Mark Allen Weiss: Data Structures and Algorithm Analysis in Java
Chapter 4: Trees
Part II - AVL Tree
Better Binary Search Trees
Prevent the degeneration of the BST :
• A BST can be set up to maintain balance during updating
operations (insertions and removals)
• Types of BST which maintain the optimal performance:
–
–
–
–
splay trees
AVL trees
Red-Black trees
B-trees
AVL Trees
• Balanced binary search
trees
• An AVL Tree is a binary
search tree such that for
every internal node v of T,
the heights of the children
of v can differ by at most 1.
4
44
2
3
17
78
1
2
32
1
88
50
1
48
62
1
Height of an AVL Tree
•
•
Proposition: The height of an AVL tree T storing n keys is O(log n).
Justification: The easiest way to approach this problem is to find n(h): the
minimum number of internal nodes of an AVL tree of height h.
•
•
n(1) = 1 and n(2) = 2
for n ≥ 3, an AVL tree of height h contains the root node, one AVL subtree of
height n-1 and the other AVL subtree of height n-2.
 n(h) = 1 + n(h-1) + n(h-2)
given n(h-1) > n(h-2)  n(h) > 2n(h-2)
n(h) > 2n(h-2)
n(h) > 4n(h-4)
…
n(h) > 2in(h-2i)
pick i = h/2 – 1  n(h) ≥ 2 h/2-1
follow h < 2log n(h) +2
 height of an AVL tree is O(log n)
•
•
•
•
•
Insertion
• A binary search tree T is called balanced if for every node v, the
height of v’s children differ by at most one.
• Inserting a node into an AVL tree involves performing an
expandExternal(w) on T, which changes the heights of some of
the nodes in T.
• If an insertion causes T to become unbalanced, we travel up the
tree from the newly created node until we find the first node x
such that its grandparent z is unbalanced node.
• Since z became unbalanced by an insertion in the subtree rooted
at its child y, height(y) = height(sibling(y)) + 2
• Need to rebalance...
Insertion: Rebalancing
• To rebalance the subtree rooted at z, we must
perform a restructuring
• we rename x, y, and z to a, b, and c based on the
order of the nodes in an in-order traversal.
• z is replaced by b, whose children are now a and c
whose children, in turn, consist of the four other
subtrees formerly children of x, y, and z.
Insertion (cont’d)
unbalanced...
5
44
z
2
17
3
1
32
1
1
50
2
1
7
78
2 y
1
48
64
3
4
62
88
x
5
T3
54
T0
...balanced
T2
44
4
3
2
17
2
1
32
1
1
48
62
2y
50
4 x
3
1
z 6
5
78
54
2
7
88
T2
T0
T1
T3
1
Restructuring
• The four ways to rotate nodes in an AVL tree, graphically represented
-Single Rotations:
a=z
b=y
single rotation
b=y
a=z
c=x
c=x
T0
T3
T1
T3
T2
T0
T1
T2
Restructuring (cont’d)
• double rotations:
double rotation
a=z
c= y
b=x
a=z
c= y
b=x
T0
T3
T2
T0
T2
T1
T3
T1
double rotation
c=z
a=y
b=x
a=y
c=z
b=x
T3
T0
T2
T1
T3
T2
T1
T0
Restructure Algorithm
Algorithm restructure(x):
Input: A node x of a binary search tree T that has both a parent y and a
grandparent z
Output: Tree T restructured by a rotation (either
single or double) involving nodes x, y, and z.
1: Let (a, b, c) be an inorder listing of the nodes x, y, and z, and let (T0, T1, T2, T3)
be an inorder listing of the the four subtrees of x, y, and z, not rooted at x, y, or
z.
2. Replace the subtree rooted at z with a new subtree rooted at b
3. Let a be the left child of b and let T0, T1 be the left and right subtrees of a,
respectively.
4. Let c be the right child of b and let T2, T3 be the left and right subtrees of c,
respectively.
Cut/Link Restructure Algorithm
• Let’s go into a little more detail on this algorithm...
