module 4: dictionaries and balanced search treesdroche/teaching/240slides/module04.pdf · reza...
TRANSCRIPT
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Module 4: Dictionaries and Balanced Search Trees
CS 240 - Data Structures and Data Management
Reza Dorrigiv, Daniel Roche
School of Computer Science, University of Waterloo
Winter 2010
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 1 / 29
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Dictionary ADT
A dictionary is a collection of items,each of which contains a key and some dataand is called a key-value pair (KVP).Keys can be compared and are typically unique.
Operations:
search(k)
insert(k , v)
delete(k)
optional: join, isEmpty, size, etc.
Examples: symbol table, license plate database
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 2 / 29
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Elementary Implementations
Common assumptions:
Dictionary has n KVPs
Each KVP uses constant space(if not, the “value” could be a pointer)
Comparing keys takes constant time
Unordered array or linked list
search Θ(n)
insert Θ(1)
delete Θ(1) (after a search)
Ordered array or linked list
search Θ(log n)
insert Θ(n)
delete Θ(n)
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 3 / 29
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Binary Search Trees (review)
Structure A BST is either empty or contains a KVP,left child BST, and right child BST.
Ordering Every key k in T .left is less than the root key.Every key k in T .right is greater than the root key.
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Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 4 / 29
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BST Search and Insert
search(k) Compare k to current node, stop if found,else recurse on subtree unless it’s empty
insert(k , v) Search for k , then insert (k, v) as new node
Example: search(24)
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Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 5 / 29
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BST Search and Insert
search(k) Compare k to current node, stop if found,else recurse on subtree unless it’s empty
insert(k , v) Search for k , then insert (k, v) as new node
Example: search(24)
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Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 5 / 29
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BST Search and Insert
search(k) Compare k to current node, stop if found,else recurse on subtree unless it’s empty
insert(k , v) Search for k , then insert (k, v) as new node
Example: search(24)
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Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 5 / 29
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BST Search and Insert
search(k) Compare k to current node, stop if found,else recurse on subtree unless it’s empty
insert(k , v) Search for k , then insert (k, v) as new node
Example: search(24)
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Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 5 / 29
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BST Search and Insert
search(k) Compare k to current node, stop if found,else recurse on subtree unless it’s empty
insert(k , v) Search for k , then insert (k, v) as new node
Example: insert(24, . . .)
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Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 5 / 29
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BST Delete
If node is a leaf, just delete it.
If node has one child, move child up
Else, swap with successor node and then delete
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Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 6 / 29
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BST Delete
If node is a leaf, just delete it.
If node has one child, move child up
Else, swap with successor node and then delete
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Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 6 / 29
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BST Delete
If node is a leaf, just delete it.
If node has one child, move child up
Else, swap with successor node and then delete
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Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 6 / 29
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BST Delete
If node is a leaf, just delete it.
If node has one child, move child up
Else, swap with successor node and then delete
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Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 6 / 29
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BST Delete
If node is a leaf, just delete it.
If node has one child, move child up
Else, swap with successor node and then delete
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Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 6 / 29
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BST Delete
If node is a leaf, just delete it.
If node has one child, move child up
Else, swap with successor node and then delete
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Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 6 / 29
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BST Delete
If node is a leaf, just delete it.
If node has one child, move child up
Else, swap with successor node and then delete
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Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 6 / 29
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Height of a BST
search, insert, delete all have cost Θ(h), whereh = height of the tree = max. path length from root to leaf
If n items are inserted one-at-a-time, how big is h?
Worst-case:
n − 1 = Θ(n)
Best-case: lg(n + 1)− 1 = Θ(log n)
Average-case: Θ(log n)(just like recursion depth in quick-sort1)
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 7 / 29
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Height of a BST
search, insert, delete all have cost Θ(h), whereh = height of the tree = max. path length from root to leaf
If n items are inserted one-at-a-time, how big is h?
Worst-case: n − 1 = Θ(n)
Best-case:
lg(n + 1)− 1 = Θ(log n)
Average-case: Θ(log n)(just like recursion depth in quick-sort1)
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 7 / 29
![Page 19: Module 4: Dictionaries and Balanced Search Treesdroche/teaching/240slides/module04.pdf · Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 3 / 29. Binary Search Trees](https://reader036.vdocuments.us/reader036/viewer/2022070923/5fbbe8f4fa74c772324ea402/html5/thumbnails/19.jpg)
Height of a BST
search, insert, delete all have cost Θ(h), whereh = height of the tree = max. path length from root to leaf
If n items are inserted one-at-a-time, how big is h?
