balanced search trees cs 302 - data structures mehmet h gunes modified from authors’ slides

Balanced Search Trees

CS 302 - Data StructuresMehmet H Gunes

Modified from authors’ slides

Contents

• Balanced Search Trees• 2-3 Trees• 2-3-4 Trees• Red-Black Trees• AVL Trees

Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013


• Height of a binary search tree sensitive to order of insertions and removals– Minimum = log2 (n + 1)

– Maximum = n

• Various search trees can retain balance despite insertions and removals



• (a) A binary search tree of maximum height; (b) a binary search tree of minimum height


2-3 Trees

• A 2-3 tree of height 3

• A 2-3 tree is not a binary tree• A 2-3 tree never taller than a minimum-

height binary tree


2-3 Trees

• Placing data items in nodes of a 2-3 tree– A 2-node (has two children) must contain single

data item greater than left child’s item(s) and less than right child’s item(s)

– A 3-node (has three children) must contain two data items, S and L , such that

• S is greater than left child’s item(s) and less than middle child’s item(s);

• L is greater than middle child’s item(s) and less than right child’s item(s).

– Leaf may contain either one or two data items.Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013

2-3 Trees

• Nodes in a 2-3 tree: (a) a 2-node; (b) a 3-node

Data Structures and Problem Solving with C++: Walls and Mirrors, Carrano and Henry, © 2013`

2-3 Trees

• A 2-3 tree

View Header file for a class of nodes for a 2-3 tree, Listing 19-1


Traversing a 2-3 Tree

• Traverse 2-3 tree in sorted order by performing analogue of inorder traversal on binary tree:


Searching a 2-3 Tree

• Retrieval operation for 2-3 tree similar to retrieval operation for binary search tree



• Possible to search 2-3 tree and shortest binary search tree with approximately same efficiency, because:– Binary search tree with n nodes cannot be shorter

than log2 (n + 1)

– 2-3 tree with n nodes cannot be taller than log2 (n + 1)

– Node in a 2-3 tree has at most two items



• A balanced binary search tree;



• a 2-3 tree with the same entries



• (a) The binary search tree of Figure 19-5a after inserting the sequence of values 32 through 39

• (b) the 2-3 tree of Figure 19-5 b after the same insertions


Inserting Data into a 2-3 Tree

• After inserting 39 into the tree



• The steps for inserting 38 into the tree: (a) The located node has no room;

(b) the node splits; (c) the resulting tree



• After inserting 37 into the tree



• (a), (b), (c) The steps for inserting 36 into the tree



• (d) the resulting tree



• The tree after the insertion of 35, 34, and 33 into the tree



• Splitting a leaf in a 2-3 tree when the leaf is a (a) left child; (b) right child



• Splitting an internal node in a 2-3 tree when the node is a (a) left child; (b) right child



• Splitting the root of a 2-3 tree



• Summary of insertion strategy


Removing Data from a 2-3 Tree

• A 2-3 tree; (b), (c), (d), (e) the steps for removing 70;



• (f) the resulting tree;



• (a), (b), (c) The steps for removing 100 from the tree; (d) the resulting tree



• The steps for removing 80 from the tree



• Results of removing 70, 100, and 80 from (a) the 2-3 tree and (b) the binary search tree



• Algorithm for removing data from a 2-3 tree



• (a) Redistributing values; (b) merging a leaf;



• (c) redistributing values and children; • (d) merging internal nodes



• (e) deleting the root


2-3-4 Trees

• A 2-3-4 tree with the same data items as the 2-3 tree


2-3-4 Trees

• Rules for placing data items in the nodes of a 2-3-4 tree– 2-node (two children), must contain a single data

item that satisfies relationships – 3-node (three children), must contain two data

items that satisfies relationships – . . .


2-3-4 Trees– 4-node (four children) must contain three data

items S , M , and L that satisfy:• S is greater than left child’s item(s) and less than

middle-left child’s item(s)• M is greater than middle-left child’s item(s) and less

than middle-right child’s item(s);• L is greater than middle-right child’s item(s) and less

than right child’s item(s).

