AVL Treeavail

7/30/2019 AVL Treeavail

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Balanced Binary Search Trees

height is O(log n), where n is the

number of elements in the tree

AVL (Adelson-Velsky and Landis)trees

red-black trees

get, put, and remove take O(log n)

time


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Balanced Binary Search Trees

Indexed AVL trees

Indexed red-black trees

Indexed operations alsotake

O(log n) time


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Balanced Search Trees

weight balanced binary search trees

2-3 & 2-3-4 trees

AA trees

B-trees

BBST

etc.


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AVL Tree

binary tree

for every node x, define its balance factor

balance factor of x = height of left subtree of x

- height of right subtree of x

balance factor of every node x is -1, 0, or1


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Balance Factors

0 0

0 0

1

0

-1 0

1

0

-1

1

-1

This is an AVL tree.


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Height

The height of an AVL tree that has n nodes is

at most 1.44 log2 (n+2).

The height of every n node binary tree is at

least log2 (n+1).


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AVL Search Tree

0 0

0 0

1

0

-1 0

1

0

-1

1

-1

10

7

83

1 5

30

40

20

25

35

45

60


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put(9)

0 0

0 0

1

0

-1 0

1

0

-1

1

-1

90

-1

0

10

7

83

1 5

30

40

20

25

35

45

60


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put(29)

0 0

0 0

1

0

0

1

0

-1

1

-1

10

7

83

1 5

30

40

20

25

35

45

60

29

0

-1

-1-2

RR imbalance => new node is in

right subtree of right subtree of

blue node (node with bf= -2)


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put(29)

0 0

0 0

1

0

0

1

0

-1

1

-1

10

7

83

1 5

30

40

2535

45

600

RR rotation.

20

0

29


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AVL Rotations

RR

LL

RL

LR


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Red Black Trees

Colored Nodes Definition

Binary search tree.

Each node is colored red or black.

Root and all external nodes are black.

No root-to-external-node path has two

consecutive red nodes. All root-to-external-node paths have the

same number of black nodes


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Example Red Black Tree

10

7

8

1 5

30

40

20

25

35

45

60

3


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Red Black Trees

Colored Edges Definition

Binary search tree.

Child pointers are colored red or black.

Pointer to an external node is black.

No root to external node path has twoconsecutive red pointers.

Every root to external node path has the

same number of black pointers.


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Example Red Black Tree

10

7

8

1 5

30

40

20

25

35

45

60

3


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Red Black Tree

The height of a red black tree that has n

(internal) nodes is between log2(n+1) and

2log2(n+1). java.util.TreeMap => red black tree


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Graphs

G = (V,E)

V is the vertex set.

Vertices are also called nodes and points.

E is the edge set.

Each edge connects two different vertices.

Edges are also called arcs and lines.

Directed edge has an orientation (u,v).

u v


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Graphs

Undirected edge has no orientation (u,v).u v

Undirected graph => no oriented edge.

Directed graph => every edge has an

orientation.


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Undirected Graph

23

8

101

45

911

67


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Directed Graph (Digraph)

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45

911

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ApplicationsCommunication Network

Vertex = city, edge = communication link.

23

8

101

45

911

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Driving Distance/Time Map

Vertex = city, edge weight = driving

distance/time.

23

8

101

45

911

67

4

8

6

6

7

5

24

4 53


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Street Map

Some streets are one way.

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101

45

911

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Complete Undirected Graph

Has all possible edges.

n = 1 n = 2 n = 3 n = 4


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Number Of EdgesUndirected Graph

Each edge is of the form (u,v), u != v.

Number of such pairs in an n vertex graph is

n(n-1).

Since edge (u,v) is the same as edge (v,u),

the number of edges in a complete

undirected graph is n(n-1)/2.

Number of edges in an undirected graph is


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Number Of EdgesDirected Graph

Each edge is of the form (u,v), u != v.

Number of such pairs in an n vertex graph is

n(n-1).

Since edge (u,v) is not the same as edge

(v,u), the number of edges in a complete

directed graph is n(n-1).

Number of edges in a directed graph is


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Vertex Degree

Number of edges incident to vertex.

degree(2) = 2, degree(5) = 3, degree(3) = 1

23

8

101

45

911

67


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Sum Of Vertex Degrees

Sum of degrees = 2e (e is number of edges)

8

10

911


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In-Degree Of A Vertex

in-degree is number of incoming edges

indegree(2) = 1, indegree(8) = 0

23

8

101

45

911

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Out-Degree Of A Vertex

out-degree is number of outbound edges

outdegree(2) = 1, outdegree(8) = 2

23

8

101

45

911

67


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Sum Of In- And Out-Degrees

each edge contributes 1 to the in-degree of

some vertex and 1 to the out-degree of some

other vertex

sum of in-degrees = sum of out-degrees = e,

where e is the number of edges in the

digraph

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