Worst-case: n − 1 = Θ(n)
Best-case: lg(n + 1)− 1 = Θ(log n)
Average-case:
Θ(log n)(just like recursion depth in quick-sort1)
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 7 / 29
![Page 20: Module 4: Dictionaries and Balanced Search Treesdroche/teaching/240slides/module04.pdf · Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 3 / 29. Binary Search Trees](https://reader036.vdocuments.us/reader036/viewer/2022070923/5fbbe8f4fa74c772324ea402/html5/thumbnails/20.jpg)
Height of a BST
search, insert, delete all have cost Θ(h), whereh = height of the tree = max. path length from root to leaf
If n items are inserted one-at-a-time, how big is h?
Worst-case: n − 1 = Θ(n)
Best-case: lg(n + 1)− 1 = Θ(log n)
Average-case: Θ(log n)(just like recursion depth in quick-sort1)
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 7 / 29
![Page 21: Module 4: Dictionaries and Balanced Search Treesdroche/teaching/240slides/module04.pdf · Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 3 / 29. Binary Search Trees](https://reader036.vdocuments.us/reader036/viewer/2022070923/5fbbe8f4fa74c772324ea402/html5/thumbnails/21.jpg)
AVL Trees
Introduced by Adel’son-Vel’skiı and Landis in 1962,an AVL Tree is a BST with an additional structural property:The heights of the left and right subtree differ by at most 1.
(The height of an empty tree is defined to be −1.)
At each non-empty node, we store height(R)− height(L) ∈ {−1, 0, 1}:−1 means the tree is left-heavy
0 means the tree is balanced
1 means the tree is right-heavy
Why not just store the actual height?It would take Θ(n log log n) space.
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 8 / 29
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AVL Trees
Introduced by Adel’son-Vel’skiı and Landis in 1962,an AVL Tree is a BST with an additional structural property:The heights of the left and right subtree differ by at most 1.
(The height of an empty tree is defined to be −1.)
At each non-empty node, we store height(R)− height(L) ∈ {−1, 0, 1}:−1 means the tree is left-heavy
0 means the tree is balanced
1 means the tree is right-heavy
Why not just store the actual height?
It would take Θ(n log log n) space.
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 8 / 29
![Page 23: Module 4: Dictionaries and Balanced Search Treesdroche/teaching/240slides/module04.pdf · Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 3 / 29. Binary Search Trees](https://reader036.vdocuments.us/reader036/viewer/2022070923/5fbbe8f4fa74c772324ea402/html5/thumbnails/23.jpg)
AVL Trees
Introduced by Adel’son-Vel’skiı and Landis in 1962,an AVL Tree is a BST with an additional structural property:The heights of the left and right subtree differ by at most 1.
(The height of an empty tree is defined to be −1.)
At each non-empty node, we store height(R)− height(L) ∈ {−1, 0, 1}:−1 means the tree is left-heavy
0 means the tree is balanced
1 means the tree is right-heavy
Why not just store the actual height?It would take Θ(n log log n) space.
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 8 / 29
![Page 24: Module 4: Dictionaries and Balanced Search Treesdroche/teaching/240slides/module04.pdf · Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 3 / 29. Binary Search Trees](https://reader036.vdocuments.us/reader036/viewer/2022070923/5fbbe8f4fa74c772324ea402/html5/thumbnails/24.jpg)
AVL insertion
To perform insert(T , k, v):
First, insert (k , v) into T using usual BST insertion
Then, move up the tree from the new leaf, updating balance factors.
If the balance factor is −1, 0, or 1, then keep going.
If the balance factor is ±2, then call the fix algorithmto “rebalance” at that node.
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 9 / 29
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How to “fix” an unbalanced AVL tree
Goal: change the structure without changing the order
y
x
A B
z
C D
Notice that if heights of A, B, C , D differ by at most 1,then the tree is a proper AVL tree.
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 10 / 29
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Right Rotation
This is a right rotation on node z :
z
y
x
A B
C
D
y
x
A B
z
C D
Note: Only two edges need to be moved, and two balances updated.
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 11 / 29
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Right Rotation
This is a right rotation on node z :
z
y
x
A B
C
D
y
x
A B
z
C D
Note: Only two edges need to be moved, and two balances updated.