• Any tree that needs to be balanced can be grouped into 7 parts: x, y, z, and
the 4 trees anchored at the children of those nodes (T0-3)
44
y
17
62
50
T0
48
x
78
54
88
T2
T1
T3
Cut/Link Restructure Algorithm
44
y
17
62
50
T0
48
x
78
54
88
T2
T1
T3
• Make a new tree which is balanced and put the 7 parts from the old tree
into the new tree so that the numbering is still correct when we do an
in-order-traversal of the new tree.
• This works regardless of how the tree is originally unbalanced.
• Let’s see how it works!
Cut/Link Restructure Algorithm
• Number the 7 parts by doing an in-order-traversal. (note that x,y, and z are now
renamed based upon their order within the traversal)
1
2z (a)
44
y (b)
17
4
62
3
x (c)
50
T0
78
54
48
5
6
88
T2
T1
T3
7
Cut/Link Restructure Algorithm
• Now create an Array, numbered 1 to 7 (the 0th element can be ignored with
minimal waste of space)
1
2
3
4
5
6
7
•Cut() the 4 T trees and place them in their inorder rank in the array
T0
1
T1
2
3
4
T2
5
6
T3
7
Cut/Link Restructure Algorithm
• Now cut x,y, and z in that order (child,parent,grandparent) and
place them in their inorder rank in the array.
T0
a
44
T1
62
b T2
78
1
2
3
4
5
6
7
•Now we can re-link these subtrees to the main tree.
•Link in rank 4 (b) where the subtree’s root formerly
4
b
62
c T
3
Cut/Link Restructure Algorithm
• Link in ranks 2 (a) and 6 (c) as 4’s children.
4b
62
a
2 44
c
78
6
Cut/Link Restructure Algorithm
• Finally, link in ranks 1,3,5, and 7 as the children of 2 and 6.
4y
62
2 44z
17
3
x
78
5
54
Now you have a balanced tree!
T2
T3
T0
•
88
50
48
6
T1
7
Cut/Link Restructure Algorithm
• This algorithm for restructuring has the exact same effect
as using the four rotation cases discussed earlier.
• Advantages: no case analysis, more elegant
• Disadvantage: can be more code to write
• Same time complexity
Removal
• We can easily see that performing a removeAboveExternal(w) can
cause T to become unbalanced.
• Let z be the first unbalanced node encountered while traveling up
the tree from w. Also, let y be the child of z with the larger height,
and let x be the child of y with the larger height.
• We can perform operation restructure(x) to restore balance at the
subtree rooted at z.
• As this restructuring may upset the balance of another node
higher in the tree, we must continue checking for balance until the
root of T is reached
Removal (cont’d)
• example of deletion from an AVL tree:
4
44
y
1
17
62
3
x
2
50
1
1
T0
48
78
0
1
54
88
T2
32
T
3
1
y
4
62
z
x
44
78
2
1
17
2
0
1
50
1
88
1
48
T0
2
54
T2
T3
Removal (cont’d)
example of deletion from an AVL tree:
z
44
4
y
1
17
62
x
2
50
1
1
T0
48
3
2
78
0
1
54
88
32
T1
T3
T2
x
4
50
2
y
z
62
44 1
1
1
17
3
48
54
0
78
2
1
88
T0
T1
T2
B-Trees
• Up to now, all data that has been stored in the tree has been
in memory.
• If data gets too big for main memory, what do we do?
• If we keep a pointer to the tree in main memory, we could
bring in just the nodes that we need.
• For instance, to do an insert with a BST, if we need the left
child, we do a disk access and retrieve the left child.
• If the left child is NIL, then we can do the insert, and store the
child node on the disk.
• Not too good for a BST
B-Trees
• The problem with BST: storing the data requires disk accesses,
which is expensive, compared to execution of machine
instructions.
• If we can reduce the number of disk accesses, then the
procedures run faster.
• The only way to reduce the number of disk accesses is to
increase the number of keys in a node.
• The BST allows only one key per leaf.
• Very good and often used for Search Engines!
– (when collection size gets very big  the index does not fit in
memory)
B-Trees
• If we increase the number of keys in the nodes, how will we
do any tree operations effectively?
