1. Retrieving Data from Disk – Access Methods Zachary G. Ives University of Pennsylvania CIS 650 – Implementing Data Management Systems January 19, 2005 Slides on B+ and R Trees courtesy of Ramakrishnan & Gehrke

2. 2 Where We Are  We’ve now seen an overview of DBMSs, and how they largely look the same as in the mid-70s  Now:  Let’s drill down in more detail  Today: disk access  Why it’s a little different from OS access (Stonebraker)  General principles of indexing (Hellerstein)  Trivia question: where is GiST used today?

3. 3 Low-Level Disk Access  What are the central operations needed by a database management system in accessing pages?  Buffering  Efficiently scannable ranges, i.e. extents  Locking and concurrency  Flushing/forcing to disk  For managing multiple users?  Context switching  Shared memory  (Avoiding convoys)

4. 4 Search Trees  We are probably all familiar with the B+ Tree  To review from CIS 550…

5. B+ Tree: The DB World’s Favorite Index  Insert/delete at log F N cost  (F = fanout, N = # leaf pages)  Keep tree height-balanced  Minimum 50% occupancy (except for root).  Each node contains d <= m <= 2d entries. d is called the order of the tree.  Supports equality and range searches efficiently. Index Entries Data Entries ("Sequence set") (Direct search)

6. Example B+ Tree  Search begins at root, and key comparisons direct it to a leaf.  Search for 5*, 15*, all data entries >= 24*...  Based on the search for 15*, we know it is not in the tree! Root 17 24 30 2* 3*5* 7*14*16* 19*20* 22*24*27* 29*33*34* 38* 39* 13

7. B+ Trees in Practice  Typical order: 100. Typical fill-factor: 67%.  average fanout = 133  Typical capacities:  Height 4: 1334 = 312,900,700 records  Height 3: 1333 = 2,352,637 records  Can often hold top levels in buffer pool:  Level 1 = 1 page = 8 Kbytes  Level 2 = 133 pages = 1 Mbyte  Level 3 = 17,689 pages = 133 MBytes

8. Inserting Data into a B+ Tree  Find correct leaf L  Put data entry onto L  If L has enough space, done!  Else, must split L (into L and a new node L2)  Redistribute entries evenly, copy up middle key  Insert index entry pointing to L2 into parent of L  This can happen recursively  To split index node, redistribute entries evenly, but push up middle key. (Contrast with leaf splits.)  Splits “grow” tree; root split increases height.  Tree growth: gets wider or one level taller at top

9. Inserting 8* into Example B+ Tree  Observe how minimum occupancy is guaranteed in both leaf and index page splits  Recall that all data items are in leaves, and partition values for keys are in intermediate nodes Note difference between copy-up and push-up.

10. 10 Inserting 8* Example: Copy up Root 17 24 30 2* 3*5* 7*14*16* 19*20* 22*24*27* 29*33*34* 38* 39* 13 Want to insert here; no room, so split & copy up: 2* 3* 5* 7* 8* 5 Entry to be inserted in parent node. (Note that 5 is copied up and continues to appear in the leaf.) 8*

11. 11 Inserting 8* Example: Push up Root 17 24 30 2* 3* 14*16* 19*20* 22*24*27* 29*33*34* 38* 39* 13 5* 7* 8* 5 Need to split node & push up 5 2430 17 13 Entry to be inserted in parent node. (Note that 17 is pushed up and only appears once in the index. Contrast this with a leaf split.)

12. Deleting Data from a B+ Tree  Start at root, find leaf L where entry belongs  Remove the entry  If L is at least half-full, done!  If L has only d-1 entries,  Try to re-distribute, borrowing from sibling (adjacent node with same parent as L)  If re-distribution fails, merge L and sibling  If merge occurred, must delete entry (pointing to L or sibling) from parent of L  Merge could propagate to root, decreasing height

13. B+ Tree Summary B+ tree and other indices ideal for range searches, good for equality searches  Inserts/deletes leave tree height-balanced; log F N cost  High fanout (F) means depth rarely more than 3 or 4  Almost always better than maintaining a sorted file  Typically, 67% occupancy on average  Note: Order (d) concept replaced by physical space criterion in practice (“at least half-full”)  Records may be variable sized  Index pages typically hold more entries than leaves

14. 14 But What about Generalizing It?  What if I had:  Multidimensional data (find a point or a polygon within a region)  Set-oriented data (find an instance containing these values)  Property-oriented queries (as above)  Let’s start with an overview of R Trees before diving into GiST

15. 15 R Trees (Guttman 84)  “Region trees” – multidimensional indices  Any number of dimensions  Generally uses containment of bounding boxes, not full polygons, for efficiency  Must deal with two issues vs. a B+ Tree:  Is there a complete ordering?  Can items overlap without being identical?  Basic idea of the R Tree:  Create a tree based on containment of polygons/“rectangles” – child polygons are contained within parent polygons

16. The R-Tree  The R-tree is a tree-structured index that remains balanced on inserts and deletes.  Each key stored in a leaf entry is intuitively a box, or collection of intervals, with one interval per dimension.  Example in 2-D: X Y Root of R Tree Leaf level

17. R-Tree Properties  Leaf entry =  Box is the tightest bounding box for a data object.  Non-leaf entry =  Box covers all boxes in child node (in fact, subtree)  All leaves at same distance from root (height balanced)  Nodes can be kept 50% full (except root)  Can choose a parameter m that is <= 50%, and ensure that every node is at least m% full

18. Example of an R-Tree R8 R9 R10 R11 R12 R17 R18 R19 R13 R14 R15 R16 R1 R2 R3 R4 R5 R6 R7 Leaf entry Index entry Spatial object approximated by bounding box R8

19. Example R-Tree (Contd.) R1R2 R3R4R5R6R7 R8R9R10R11R12R13R14R15 R16 R17R18R19

20. Search for Objects Overlapping Box Q Start at root. 1. If current node is non-leaf, for each entry, if box E 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 several subtrees at each node! (In contrast, a B-tree equality search goes to just one leaf.)