– A leaf may contain either one, two, or three data items


2-3-4 Trees

• A 4-node in a 2-3-4 tree


2-3-4 Trees

• Has more efficient insertion and removal operations than a 2-3 tree

• Has greater storage requirements due to the additional data members in its 4-nodes


2-3-4 Trees

• Searching and Traversing a 2-3-4 Tree– Simple extensions of the corresponding algorithms

for a 2-3 tree

• Inserting Data into a 2-3-4 Tree– Insertion algorithm splits a node by moving one of

its items up to its parent node– Splits 4-nodes as soon as it encounters them on

the way down the tree from the root to a leaf


2-3-4 Trees

• Inserting 20 into a one-node 2-3-4 tree (a) the original tree; (b) after splitting the node; (c) after inserting 20


2-3-4 Trees

• After inserting 50 and 40 into the tree


2-3-4 Trees

• The steps for inserting 70 into the tree : (a) after splitting the 4-node; (b) after inserting 70


2-3-4 Trees

• After inserting 80 and 15 into the tree


2-3-4 Trees

• The steps for inserting 90 into the tree


2-3-4 Trees

• The steps for inserting 100 into the tree


2-3-4 Trees

• Splitting a 4-node root during insertion into a 2-3-4 tree


2-3-4 Trees

• Splitting a 4-node whose parent is a 2-node during insertion into a 2-3-4 tree, when the 4-node is a (a) left child; (b) right child


2-3-4 Trees

• Splitting a 4-node whose parent is a 3-node during insertion into a 2-3-4 tree, when the 4-node is a (a) left child


2-3-4 Trees

• Splitting a 4-node whose parent is a 3-node during insertion into a 2-3-4 tree, when the 4-node is a (b) middle child


2-3-4 Trees

• Splitting a 4-node whose parent is a 3-node during insertion into a 2-3-4 tree, when the 4-node is a (c) right child


2-3-4 Trees

• Removing Data from a 2-3-4 Tree– Removal algorithm has same beginning as removal

algorithm for a 2-3 tree– Locate the node n that contains the item I you

want to remove– Find I ’s inorder successor and swap it with I so

that the removal will always be at a leaf– If leaf is either a 3-node or a 4-node, remove I .


Modified from Dr George Bebis and Dr Monica Nicolescu

Red-Black Trees

• Use a special binary search tree—a red-black tree —to represent a 2-3-4 tree

• Retains advantages of a 2-3-4 tree without storage overhead

• The idea is to represent each 3-node and 4-node as an equivalent binary search tree


Red-Black Trees

• Red-black representation of (a) a 4-node; (b) a 3-node


Red-Black Trees

• A red-black tree that represents the 2-3-4 tree


Red-Black Trees

• Searching and traversing– Red-black tree is a binary search tree, search and

traverse it by using algorithms for binary search tree

• Inserting, removing with a red-black tree– Adjust the 2-3-4 insertion algorithms to

accommodate the red-black representation


Red-Black Trees

• Splitting a red-black representation of a 4-node that is the root


Red-Black Trees

• Splitting a red-black representation of a 4-node whose parent is a 2-node, when the 4-node is a (a) left child


Red-Black Trees

• Splitting a red-black representation of a 4-node whose parent is a 2-node, when the 4-node is a (b) right child


Red-Black Trees

• Splitting a red-black representation of a 4-node whose parent is a 3-node


Red-Black-Trees Properties(**Binary search tree property is satisfied**)

1. Every node is either red or black

2. The root is black

3. Every leaf (NIL) is black

4. If a node is red, then both its children are black

• No two consecutive red nodes on a simple path from the root to a leaf

5. For each node, all paths from that node to a leaf contain the same number of black nodes

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Example: RED-BLACK-TREE

• For convenience, we add NIL nodes and refer to them as the leaves of the tree.– Color[NIL] = BLACK

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30 47

38 50

NIL NIL

NIL

NIL NIL NIL NIL

NIL

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Definitions

• Height of a node: the number of edges in the longest path to a leaf

• Black-height bh(x) of a node x: the number of black nodes (including NIL) on the path from x to a leaf, not counting x

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30 47

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NIL NIL

NIL

NIL NIL NIL NIL

NIL

h = 4bh = 2

h = 3bh = 2

h = 2bh = 1

h = 1bh = 1

h = 1bh = 1

h = 2bh = 1 h = 1

bh = 1

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Height of Red-Black-Trees

A red-black tree with n internal nodes has height at most 2log(N+1)

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Insert Item

What color to make the new node?• Red?

