1. Multiway Trees Searching and B-Trees Advanced Tree Structures
CIS265
Multiway Trees
Searching and B-Trees
Advanced Tree Structures
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2. Multiway Trees
We now look at trees that are not restricted to one entry per node
Used for internal and external indices and search trees, among other things
Multiway Trees

3. Multiway search trees Basic Theory
“A multi-way search tree of order n is a general tree in which each node refers to n or few sub-trees and contains one few keys than it has sub-trees.”
Pointer to Keys <
Key1
Key 1
Pointer
to keys
between
Key1 and Key2
Key 2
Key2 and Key3
Key 3
Pointer to
Keys > Key3
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4. Multiway search trees Basic Theory
A node in a serach tree with pointers to sub-trees below it
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5. Say That Again, Please… Each node has n children and n-1 keys
The keys in each node are in ascending order
The keys in the first i children are smaller than the ith key
The keys in the last n-i children are larger than the ith key
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6. How about an example…?
This is a 4-way tree. n=4
This node has 4 children, and 3 keys
58 59
100
Each node has n children and n-1 keys.
61 62
After that, it depends on your data. Of course, you also have to follow all the other rules. 57
55 56
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7. The keys in each node are in ascending order 52 54 61 62
58 59
100
The keys in each node are in ascending order
61 62
This holds true for each node. 57
55 56
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8. i=2. The first & second child all have keys smaller than this one.
i=3. The first three children all have keys smaller than this key.
i=1. The first child has keys all smaller than this key.
58 59
100
The keys in the first i children are smaller than the ith key.
61 62 57
55 56
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9. i=1. The last three children have keys all larger than this key.
i=2. The third & fourth children all have keys larger than this one.
i=3. The last child has keys larger than this key.
58 59
100
The keys in the last n-i children are larger than the ith key.
61 62 57
55 56
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10. Implementing a Multiway Tree
When nodes are not full, multiway search trees waste storage.
Used primarily to store data on external direct-access device such as a disk.
Also used as indices for non-ordered files stored on direct access devices.
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11. Implementing a Multiway Tree
Simple rule to improve operations:
Try & keep as many keys as possible in each node (faster to search)
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12. B-Trees Reduces the time required for accessing secondary storage
The nodes can become very large - typically the size of a block
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13. B-Trees
A B-tree of order n is a multi-way search tree with the following properties:
The root has at least one key.
Each non-root node holds (k-1) keys and k pointers to sub-trees where: n/2  k  n (ceiling function)
All leaves (terminal nodes) are at the same level.
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14. B-Trees According to these conditions, a B-Tree is
always at least half full,
has relatively few levels, and
‘perfectly’ balanced.
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15. B-Trees B3-Tree (non-root nodes at full and lowest capacity)
ptr1
key1
ptr2
key2
ptr3
ptr1
key1
ptr2
ptr1
key1
ptr2
key2
ptr3
key3
ptr4
key4
ptr5
ptr1
key1
ptr2
key2
ptr3
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16. B-Trees
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17. A B-Tree of Order 5 root Minimum Entries 42 16 21 58 76 81 93 11 14 17
The 5 Subtrees are not shown.. 11 14 17 19 20 21 22 23 24
Maximum Entries
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18. What’s inside the node? 42 A very simplistic drawing. Key Data K D K D
Num Entries K D K D
entry
Entry key <key type> data <data type> rightPtr <node pointer> end entry
node
node firstPtr <pointer to node> numEntries <integer> entries <array [1 .. M-1] of entry> end node
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19. Inserting a key in a B-Tree
Remember that all leaves in a B-Tree have to be at the same level
This can present several challenges
There are three cases we need to consider
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20. Case 1: B-tree Insertion
A key is placed in a leaf which still has some room. 12 12
13 15
need to preserve order, so move 8 down by 1, and insert the 7.
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21. Case 2: B-tree Insertion
The leaf in which a key should be placed is full.
In this case, the leaf is split, creating a new leaf, and half the keys are moved from the full leaf to the new leaf.