21. Insert Entry  Start at root and go down to “best-fit” leaf L  Go to child whose box needs least enlargement to cover B; resolve ties by going to smallest area child  This corresponds to the “penalty” in GiST  If best-fit leaf L has space, insert entry and stop  Otherwise, split L into L1 and L2  Adjust entry for L in its parent so that the box now covers (only) L1  Add an entry (in the parent node of L) for L2  (This could cause the parent node to recursively split)

22. Splitting a Node During Insertion  The entries in node L plus the newly inserted entry must be distributed between L1 and L2  Goal is to reduce likelihood of both L1 and L2 being searched on subsequent queries  Redistribute to minimize area of L1 plus area of L2 Exhaustive algorithm is too slow; quadratic and linear heuristics are used GOOD SPLIT! BAD!

23. R-Tree Variants  The R* tree uses forced reinserts to reduce overlap in tree nodes  When a node overflows, instead of splitting:  Remove some (say, 30% of the) entries and reinsert them into the tree  Could result in all reinserted entries fitting on some existing pages, avoiding a split  R* trees also use a different heuristic, minimizing box perimeters rather than box areas during insertion  Another variant, the R+ tree, avoids overlap by inserting an object into multiple leaves if necessary  Searches now take a single path to a leaf, at cost of redundancy

24. 24 R Trees in Practice  One of many (dozens of!) multidimensional index structures  Perhaps the most popular, though not as ubiquitous as the B+ Tree  Implemented in a few DBMSs, such as Oracle, Illustra (Stonebraker startup, acquired by Informix, then IBM) and MySQL

25. 25 Generalizing B+ Trees and R Trees: GiST  Fundamental goal: can we create an index creation toolkit that can be customized for any kind of index we’d like?  Doesn’t quite get there – but a nice framework for understanding the concepts  Clearly, there seem to be some aspects of the B+ Tree and R Tree that can be drawn out:  Height balanced – requires re-organization  Data on leaves; intermediate nodes help focus on a child  High fan-out  Used in PostgreSQL, SHORE

26. 26 Similarities and Differences  Key differences between B+ Trees and R Trees:  B+ Tree:  one-dimensional  full ordering among data  R Tree:  many-dimensional  no complete ordering among the data  means that intermediate nodes’ children may fit in more than one place  also, intermediate node’s bounding box may have lots of space that’s not occupied by child nodes (relates to “curse of dimensionality”)  needs to rely on heuristics about where to insert a node  may need to search many possible paths along the tree

27. 27 Basic Operations that Need Customizing for Search  Contains(node, predicate)  Returns FALSE only if the node or its children can’t contain the predicate  Can return false positives  What does this correspond to in a B+ Tree? An R Tree?  What additional work needs to be done if contains() returns true?

28. 28 Insertion – Simple Case  Union(entrySet) returns predicate  Returns a predicate (e.g., bounding box) that holds over a set (e.g., of children) – basically a disjunction  Compress(entry) returns entry’  Simplifies the predicate of the entry, potentially increasing the chance of false positives  Decompress(entry) returns entry’  The inverse of the above – may have false positives  Penalty(entry1, entry2)  Gives the cost of inserting entry2 into entry1

29. 29 Insertion – Splitting  PickSplit(entrySet) returns  Splits the set of children in an intermediate node into two sets  Typically split according to a “badness metric”

30. 30 Basic Routines  How do we search for a node set satisfying a predicate?  Search(node, predicate)  For every nonleaf, if Consistent, call recursively on children; return union of result sets from children  For leaf, if Consistent, return singleton set, else empty set  Ordered domains: FindMin(root, predicate), Next(root, predicate, current)  Used to do a linear scan of the data at the leaves, if ordered

31. 31 Basic Routines II  How do we insert a node?  Insert(node, new, level)  L = ChooseSubtree(node, new, level)  If room in L, insert new as a child, else invoke Split(node, L, new)  AdjustKeys(node, L)  ChooseSubtree(node, new, level) returns node at level  Recursively descend tree, minimizing Penalty  If at desired level, return node  Among child entries in node, find one with minimal Penalty, return ChooseSubtree on that child

32. 32 Helper Functions  Split(root, node, new)  Invoke PickSplit on union of children of node and new  Put one partition into node and create a new node’ with the remainder  Insert all of node’ children into Parent(node) – if insufficient space, invoke PickSplit on this parent  Modify the predicate describing node  AdjustKeys(root, node)  If node = root, or predicate referring to node is correct, return  Otherwise,  modify predicate for node to contain the union of all keys of node  recursively call AdjustKeys(root, parent(node))

33. 33 Exercise  In groups of 2, describe how to implement the following for B+ Trees:  Contains()  Consistent()  Union()  PickSplit()  Assume that predicates in B+ Trees will be ranges  What are the major differences in the R Tree implementation?

34. 34 Next Time  Query execution principles and strategies  Should be largely review!  Sections 1-5, 7  No review required for this paper  For Wednesday:  Read Chaudhuri survey as an overview  Read and review Selinger et al. paper