Data Structure II

So Pak Yeung

26-2-2011

Outline

Review Array Sorted Array Linked List

Binary Search Tree Heap Hash Table

Review

Operation Find an element Find the min/max element Insert an element Remove an element

Time complexity? O(1)? O(lg N)? O(N)?

Array

Find an element O(N)

Find the smallest element O(N)

Insert an element O(1)

Remove an element O(1)

Sorted Array

Find an element O(lg N)

Find the smallest element O(1)

Insert an element O(N)

Remove an element O(N)

Linked List

Find an element O(N)

Find the smallest element O(N)

Insert an element O(1)

Remove an element O(1)

Binary Search Tree

Binary Tree At most 2 children for each node

For each Node i, Node j <= Node i, for all Node j in left subtree of N

ode i Node j > Node i, for all Node j in right subtree of N

ode i

Binary Search Tree

5 13 34

8

212

Binary Search Tree

5 13 34

8

212

• Find 13

Binary Search Tree

5 13 34

8

212

• Find 3

???

Binary Search Tree

5 13 34

8

212

• Insert 1

1

Binary Search Tree

5 13 34

8

212

• Insert 3

1

3

Binary Search Tree

Find an element Seems O(lg N)?

Find min/max Seems O(lg N)?

Insert an element Seems O(lg N)?

Binary Search Tree

5

13

34

8

21

2

•Worst Case: O(N)!!!

Binary Search Tree

Remove a leaf node Just Do it!

Remove a node with single child Replace the node by its child

Remove a node with 2 children Replace the node by the max node of left subtree /

min node of right subtree Lazy Deletion

Mark the node as deleted instead of deleting it

Binary Search Tree

5

13

34

8

21

2

•Again, seems O(lg N), worst Case: O(N)!!!

Heap

Priority Queue Binary Tree For each node, it is no greater/less than all n

odes of its subtree Operation

Extract min/max Insert

Heap

5 21 34

1

132

8

Heap

Extract min/max Get the value from the root (O(1) to find min) