Optimizing Multidimensional Index Trees for Main Memory

Access

Author: Kihong Kim, Sang K. Cha, Keunjoo Kwon

Members: Iris Zhang, Grace Yung, Kara Kwon, Jessica Wong

Outline

1. Abstraction

2. Introduction

3. Motivation

4. MBR Compression

5. CR-tree

6. Analysis

7. Conclusion

Abstraction

• CR-tree – Cash Conscious version of the R-tree

• Compress MBR key – Remove reading and trailing less significant bits

• CR-tree becomes wider and smaller – Faster searching, less memory consuming than

ordinary R-tree

Introduction

• DB tables and indexes in Main Memory

• How to search faster in R-tree ???– MBR key compression

Motivation1. Memory Hierarchy

-> Reduce Cache miss !!!

Motivation (Cont’d)2. Basic Idea

- Compression scheme

Motivation (Cont’d)

2. Basic Idea (Cont’d)• Quantize into 16 levels or 4

bits by cutting off trailing insignificant bits

• the Result MBR -> QRMBR

• CR- tree use QRMBR as index key

Motivation (Cont’d)2. Basic Idea (Cont’d)

• Power of Quantize

Original coordinates

(43166,27102),(43178,27190)

After Quantize

=>(8,11),(14,15)

8bytes of entry size become 2bytes

=>each node can pack 4 times more entry!!!

Motivation (Cont’d)

2. Basic Idea (Cont’d)

-Structure of CR-tree node

Motivation (Cont’d)

3. Problem Formulation (reduce index search time)

c->node size, Nnode access-> # of node access

Tindex search = c · Nnode access · (Ckey compare + Ccache miss + CTLB miss / c)

Index search time mostly depends on c · Nnode ccess

Motivation (Cont’d)

3. Problem Formulation (reduce index search time)

How to minimize c · Nnode access- ??

• Change node size c · Nnode access- become minimal

• Packing more entries into a fixed-size node• Clustering index entries

MBR Compression

Two Desirable properties of MBR Compression

• Overlap Check without Decompression

• Simplicity

RMBR( Relative Representation of MBR)

- Represent the coordinates of an MBR relatively to the left corner of its parents MBR

- Cut off leading non-discriminating bits

- Can save only 32 bits per MBR

QRMBR (Quantized Relative Representation of MBR)

• Cannot obtain a sufficient compression ratio form the RMBR

• Cut off trailing insignificant bits form an RMBR

• There’s overhead but it’s paid off by the significant savings in cache misses

QRMBR (Cont’d)Correctness

• If two MBRs overlap, the resulting QRMBR must overlap also

• Two non-overlapping MBRs may overlap

CR-tree Operations-Search

• Query rectangle need to change to QRMBR using MBR of each node as the reference MBR

• Compare Query QRMBR and object QRMBR whether they overlap

CR-tree Operations-Search

MBR of R3 Entry of R4

Entry of R5

MBR of R0 Entry of R1

Entry of R2

Entry of R3

Node of R0 level

Node of R3 level

R4R5

{(43166,27102),(43178,27112)}

CR-tree Operations-Insertion

• Choosing child node that needs the least enlargement to enclose the object MBR

• In Internal node, Object MBR is first transformed into the QRMBR using the reference MBR

• Enlargement calculated between a pair of QRMBR

• In leaf node, node MBR is first adjusted such that it encloses the object MBR

• Index entry for the object is created in the node

• QRMBR in the node are recalculated because their reference MBR has changed

CR-tree Operations-Insertion

R4R5

R6 MBR of R3 Entry of R4

Entry of R6

Node of R3 level

Entry of R5

1.Change!!! 2. Insert !!!

3. Recalculate QRMBR

CR-tree Operations-Insertion(Cont’d)

Algorithm SplitNode..

Algorithm AdjustTree. Ascend from a leaf node L up to the root, adjusting MBRs of nodes and propagating node splits as necessary. When a node MBR has been adjusted, recalculate the QRMBRs in the node.

CR-tree Operations-Deletion

Algorithm Delete. Remove index record E from a CR-tree

Algorithm Condense Tree.

- Eliminate the node if it has too few entries and relocate its entries .

- Adjust all MBRs, making them smaller if possible.

- Recalculate the QRMBR in the node.

CR-tree Operations-Bulkload

• Similar with other R-tree variants

Conclusion• How to optimize cache behavior of indexes

in main memory DB environment?• QRMBR – Pack more entries in the node• CR-tree based on QRMBR increases the

fanout of the R-tree and decreases the index size for cache behavior

• 2.5 times search faster and 60% less memory use than R-tree

References

• Jun Rao, Kenneth A. Ross: Making B+-Trees Cache Conscious in Main Memory, ACM SIGMOD 2000

• Shimin Chen, Phillip B. Gibbons, Todd C. Mowry, Gary Valentin: Fractal Prefetching B+trees: Optimizing Both Cache and Disk Performance. ACM SIGMOD 2002

• Kihong Kim, Sang K. Cha, Keunjoo Kwon: Optimizing Multidimensional Index Trees for Main Memory Access. ACM SIGMOD 2001

Q&A