ITCS 6163

Lecture 5

Indexing datacubes

Objective: speed queries up.

Traditional databases (OLTP): B-Trees

• Time and space logarithmic to the amount of indexed keys.

• Dynamic, stable and exhibit good performance under updates. (But OLAP is not about updates….)

Bitmaps:

• Space efficient

• Difficult to update (but we don’t care in DW).

• Can effectively prune searches before looking at data.

BitmapsR = (…., A,….., M)

R (A) B8 B7 B6 B5 B4 B3 B2 B1 B0

3 0 0 0 0 0 1 0 0 0 2 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 2 0 0 0 0 0 0 1 0 0 8 1 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 1 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 7 0 1 0 0 0 0 0 0 0 5 0 0 0 1 0 0 0 0 0 6 0 0 1 0 0 0 0 0 0 4 0 0 0 0 1 0 0 0 0

Query optimization

Consider a high-selectivity-factor query with predicates on two attributes.

Query optimizer: builds plans(P1) Full relation scan (filter as you go).(P2) Index scan on the predicate with lower selectivity

factor, followed by temporary relation scan, to filter out non-qualifying tuples, using the other predicate. (Works well if data is clustered on the first index key).

(P3) Index scan for each predicate (separately), followed by merge of RID.

Query optimization (continued)

(P2)

Blocks of data

Pred. 2

answer

t1

tn

Index Pred1

(P3)

t1

tn

Index Pred2

Tuple list1

Tuple list2

Merged list

Query optimization (continued)

When using bitmap indexes (P3) can be an easy winner!

CPU operations in bitmaps (AND, OR, XOR, etc.) are more efficient than regular RID merges: just apply the binary operations to the bitmaps

(In B-trees, you would have to scan the two lists and select tuples in both -- merge operation--)

Of course, you can build B-trees on the compound key, butwe would need one for every compound predicate (exponential number of trees…).

Bitmaps and predicates

A = a1 AND B = b2

Bitmap for a1 Bitmap for b2

AND =

Bitmap for a1 and b2

Tradeoffs

Dimension cardinality small dense bitmaps

Dimension cardinality large sparse bitmaps

Compression

(decompression)

Bitmap for prod

Bitmap for prod

…..

Query strategy for Star joinsMaintain join indexes between fact table and dimension tables

Prod.

Fact tableProduct Type Location

Dimension table

a ... k

…

…

Bitmap for type a

Bitmap for type k

…..Bitmap for loc.

Bitmap for loc.

…..

Strategy exampleAggregate all sales for products of location , or

Bitmap for Bitmap for Bitmap for

OR OR =

Bitmap for predicate

Star-Joins

Select F.S, D1.A1, D2.A2, …. Dn.An

from F,D1,D2,Dn where F.A1 = D1.A1

F.A2 = D2.A2 … F.An = Dn.An

and D1.B1 = ‘c1’ D2.B2 = ‘p2’ ….

Likely strategy:

For each Di find suitable values of Ai such that Di.Bi = ‘xi’ (unless you have a bitmap index for Bi). Use bitmap index on Ai’ values to form a bitmap for related rows of F (OR-ing the bitmaps).

At this stage, you have n such bitmaps, the result can be found AND-ing them.

Example

Selectivity/predicate = 0.01 (predicates on the dimension tables) n predicates (statistically independent)Total selectivity = 10 -2n

Facts table = 108 rows, n = 3, tuples in answer = 108/ 106 = 100 rows. In the worst case = 100 blocks… Still better than all the blocks in the relation (e.g., assuming 100 tuples/block, this would be 106 blocks!)

Design Space of Bitmap Indexes

The basic bitmap design is called Value-list index. The focus there is on the columns. If we change the focus to the rows, the index becomes a set of attribute values (integers) in each tuple (row), that can be represented in a particular way.

5 0 0 0 1 0 0 0 0 0

We can encode this row in many ways...

Attribute value decompositionC = attribute cardinality Consider a value of the attribute, v, and a sequence of numbers <bn-1, bn-2 , …,b1>. Also, define bn = C / bi , then v can be decomposed into a sequence of n digits <vn, vn-1, vn-2 , …,v1> as follows:

v = V1

= V2 b1 + v1

= V3(b2b1) + v2 b1 + v1

… n-1 i-1 = vn ( bj) + …+ vi ( bj) + …+ v2b1 + v1

where vi = Vi mod bi and Vi = Vi-1/bi-1

<10,10,10> (decimal system!)