10 20 30 40 50 60 70
• Above is a node with 7 keys. How do we add children?
10 20 30 40 50 60 70
123456789
11 22 33 44 55 66 77
B-Trees
• Clearly, the tree below is useless.
10 20 30 40 50 60 70
123456789
•
•
11 22 33 44 55 66 77
How many pointers do we need?
Using the idea of BST, we need to be able to put nodes into the tree
that have smaller, same and larger values than the node we are
currently examining.
B-Trees: A General Case of Multi-Way
Search Trees
10
4
8
6
30
7
8
9
40
50
60
70
1
2
•
•
We can easily find any value.
We need to create operations, which require rules on what makes a tree a
B-Tree.
Clearly, having one key per node would be very bad.
We need a mechanism to increase the height of the tree (since the number
of keys in any node can get very high) so we can shift keys out of a node,
making the nodes smaller.
•
•
3
20
11
22
33
44
55
66
77
B-Trees: Fields in a Node
A B-Tree is a rooted tree (whose root is root[T]))
having the following properties:
1. Every internal node x has the following fields:
n[x]
c1[x]
key1[x key2[x key3[x key4[x key5[x key6[x key7[x key8[x leaf[x
]
]
]
]
]
]
]
]
]
c2[x]
c3[x]
c4[x]
c5[x]
c6[x]
c7[x]
c8[x]
c9[x]
n[x] is the number of keys in the node. n[x] = 8 above.
leaf[x] = false for internal nodes, since x is not a leaf.
The keyi[x] are the values of the keys, where keyi[x]  keyi+1[x].
ci[x] are pointers to child nodes. All the keys in ci[x] have values that are between
keyi-1[x] and keyi[x].
B-Trees
• Leaf nodes have no child pointers
• leaf[x] = true for leaf nodes.
• All leaf nodes are at the same level
n[x]
key1[x key2[x key3[x key4[x key5[x key6[x key7[x key8[x leaf[x
]
]
]
]
]
]
]
]
]
B-Trees
•
There are lower and upper bounds on the number of keys a
node can contain. This depends on the “minimum degree” t
 2, which we must specify for any given B-Tree.
a. Every node other than the root must have at least t-1 keys.
Every internal node other than the root thus has t children. If
the tree is nonempty, the root must have at least one key.
b. Every node can contain at most 2t-1 keys. Therefore, an
internal node can have at most 2t children. A node is full if it
contains exactly 2t-1 keys.
Height of B-Tree
• If n  1, then for any n-key B-tree of height h and mimimum
degree t  2,
height = h  logt[(n+1)/2]
• The important thing to notice is that the height of the tree is log
base t. So, as t increases, the height, for any number of nodes n,
will decrease.
• Using the formula logax = (logbx)/(logba), we can see that
– log2106 = (log10106)/(log102)  6/0.30102999566398  19
– log10106 = 6
So, 13 less disk accesses to get to the leafs!
Basic Operations
• The root of the B-tree is always in main
memory, so that a Disk-Read on the root is
never required; a Disk-Write of the root is
required, however, whenever the root node is
changed.
• Any nodes that are passed as parameters must
already have had a Disk-Read operation
performed on them.
Searching a B-Tree
Start at the leftmost key
in the node, and go to the
right until you go too far.
If it is a leaf node, then you
are done, as there is no leaf
to inspect
Otherwise, retrieve the child node
from the disk, and put it into
memory
Inserting into B-trees
• Really very easy. Very similar with (2,4) trees.
• Just keep in mind that you are starting at the
root, and then finding the subtree where the key
should be inserted, and following the pointer.
• A deletion may eventually occur, and sometimes
deletions force keys into their parents. So, if we
encounter a full node on our way to the node
where the insertion will take place, we must split
that node into two.
Inserting into B-trees (cont’d)
B-Tree-Insert
If the node has
2t-1 keys, it can’t
accept any more
keys, so you need
to split it into
2 nodes before
doing the insert.
Otherwise, call Nonfull()
Deleting Keys from Nodes
Deleting Keys from Nodes