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 11 / 29
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Left Rotation
This is a left rotation on node x :
x
A
y
B
z
C D
y
x
A B
z
C D
Again, only two edges need to be moved and two balances updated.
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 12 / 29
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Double Right Rotation
This is a double right rotation on node z :
z
x
A
y
B C
D
z
y
x
A B
C
D
y
x
A B
z
C D
First, a left rotation on the left subtree (x).
Second, a right rotation on the whole tree (z).
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 13 / 29
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Double Right Rotation
This is a double right rotation on node z :
z
x
A
y
B C
D
z
y
x
A B
C
D
y
x
A B
z
C D
First, a left rotation on the left subtree (x).Second, a right rotation on the whole tree (z).
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 13 / 29
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Double Left Rotation
This is a double left rotation on node x :
x
A
z
y
B C
D
y
x
A B
z
C D
Right rotation on right subtree (z),followed by left rotation on the whole tree (x).
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 14 / 29
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Fixing a slightly-unbalanced AVL tree
Idea: Identify one of the previous 4 situations, apply rotations
fix(T )T : AVL tree with T .balance = ±21. if T .balance = −2 then2. if T .left.balance = 1 then3. rotate-left(T .left)4. rotate-right(T )5. else if T .balance = 2 then6. if T .right.balance = −1 then7. rotate-right(T .right)8. rotate-left(T )
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 15 / 29
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AVL Tree Operations
search: Just like in BSTs, costs Θ(height)
insert: Shown already, total cost Θ(height)fix will be called at most once.
delete: First search, then swap with successor (as with BSTs),then move up the tree and apply fix (as with insert).fix may be called Θ(height) times.Total cost is Θ(height).
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 16 / 29
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AVL tree examples
Example: insert(8) 22
-1
10
1
4
1
6
0
14
1
13
0
18
-1
16
0
31
1
28
0
37
1
46
0
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 17 / 29
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AVL tree examples
Example: insert(8) 22
-1
10
1
4
1
6
0
8
0
14
1
13
0
18
-1
16
0
31
1
28
0
37
1
46
0
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 17 / 29
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AVL tree examples
Example: insert(8) 22
-1
10
1
4
1
6
1
8
0
14
1
13
0
18
-1
16
0
31
1
28
0
37
1
46
0
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 17 / 29
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AVL tree examples
Example: insert(8) 22
-1
10
1
4
2
6
1
8
0
14
1
13
0
18
-1
16
0
31
1
28
0
37
1
46
0
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 17 / 29
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AVL tree examples
Example: insert(8) 22
-1
10
1
6
0
4
0
8
0
14
1
13
0
18
-1
16
0
31
1
28
0
37
1
46
0
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 17 / 29
![Page 39: Module 4: Dictionaries and Balanced Search Treesdroche/teaching/240slides/module04.pdf · Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 3 / 29. Binary Search Trees](https://reader036.vdocuments.us/reader036/viewer/2022070923/5fbbe8f4fa74c772324ea402/html5/thumbnails/39.jpg)
AVL tree examples
Example: delete(22) 22
-1
10
1
6
0
4
0
8
0
14
1
13
0
18
-1
16
0
31
1
28
0
37
1
46
0
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 17 / 29
![Page 40: Module 4: Dictionaries and Balanced Search Treesdroche/teaching/240slides/module04.pdf · Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 3 / 29. Binary Search Trees](https://reader036.vdocuments.us/reader036/viewer/2022070923/5fbbe8f4fa74c772324ea402/html5/thumbnails/40.jpg)
AVL tree examples
Example: delete(22) 28
-1
10
1
6
0
4
0
8
0
14
1
13
0
18
-1
16
0
31
1
37
1
46
0
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 17 / 29
![Page 41: Module 4: Dictionaries and Balanced Search Treesdroche/teaching/240slides/module04.pdf · Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 3 / 29. Binary Search Trees](https://reader036.vdocuments.us/reader036/viewer/2022070923/5fbbe8f4fa74c772324ea402/html5/thumbnails/41.jpg)
AVL tree examples
Example: delete(22) 28
-1
10
1
6
0
4
0
8
0
14
1
13
0
18
-1
16
0
31
2
37
1
46
0
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 17 / 29
![Page 42: Module 4: Dictionaries and Balanced Search Treesdroche/teaching/240slides/module04.pdf · Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 3 / 29. Binary Search Trees](https://reader036.vdocuments.us/reader036/viewer/2022070923/5fbbe8f4fa74c772324ea402/html5/thumbnails/42.jpg)
AVL tree examples
Example: delete(22) 28
-2
10
1
6
0
4
0
8
0
14
1
13
0
18
-1
16
0
37
0
31
0
46
0
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 17 / 29
![Page 43: Module 4: Dictionaries and Balanced Search Treesdroche/teaching/240slides/module04.pdf · Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 3 / 29. Binary Search Trees](https://reader036.vdocuments.us/reader036/viewer/2022070923/5fbbe8f4fa74c772324ea402/html5/thumbnails/43.jpg)
AVL tree examples
Example: delete(22) 14
0
10
-1
6
0
4
0
8
0
13
0
28
0
18
-1
16
0
37
0
31
0
46
0
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 17 / 29
![Page 44: Module 4: Dictionaries and Balanced Search Treesdroche/teaching/240slides/module04.pdf · Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 3 / 29. Binary Search Trees](https://reader036.vdocuments.us/reader036/viewer/2022070923/5fbbe8f4fa74c772324ea402/html5/thumbnails/44.jpg)
Height of an AVL tree
Define N(h) to be the least number of nodes in a height-h AVL tree.