Then, the last key of the new left leaf is moved to the parent, and a pointers to the new left and right leaves is placed in the parent as well.
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22. 12
we want to insert a 6 in this leaf 12
split the leaf, and add the new value
2 5
the last key of the old leaf is moved to the parent and a pointer to the new leaf is added to the parent as well.
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23. Case 3: B-tree Insertion
A special case arises if the root of the B-tree is full.
In this case, a new root and a new sibling of the existing overfloding root is created.
This is the only case in which a B-tree increases in height.
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24.
we want to insert a 13
18 19
split the node as case #2
the last key of the old leaf is moved to the parent and a pointer to the new leaf is added to the parent as well.
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25.
18 19
13 14
split the root - as there will be more keys than slots 15
18 19
13 14
add a new root, increasing the tree’s depth
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26. B-Tree Insert
A B-Tree grows from the bottom up
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27. Deleting a Key from a B-Tree
Deleting a key is similar to inserting a key
More special cases
Merging rather than splitting nodes
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28. Deleting a node - Simple Case
If, after deleting a key K, the leaf is at least half full and only keys greater than K are moved to the left to fill the hole.
Opposite of insertion
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29. Deleting a node - Underflow Case
If after deleting K, the number of keys in the leaf is less than [m/2]-1, causing an underflow
If there is a left or right sibling with the number of keys exceeding the minimum ([m/2]-1), then all keys from this leaf and this sibling are redistributed between them by moving the separator key from the parent to the leaf and moving one key from the sibling to the parent
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30. Deleting a node - Underflow Case (Cont.)
If the leaf underflows and the number of siblings is [m/2]-1, then the leaf and a sibling are merged; the keys from the leaf, from its sibling, and the separating key from the parent are all put in the leaf, and the sibling node is discarded.
If we merge siblings, and the parent is the root with only one key, all the keys are combined in to a new root, and the old root and old nodes are discarded
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31. 16
1 2
18 20
23 24
27 37
Delete 6 16
1 2
5 7
18 20
23 24
27 37
Remove the 6, move the 7 over
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32. 16
1 2
5 7
18 20
23 24
27 37
Delete 7 16
1 2
5 8
18 20
23 24
27 37
14 15
Note the re-arrangement! Nodes can not be less than 50% full
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33. part 1. We merge 2 cells - deleting the empty one16
1 2
5 8
18 20
23 24
27 37
14 15
Delete 8 16 3
1 2
18 20
23 24
27 37
part 1. We merge 2 cells - deleting the empty one
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34. part 2. We merge 3 cells - deleting the empty ones16 3
22 25
1 2
18 20
23 24
27 37
continuing...
1 2
18 20
23 24
27 37
part 2. We merge 3 cells - deleting the empty ones
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35. We adjust keys so that the root remains full
1 2
18 20
23 24
27 37
Delete 16
1 2
18 20
23 24
27 37
We adjust keys so that the root remains full
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36. Another look at Deleting Nodes from a B-Tree
We will look at 5 more examples to make sure this is all clear.
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37. 21 42
Delete 11: We need to borrow from the right, and move the 21 down, and the 42 up. This insures all the B-Tree rules are still being followed. 45
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38. 78 74 45
Delete 97: We need to borrow from the left, and move the 78 down, and the 74 up. This insures all the B-Tree rules are still being followed. 45 63
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39. 45 63
Delete 45: We need to “combine” 63, 74, 78, 85. This insures all the B-Tree rules are still being followed. 42 63 74
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40. 42 63 74
Delete 63 & 78: Shift 74 and 85 down to positions 1 & 2 in the node. This insures all the B-Tree rules are still being followed. 42 74 85
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41. Step 1: Copy 21 to the parent and delete from the leaf42 74 85
Finally, we delete 42: We need to remove the original root, and combine all the remaining entries in to one node. This insures all the B-Tree rules are still being followed. 42 21 74 85 14 74 85
Step 1: Copy 21 to the parent and delete from the leaf
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42. Step 2: Combine 14, 21, 74, 85 and delete original root.
