1. Fractal Prefetching B + -Trees: Optimizing Both Cache and Disk Performance Author: Shimin Chen, Phillip B. Gibbons, Todd C. Mowry, Gary Valentin Members: Iris Zhang, Grace Yung, Kara Kwon, Jessica Wong

2. Outline 1.Introduction 2.Optimizing I/O Performance a.Searches b.Range Scans 3.Optimizing Cache Performance a.Disk-First fpB + -Trees b.Cache-First fpB + -Trees 4.Conclusion

3. Introduction Traditional B + -Trees –Optimized for I/O performance –tree nodes = disk pages Recent new types of B + -Trees –Optimized for CPU cache performance –tree nodes sizes = one or few cache lines –Introduce concept of prefetching

4. Introduction (cont’d) Figure 1: Traditional B + -Trees Page Control Info Index entry (key and page/tuple ID)

5. Introduction (cont’d) Problem (due to large discrepancy in optimal node sizes) 1.Disk-optimized B + -Trees suffer from poor cache performance 2.Cache-optimized B + -Trees suffer from poor disk performance

6. Introduction (cont’d) Proposal: Fractal Prefetching B + -Trees (fpB + -Trees) 1.Embed “cache-optimized” trees within “disk- optimized” trees 2.Optimize both cache and I/O performance 3.Two approaches: -> disk-first -> cache-first

7. Introduction (cont’d) Figure 2: Self-similar “tree within a tree” structure

8. Introduction (cont’d) Disk-first and Cache-first What is done to optimize performance How to process operations efficiently –Bulkload –Search –Insertion –Deletion

9. Optimizing I/O Performance fpB + -Trees combine features of disk- and cache-optimized B + -Trees to achieve best of both structures Consider two concepts from pB + -Trees –Searches: Prefetching and node sizes –Range Scans: Prefetching via jump-pointer arrays

10. Optimizing I/O Performance (cont’d) Prefetching: –Modern db servers are composed of multiple disks per processor –Goal: effectively exploit I/O parallelism Explicitly prefetching disk pages even when the access patterns are not sequential

11. Searches: Prefetching and Node Sizes (cont’d) For disk-resident data –Increase the B + -Tree node size to be a multiple of the disk page size –Prefetch all pages of a node when accessing it Pages are placed on different disks so that requests can be serviced in parallel Result: faster search

12. Searches: Prefetching and Node Sizes (cont’d) Problem –I/O latency improves for a single search, but may become worse when there are extra seeks for a node –Additional seeks may degrade performance Conclusion: target node-size for fpB + -Tree will be a single disk page

13. Range Scans: Prefetching via Jump-Pointer Arrays Range scan –searching for the starting key of the range, then reading consecutive leaf nodes in the tree Jump-pointer array helps leaves to be effectively prefetched One implementation: add sibling pointers to each node that is a parent of leaves

14. Range Scans: Prefetching via Jump-Pointer Arrays (cont’d) Figure 3: Internal jump-pointer array Tree Leaf Parent

15. Range Scans: Prefetching via Jump-Pointer Arrays (cont’d) This technique can be applied to fpB + - Tree Enhancement to avoid overshooting: –fpB + -Trees begin by searching for both start and end key in order to remember the range end page –This technique does not decrease throughput

16. Optimizing Cache Performance The search operation of B + -Trees suffers poor cache performance –During a search, each page on the path to a key is visited –In each page, binary search is performed on the large continuous array –Costly in terms of cache misses

17. Optimizing Cache Performance (cont’d) Example: –Key, page ID and tuple ID are all 4 bytes –With a 8KB page, can hold over 1000 entries –Cache line is 64 bytes => hold 8 entries –Suppose page has 1023 entries (1 to 1023) –Locate a matching entry 71, requires 10 probes with binary search 512, 256, 128, 64, 96, 80, 72, 68, 70, 71

18. Optimizing Cache Performance (cont’d) The update operation of B+-Trees is costly –Insertion and deletion both begin with search –To insert an entry in a sorted array, on average half of the page must be copied to make room for the new entry

19. Disk-First fpB + -Trees Start with disk-optimized B + -Trees Organize keys and pointers in each page- sized node into a cache-optimized tree In each node - small cache-optimized tree: in-page tree –Modeled after pB + -Trees, which is shown to have best cache performance

20. Disk-First fpB + -Trees (cont’d) Figure 4: Disk-optimized fpB + -Trees : a cache-optimized tree inside each page page control info

21. Disk-First fpB + -Trees (cont’d) In-page tree has nodes aligned on cache line boundaries Each node is several cache lines wide –When a node is visited as part of a search, all cache lines in the node are prefetched Increases fan-out of the node and reduce height of the in-page tree Result: better overall performance

22. Disk-First fpB + -Trees (cont’d) Non-leaf nodes –Contains pointers to other in-page nodes within the same page –To further pack more entries into each node, use short in-page offsets instead of full pointers Leaf nodes –Contains pointers to nodes external to their in-page tree