One subtree must have height at least h − 1, the other at least h − 2:
N(h) =
1 + N(h − 1) + N(h − 2), h ≥ 11, h = 00, h = −1
What sequence does this look like?
The Fibonacci sequence!
N(h) = Fh+3 − 1 =
⌈ϕh+3
√5
⌋− 1, where ϕ =
1 +√
5
2
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 18 / 29
![Page 45: Module 4: Dictionaries and Balanced Search Treesdroche/teaching/240slides/module04.pdf · Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 3 / 29. Binary Search Trees](https://reader036.vdocuments.us/reader036/viewer/2022070923/5fbbe8f4fa74c772324ea402/html5/thumbnails/45.jpg)
Height of an AVL tree
Define N(h) to be the least number of nodes in a height-h AVL tree.
One subtree must have height at least h − 1, the other at least h − 2:
N(h) =
1 + N(h − 1) + N(h − 2), h ≥ 11, h = 00, h = −1
What sequence does this look like? The Fibonacci sequence!
N(h) = Fh+3 − 1 =
⌈ϕh+3
√5
⌋− 1, where ϕ =
1 +√
5
2
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 18 / 29
![Page 46: Module 4: Dictionaries and Balanced Search Treesdroche/teaching/240slides/module04.pdf · Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 3 / 29. Binary Search Trees](https://reader036.vdocuments.us/reader036/viewer/2022070923/5fbbe8f4fa74c772324ea402/html5/thumbnails/46.jpg)
AVL Tree Analysis
Easier lower bound on N(h):
N(h) > 2N(h− 2) > 4N(h− 4) > 8N(h− 6) > · · · > 2iN(h− 2i) ≥ 2bh/2c
Since n > 2bh/2c, h ≤ 2 lg n,and an AVL tree with n nodes has height O(log n).Also, n ≤ 2h+1 − 1, so the height is Θ(log n).
⇒ search, insert, delete all cost Θ(log n).
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 19 / 29
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AVL Tree Analysis
Easier lower bound on N(h):
N(h) > 2N(h− 2) > 4N(h− 4) > 8N(h− 6) > · · · > 2iN(h− 2i) ≥ 2bh/2c
Since n > 2bh/2c, h ≤ 2 lg n,and an AVL tree with n nodes has height O(log n).Also, n ≤ 2h+1 − 1, so the height is Θ(log n).
⇒ search, insert, delete all cost Θ(log n).
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 19 / 29
![Page 48: Module 4: Dictionaries and Balanced Search Treesdroche/teaching/240slides/module04.pdf · Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 3 / 29. Binary Search Trees](https://reader036.vdocuments.us/reader036/viewer/2022070923/5fbbe8f4fa74c772324ea402/html5/thumbnails/48.jpg)
2-3 Trees
A 2-3 Tree is like a BST with additional structual properties:
Every node either contains one KVP and two children,or two KVPs and three children.
All the leaves are at the same level.(A leaf is a node with empty children.)
Searching through a 1-node is just like in a BST.For a 2-node, we must examine both keys and follow the appropriate path.
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 20 / 29
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Insertion in a 2-3 tree
First, we search to find the leaf where the new key belongs.
If the leaf has only 1 KVP, just add the new one to make a 2-node.