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43. B* Trees Variant of the B-Tree
Introduced by Donald Knuth, named by Douglas Comer
All nodes required to bit at least 2/3 full
Two nodes are split in to three, rather than one into two
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44. B+ Trees Enhancement of B-Trees
Allows all data to be accessed sequentially
References to data are made only from leaves
Internal nodes are indexed for faster access
Leaves are indexed sequentially
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45. B+ Trees Internal Nodes store keys, pointers and a key count
Leaves store keys, references to records in a data file associated with the keys, and a pointer to the next leaf
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46. index set leaves A B+ tree, order of 4 1 CD244 2 BF90 BQ322
2 CF DR300
index set
2 DR300 DR305
2 BQ322 CD123
2 BF90 BF130
3 CF04 CF DP102
3 AB203 AS09 BC26
2 CD244 CF03
leaves
A B+ tree, order of 4
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47. Inserting in to a B+ tree
Case 1: There is still room in the leaf:
The keys still must be in order
No changes are made to the index set
Case 2: The leaf is full
The leaf is split
The new leaf node is included in the sequence set
keys are distributed evenly
first key from the new node is copied (not moved) to the parent
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48. 29
part of a B+ tree 29
after inserting a note the copy of the new key in the index set
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49. Deleting from a B+Tree Case 1: No underflow after removal
Insure keys are still in order
No changes made to index set
Case 2: Removal causes underflow
Keys from leaf & keys from sibling are redistributed
-or-
leaf is deleted and its keys are added to sibling
Need to update parent
May require changes to index set
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50. Delete the 6 - note there is no change made to the index set!29
Delete the 6 - note there is no change made to the index set! 29
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51. 29
Delete the 2 - note we need to merge 2 leaves, and update the index set 29
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52. 2-3-4 Trees 27

53. Multiway Trees

54. Prefix B+ Trees
A prefix B+ tree is a B+ tree in which the chosen separators are the shortest prefixes that allow us to distinguish two neighboring index keys
Operations are generally the same as “standard” B+ trees
Not much difference in execution times between B+ and Prefix B+ trees
Largely of theoretical interest
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55. a B+ tree a Prefix B+ tree 1 CD244 2 BF90 BQ322 2 CF04 DR300
2 DR300 DR305
2 BQ322 CD123
2 BF90 BF130
3 CF04 CF05 DP102
3 AB203 AS09 BC26
2 CD244 CF03
a Prefix B+ tree
1 CD2
2 BF BQ
2 CF DR
2 DR300 DR305
2 BQ322 CD123
2 BF90 BF130
3 CF04 CF05 DP102
3 AB203 AS09 BC26
2 CD244 CF03
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56. Bit Trees Bit Trees take Prefix B+ Trees to the extreme
Rather than using bytes of data as separators, we use bits of data
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57. R-Trees R-Trees are used to store spatial data
The “R” is for rectangle
Based on B-trees
A leaf in an R-Tree contains entries of the form (rect, id) where rect = ([x0, y0], … [xn-1, yn-1]) and is an n-dimensional rectangle and id is a pointer to the data file
A non-leaf has the form (rect, id) where rect is the smallest rectangle encompassing all the rectangles found in child
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58. 2-4 Trees A special case of B Trees
Rather than storing an entire block of data, we store 1, 2, or at most 3 elements.
Wastes space
Transform to binary trees
Uses two types of links
links between nodes representing keys belong to same node
links representing regular parent-child links
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59. Digital Search Trees
Forms a general tree based on the symbols of which the keys are composed
Example: If keys are integers, each digit position determines one of ten possible sons of a given node
If the keys are alphabetic, each letter of the alphabet determines a branch in the tree
Every leaf contains the special symbol “eok” - signifying end of key
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60. Digital Search Trees Keys in this tree: 180 185 186 195 1 8 9 5 6 55 6 5
eok
eok
eok
eok
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