576 = 5 x 10 x 10 + 7 x 10 + 6

576/100 = 5 | 76

76/10 = 7 | 6

6

Number systems

How do you write 576 in:

<2,2,2,2,2,2,2,2,2>

576 = 1 x 29 + 0 x 28 + 0 x 27 + 1 x 26 + 0 x 25 + 0 x 24 + 0 x 23 +

0 x 22+ 0 x 21 + 0 x 20

576/ 29 = 1 | 64, 64/ 28 = 0|64, 64/ 27 = 0|64, 64/ 26 = 1|0,

0/ 25 = 0|0, 0/ 24= 0|0, 0/ 23= 0|0, 0/ 22 = 0|0, 0/ 21 = 0|0, 0/

20 = 0|0

< 7,7,5,3>

576/(7x7x5x3) = 576/735 = 0 | 576, 576/(7x5x3)=576/105=5|51

576 = 5 x (7x5x3)+51

51/(5x3) = 51/15 = 3 | 6

576 = 5 x (7x5x3) + 3 (5 x 3) + 16

6/3 =2 | 0

576 = 5 x (7x 5 x 3) + 3 x (5 x 3 ) + 2 x (3)

BitmapsR = (…., A,….., M) value-list index

R (A) B8 B7 B6 B5 B4 B3 B2 B1 B0

3 0 0 0 0 0 1 0 0 0 2 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 2 0 0 0 0 0 0 1 0 0 8 1 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 1 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 7 0 1 0 0 0 0 0 0 0 5 0 0 0 1 0 0 0 0 0 6 0 0 1 0 0 0 0 0 0 4 0 0 0 0 1 0 0 0 0

Examplesequence <3,3> value-list index (equality)

R (A) B22

B12

B02 B2

1 B11 B0

1

3 (1x3+0) 0 1 0 0 0 1 2 0 0 1 1 0 0 1 0 0 1 0 1 0 2 0 0 1 1 0 0 8 1 0 0 1 0 0 2 0 0 1 1 0 0 2 0 0 1 1 0 0 0 0 0 1 0 0 1 7 1 0 0 0 1 0 5 0 1 0 1 0 0 6 1 0 0 0 0 1 4 0 1 0 0 1 0

Encoding scheme

Equality encoding: all bits to 0 except the one that corresponds to the value

Range Encoding: the vi righmost bits to 0, the remaining to 1

Range encodingsingle component, base-9

R (A) B8 B7 B6 B5 B4 B3 B2 B1 B0

3 1 1 1 1 1 1 0 0 0 2 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 8 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 7 1 1 0 0 0 0 0 0 0 5 1 1 1 1 0 0 0 0 0 6 1 1 1 0 0 0 0 0 0 4 1 1 1 1 1 0 0 0 0

Example (revisited)sequence <3,3> value-list index(Equality)

R (A) B22

B12

B02 B2

1 B11 B0

1

3 (1x3+0) 0 1 0 0 0 1 2 0 0 1 1 0 0 1 0 0 1 0 1 0 2 0 0 1 1 0 0 8 1 0 0 1 0 0 2 0 0 1 1 0 0 2 0 0 1 1 0 0 0 0 0 1 0 0 1 7 1 0 0 0 1 0 5 0 1 0 1 0 0 6 1 0 0 0 0 1 4 0 1 0 0 1 0

Examplesequence <3,3> range-encoded index

R (A) B12

B02 B1

1 B01

3 1 0 1 1 2 1 1 0 0 1 1 1 1 0 2 1 1 0 0 8 0 0 0 0 2 1 1 0 0 2 1 1 0 0 0 1 1 1 1 7 0 0 1 0 5 1 0 0 0 6 0 0 1 1 4 1 0 1 0

Design Space

b Value-list

log2C b,b,…,b

Bit-Sliced

<b2,b1>

….

equality range

RangeEval

Evaluates each range predicate by computing two bitmaps: BEQ bitmap and either BGT or BLT

RangeEval-Opt uses only <=

A < v is the same as A <= v-1

A > v is the same as Not( A <= v)

A >= v is the same as Not (A <= v-1)

RangeEval-OPT