23. Disk-First fpB + -Trees (cont’d) Optimal in-page node size is determined by memory system parameters and key and pointer sizes Optimal page size is determined by I/O parameters and disk and memory prices With a mismatch between the two sizes, tree may have overflow or underflow

24. Disk-First fpB + -Trees (cont’d) page control info Unused Space Figure 5: Overflow and Underflow

25. Disk-First fpB + -Trees (cont’d) page control info Figure 6: Fitting cache-optimized trees in a page - use smaller nodes when overflow - use larger nodes when underflow

26. Disk-First fpB + -Trees: Operations Bulkload: operations at two granularities –At a page granularity: follow common B + - Tree bulkload algorithm –For in-page trees of non-leaf pages, pack entries into one in-page leaf node after another –For in-page trees of leaf pages, try to distribute entries across all in-page leaf nodes Maintain a linked list of all in-page leaf nodes

27. Disk-First fpB + -Trees: Operations (cont’d) Search –Straightforward search done for each granularity

28. Disk-First fpB + -Trees: Operations (cont’d) Insertion: operations at two granularities –If there are empty slots in the in-page leaf node, insert the entry into the sorted array for the node

29. Disk-First fpB + -Trees: Operations (cont’d) Insertion: operations at two granularities –Otherwise, split the leaf node into two a.Allocate new nodes in the same page b.Reorganize in-page tree if number of entries is fewer than page maximum fan-out c.Split the page by copying half of the in-page leaf nodes to a new page, and rebuild the two in-page trees in their respective pages

30. Disk-First fpB + -Trees: Operations (cont’d) Deletion –A search for the entry –Follow by a lazy deletion of entry in a leaf node –Do not merge leaf nodes that become half empty

31. Cache-First fpB + -Trees Start with cache-optimized B + -Trees Ignore page boundaries Then try to intelligently place cache- optimized nodes into disk pages

32. Cache-First fpB + -Trees (cont’d) Non-leaf node –Contains an array of keys and pointers –A pointer is a combination of a page ID and an offset in the page Use the page ID to retrieve a disk page Visit a node in the page by the offset Leaf node –Contains an array of keys and tuple ids

33. Cache-First fpB + -Trees: Node Placement Goal 1: group sibling leaf nodes together into the same page to reduce disk operations during range scans Approach: designate certain pages as leaf pages that contain only leaf nodes –Leaf nodes in the same page are siblings

34. Cache-First fpB + -Trees: Node Placement (cont’d) Goal 2: group a parent node and its children together into the same page to ensure searches only need one disk operation for a parent and its child Problems: –Not possible for all nodes –Node size mismatch (overflow and underflow)

35. Cache-First fpB + -Trees: Node Placement (cont’d) For underflow (i.e. “not enough” children) –Place grandchildren, great grandchildren, etc onto the same page For overflow: two approaches a.Place overflowed child into its own page as top-level node with its own children b.Store overflowed child in special overflow pages

36. Cache-First fpB + -Trees: Node Placement (cont’d) Figure 8: Cache-first fpB + -Tree design Nonleaf nodes Aggressive Placement Overflow pages for leaf node parents

37. Cache-First fpB + -Trees: Operations Bulkload: Leaf nodes –Placed consecutively in leaf pages, and linked together with sibling links

38. Cache-First fpB + -Trees: Operations Bulkload: Non-leaf nodes –Determine whether there is space for the node to fit into the same page as its parent –If not, then Allocate the node as the top level node in a new page, or If the non-leaf node is a parent of a leaf node, place it into the overflow page

39. Cache-First fpB + -Trees: Operations (cont’d) Search –Straightforward with one thing to note –When proceeding from a parent to one of its children, compare the page ID –Same page ID indicates parent and child are on the same page Can directly access the node in the page without retrieving the page from buffer manager

40. Cache-First fpB + -Trees: Operations (cont’d) Insertion: –If there are empty slots in the leaf node, simply insert the entry; else need to split node into two –If leaf page has space, accommodate the new node; else need to split the leaf page Move second half of the leaf nodes to a new page Update corresponding child pointers in their parents

41. Cache-First fpB + -Trees: Operations (cont’d) Insertion: –After leaf node split, need to insert an entry into the parent node –If parent node is full, it needs to be split For leaf parent node, the new node may be allocated from overflow pages If further splits up the tree are needed, the new node must be allocated as described in bulkload

42. Cache-First fpB + -Trees: Operations (cont’d) Deletion –Similar to disk-first fpB + -Trees

43. Conclusion 1.Problems of traditional B + -Trees 2.In optimizing I/O performance, considered two concepts from pB + -Trees: searches and range scans 3.How disk-first and cache-first fpB + -Trees performances better traditional B + -Trees 4.Operations (bulkload, search, insertion, deletion)