Otherwise, order the three keys as a < b < c .Split the leaf into two 1-nodes, containing a and c ,and (recursively) insert b into the parent along with the new link.
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 21 / 29
![Page 50: Module 4: Dictionaries and Balanced Search Treesdroche/teaching/240slides/module04.pdf · Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 3 / 29. Binary Search Trees](https://reader036.vdocuments.us/reader036/viewer/2022070923/5fbbe8f4fa74c772324ea402/html5/thumbnails/50.jpg)
2-3 Tree Insertion
Example: insert(19)
25 43
18
12 21 24
31 36
28 33 39 42
51
48 56 62
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 22 / 29
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2-3 Tree Insertion
Example: insert(19)
25 43
18
12 21 24
31 36
28 33 39 42
51
48 56 62
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 22 / 29
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2-3 Tree Insertion
Example: insert(19)
25 43
18
12 19 21 24
31 36
28 33 39 42
51
48 56 62
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 22 / 29
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2-3 Tree Insertion
Example: insert(19)
25 43
18 21
12 19 24
31 36
28 33 39 42
51
48 56 62
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 22 / 29
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2-3 Tree Insertion
Example: insert(41)
25 43
18 21
12 19 24
31 36
28 33 39 42
51
48 56 62
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 22 / 29
![Page 55: Module 4: Dictionaries and Balanced Search Treesdroche/teaching/240slides/module04.pdf · Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 3 / 29. Binary Search Trees](https://reader036.vdocuments.us/reader036/viewer/2022070923/5fbbe8f4fa74c772324ea402/html5/thumbnails/55.jpg)
2-3 Tree Insertion
Example: insert(41)
25 43
18 21
12 19 24
31 36
28 33 39 41 42
51
48 56 62
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 22 / 29
![Page 56: Module 4: Dictionaries and Balanced Search Treesdroche/teaching/240slides/module04.pdf · Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 3 / 29. Binary Search Trees](https://reader036.vdocuments.us/reader036/viewer/2022070923/5fbbe8f4fa74c772324ea402/html5/thumbnails/56.jpg)
2-3 Tree Insertion
Example: insert(41)
25 43
18 21
12 19 24
31 36 41
28 33 39 42
51
48 56 62
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 22 / 29
![Page 57: Module 4: Dictionaries and Balanced Search Treesdroche/teaching/240slides/module04.pdf · Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 3 / 29. Binary Search Trees](https://reader036.vdocuments.us/reader036/viewer/2022070923/5fbbe8f4fa74c772324ea402/html5/thumbnails/57.jpg)
2-3 Tree Insertion
Example: insert(41)
25 36 43
18 21
12 19 24
31
28 33
41
39 42
51
48 56 62
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 22 / 29
![Page 58: Module 4: Dictionaries and Balanced Search Treesdroche/teaching/240slides/module04.pdf · Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 3 / 29. Binary Search Trees](https://reader036.vdocuments.us/reader036/viewer/2022070923/5fbbe8f4fa74c772324ea402/html5/thumbnails/58.jpg)
2-3 Tree Insertion
Example: insert(41)
36
25
18 21
12 19 24
31
28 33
43
41
39 42
51
48 56 62
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 22 / 29
![Page 59: Module 4: Dictionaries and Balanced Search Treesdroche/teaching/240slides/module04.pdf · Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 3 / 29. Binary Search Trees](https://reader036.vdocuments.us/reader036/viewer/2022070923/5fbbe8f4fa74c772324ea402/html5/thumbnails/59.jpg)
Deletion from a 2-3 Tree
As with BSTs and AVL trees, we first swap the KVP with its successor,so that we always delete from a leaf.
Say we’re deleting KVP x from a node V :
If X is a 2-node, just delete x .
ElseIf X has a 2-node sibling U, perform a transfer :Put the “intermediate” KVP in the parent between V and U into V ,and replace it with the adjacent KVP from U.
Otherwise, we merge V and a 1-node sibling U:Remove V and (recursively) delete the “intermediate” KVPfrom the parent, adding it to U.