– Let’s insert 35!

• Property 4 is violated: if a node is red, then both children are black• Black?

– Let’s insert 14!

• Property 5 is violated: all paths from a node to its leaves contain the same number of black nodes

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30 47

38 50

Delete Item

What color was the node that was removed? Red? 1. Every node is either red or black2. The root is black3. Every leaf (NIL) is black4. If a node is red, then both its children are black

5. For each node, all paths from the node to descendant leaves contain the same number of black nodes

OK!

OK!

OK!

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30 47

38 50

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OK!

OK!

Delete Item

What color was the node that was removed? Black? 1. Every node is either red or black2. The root is black3. Every leaf (NIL) is black4. If a node is red, then both its children are black

5. For each node, all paths from the node to descendant leaves contain the same number of black nodes

OK!

OK!

Not OK! Could createtwo red nodes in a row

Not OK! Could change theblack heights of some nodes

Not OK! If removing the root and the child that replaces it is red

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Rotations• Operations for re-structuring the tree after

insert and delete operations– Together with some node re-coloring, they help

restore the red-black-tree property

– Change some of the pointer structure

– Preserve the binary-search tree property

• Two types of rotations:– Left & right rotations

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Left Rotations• Assumptions for a left rotation on a node x:

– The right child y of x is not NIL

• Idea:– Pivots around the link from x to y– Makes y the new root of the subtree– x becomes y ’s left child– y ’s left child becomes x ’s right child

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Example: LEFT-ROTATE

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LEFT-ROTATE(T, x)1. y ← right[x] ►Set y

2. right[x] ← left[y] ► y’s left subtree becomes x’s right subtree

3. if left[y] NIL4. then p[left[y]] ← x ► Set the parent relation from left[y] to x

5. p[y] ← p[x] ► The parent of x becomes the parent of y

6. if p[x] = NIL7. then root[T] ← y8. else if x = left[p[x]]9. then left[p[x]] ← y10. else right[p[x]] ← y11.left[y] ← x ► Put x on y’s left

12.p[x] ← y ► y becomes x’s parent

• Assumptions for a right rotation on a node x:– The left child x of y is not NIL

• Idea:– Pivots around the link from y to x– Makes x the new root of the subtree– y becomes x ’s right child– x ’s right child becomes y ’s left child

Right Rotations

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Insert Item• Goal:

– Insert a new node z into a red-black tree

• Idea:– Insert node z into the tree as for an ordinary

binary search tree– Color the node red– Restore the red-black tree properties

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RB-INSERT(T, z)1. y ← NIL

2. x ← root[T]

3. while x NIL

4. do y ← x

5. if key[z] < key[x]

6. then x ← left[x]

7. else x ← right[x]

8. p[z] ← y

• Initialize nodes x and y• Throughout the algorithm y points to the parent of x

• Go down the tree untilreaching a leaf• At that point y is theparent of the node to beinserted

• Sets the parent of z to be y

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RB-INSERT(T, z)

9. if y = NIL

10. then root[T] ← z

11. else if key[z] < key[y]

12. then left[y] ← z

13. else right[y] ← z

14. left[z] ← NIL

15. right[z] ← NIL

16. color[z] ← RED

17. RB-INSERT-FIXUP(T, z)

The tree was empty: set the new node to be the root

Otherwise, set z to be the left orright child of y, depending on whether the inserted node is smaller or larger than y’s key

Set the fields of the newly added node

Fix any inconsistencies that could have been introduced by adding this new red node

RB Properties Affected by Insert

1. Every node is either red or black

2. The root is black

3. Every leaf (NIL) is black

4. If a node is red, then both its children are black

5. For each node, all paths

from the node to descendant

leaves contain the same number

of black nodes

OK!

