1 values: >1 2 5 10 9 29 16 Start position: 4 values: >22 25 35 31 57 42 Start position: 7 values: >35 53 M.Kersten 2008

The Ripple Touch only the pieces that are relevant for the current query Select 7<= A< 15 7 Pending insertions Start position: 1 values: >1 2 5 10 9 29 16 Start position: 4 values: >22 25 35 31 57 Avoid shifting down non interesting pieces 42 Start position: 7 values: >35 53 M.Kersten 2008

The Ripple Touch only the pieces that are relevant for the current query Select 7<= A< 15 7 Pending insertions Start position: 1 values: >1 2 5 10 9 29 16 Start position: 4 values: >22 25 35 31 57 Avoid shifting down non interesting pieces 42 Start position: 7 values: >35 53 Immediately make room for the new tuples M.Kersten 2008

The Ripple Touch only the pieces that are relevant for the current query Select 7<= A< 15 7 Pending insertions Start position: 1 values: >1 2 5 10 16 9 29 25 Start position: 5 values: >22 35 31 57 Avoid shifting down non interesting pieces 42 Start position: 7 values: >35 53 Immediately make room for the new tuples M.Kersten 2008

The Ripple Touch only the pieces that are relevant for the current query Select 7<= A< 15 7 Pending insertions 2 Start position: 1 values: >1 5 10 16 9 29 25 Start position: 5 values: >22 35 31 57 Avoid shifting down non interesting pieces 42 Start position: 7 values: >35 53 Immediately make room for the new tuples M.Kersten 2008

The Ripple Maintain high performance through the whole query
sequence in a self-organizing way M.Kersten 2008

The Ripple Merge Ripple Maintain high performance
through the whole query sequence in a self-organizing way Merge Gradually Merge Completely M.Kersten 2008

MonetDB Research & Development line
XQuery and Information Retrieval Astronomy databases (SkyServer) Event Stream Engine Relational Cache Effectiveness Multi-core Parallelism DataStorage Rings RDF Engine …. Array Databases become en vogue M.Kersten 2008

Indices in a DBMS.

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