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 23 / 29
![Page 60: Module 4: Dictionaries and Balanced Search Treesdroche/teaching/240slides/module04.pdf · Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 3 / 29. Binary Search Trees](https://reader036.vdocuments.us/reader036/viewer/2022070923/5fbbe8f4fa74c772324ea402/html5/thumbnails/60.jpg)
2-3 Tree Deletion
Example: delete(43)
36
25
18 21
12 19 24
31
28 33
43
41
39 42
51
48 56 62
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 24 / 29
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2-3 Tree Deletion
Example: delete(43)
36
25
18 21
12 19 24
31
28 33
48
41
39 42
51
56 62
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 24 / 29
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2-3 Tree Deletion
Example: delete(43)
36
25
18 21
12 19 24
31
28 33
48
41
39 42
56
51 62
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 24 / 29
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2-3 Tree Deletion
Example: delete(19)
36
25
18 21
12 19 24
31
28 33
48
41
39 42
56
51 62
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 24 / 29
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2-3 Tree Deletion
Example: delete(19)
36
25
18
12 21 24
31
28 33
48
41
39 42
56
51 62
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 24 / 29
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2-3 Tree Deletion
Example: delete(19)
36
25
18
12 21 24
31
28 33
48
41
39 42
56
51 62
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 24 / 29
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2-3 Tree Deletion
Example: delete(42)
36
25
18
12 21 24
31
28 33
48
41
39 42
56
51 62
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 24 / 29
![Page 67: Module 4: Dictionaries and Balanced Search Treesdroche/teaching/240slides/module04.pdf · Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 3 / 29. Binary Search Trees](https://reader036.vdocuments.us/reader036/viewer/2022070923/5fbbe8f4fa74c772324ea402/html5/thumbnails/67.jpg)
2-3 Tree Deletion
Example: delete(42)
36
25
18
12 21 24
31
28 33
48
39 41
56
51 62
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 24 / 29
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2-3 Tree Deletion
Example: delete(42)
36
25
18
12 21 24
31
28 33
48 56
39 41 51 62
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 24 / 29
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2-3 Tree Deletion
Example: delete(42)
25 36
18
12 21 24
31
28 33
48 56
39 41 51 62
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 24 / 29
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2-3 Tree Deletion
Example: delete(42)
25 36
18
12 21 24
31
28 33
48 56
39 41 51 62
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 24 / 29
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B-Trees
The 2-3 Tree is a specific type of B-tree:
A B-tree of minsize d is a search tree satisfying:
Each node contains at most 2d KVPs.Each non-root node contains at least d KVPs.
All the leaves are at the same level.
Some people call this a B-tree of order (2d + 1), or a (d + 1, 2d + 1)-tree.A 2-3 tree has d = 1.
search, insert, delete work just like for 2-3 trees.
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 25 / 29
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Height of a B-treeWhat is the least number of KVPs in a height-h B-tree?
Level Nodes Node size KVPs0 1 1 11 2 d 2d2 2(d + 1) d 2d(d + 1)3 2(d + 1)2 d 2d(d + 1)2
· · · · · · · · · · · ·h 2(d + 1)h−1 d 2d(d + 1)h−1
Total: 1 +h−1∑i=0
2d(d + 1)i = 2(d + 1)h − 1
Therefore height of tree with n nodes is Θ((log n)/(log d)
).
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 26 / 29
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Analysis of B-tree operations
Assume each node stores its KVPs and child-pointers in a dictionarythat supports O(log d) search, insert, and delete.
Then search, insert, and delete work just like for 2-3 trees, and eachrequire Θ(height) node operations.
Total cost is O
(log n
log d· (log d)
)= O(log n).
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 27 / 29
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Dictionaries in external memory
Tree-based data structures have poor memory locality :If an operation accesses m nodes, then it must accessm spaced-out memory locations.
Observation: Accessing a single location in external memory(e.g. hard disk) automatically loads a whole block (or “page”).
In an AVL tree or 2-3 tree, Θ(log n) pages are loaded in the worst case.
If d is small enough so a 2d-node fits into a single page,then a B-tree of minsize d only loads Θ
((log n)/(log d)
)pages.
This can result in a huge savings:memory access is often the largest time cost in a computation.
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 28 / 29
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B-tree variations
Max size 2d + 1: Permitting one additional KVP in each nodeallows insert and delete to avoid backtracking viapre-emptive splitting and pre-emptive merging .
Red-black trees: Identical to a B-tree with minsize 1 and maxsize 3,but each 2-node or 3-node is represented by 2 or 3 binary nodes,and each node holds a “color” value of red or black.
B+-trees: All KVPs are stored at the leaves(interior nodes just have keys),and the leaves are linked sequentially.
Reza Dorrigiv, Daniel Roche (CS, UW) CS240 - Module 4 Winter 2010 29 / 29