If z is the root not OK

OK!

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4738

50

If p(z) is red not OKz and p(z) are both redOK!

RB-INSERT-FIXUP

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Case 1: z ’s “uncle” (y) is red (z could be either left or right child)

Idea: • p[p[z]] (z ’s grandparent) must be black

• color p[z] black

• color y black

• color p[p[z]] red

• z = p[p[z]]

– Push the “red” violation up the tree

RB-INSERT-FIXUP

Case 2: • z ’s “uncle” (y) is black• z is a left child

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Case 2

Idea:• color p[z] black • color p[p[z]] red• RIGHT-ROTATE(T, p[p[z]])• No longer have 2 reds in a row• p[z] is now black

RB-INSERT-FIXUP

Case 3: • z ’s “uncle” (y) is black• z is a right childIdea:• z p[z]• LEFT-ROTATE(T, z) now z is a left child, and both z and p[z] are red case 2

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Case 3 Case 2

Example

11Insert 4

2 14

1 157

85

4

y

11

2 14

1 157

85

4

z

Case 1

y

z and p[z] are both redz’s uncle y is redz

z and p[z] are both redz’s uncle y is blackz is a right child

Case 3

11

2

14

1

15

7

8

5

4

z

y Case 2

z and p[z] are redz’s uncle y is blackz is a left child

112

141

15

7

85

4

z

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RB-INSERT-FIXUP(T, z)1. while color[p[z]] = RED

2. if p[z] = left[p[p[z]]]

3. then y ← right[p[p[z]]]

4. if color[y] = RED

5. then Case1

6. else if z = right[p[z]]

7. then Case3

8. Case2

9. else (same as then clause with “right” and “left” exchanged for lines 3-4)

10. color[root[T]] ← BLACK

The while loop repeats only whencase1 is executed: O(logN) times

Set the value of x’s “uncle”

We just inserted the root, orThe red violation reached the root

Analysis of InsertItem

• Inserting the new element into the tree O(logN)

• RB-INSERT-FIXUP– The while loop repeats only if CASE 1 is executed– The number of times the while loop can be

executed is O(logN)

• Total running time of Insert Item: O(logN)

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Delete Item• Delete as usually, then re-color/rotate

• A bit more complicated though …

• Demo– http://gauss.ececs.uc.edu/RedBlack/redblack.html

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Problems

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• What red-black tree property is violated in the tree below? How would you restore the red-black tree property in this case?– Property violated: if a node is red, both its children are black– Fixup: color 7 black, 11 red, then right-rotate around 11

Problems

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112

141

15

7

85

4

z

Problems

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• Let a, b, c be arbitrary nodes in subtrees , , in the tree below.

• How do the depths of a, b, c change when a left rotation is performed on node x?– a: increases by 1– b: stays the same– c: decreases by 1

Problems

• When we insert a node into a red-black tree, we initially set the color of the new node to red.

Why didn’t we choose to set the color to black?

• Would inserting a new node to a red-black tree and then immediately deleting it, change the tree?

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AVL Trees

• Named for inventors– Adel’son-Vel’skii and Landis

• A balanced binary search tree– Maintains height close to the minimum– After insertion or deletion, check the tree is still

AVL tree • determine whether any node in tree has left and right

subtrees whose heights differ by more than 1

• Can search AVL tree almost as efficiently as minimum-height binary search tree.


AVL Trees

• (a) An unbalanced binary search tree; (b) a balanced tree after rotation; (c) a balanced tree after insertion


• (a) Before;• (b) and after a single left rotation that decreases the

tree’s height; • (c) the rotation in general


AVL Trees

• (a) Before; • (b) and after a single left rotation that does not affect the tree’s height; • (c) the rotation in general


AVL Trees

AVL Trees

• (d) the double rotation in general


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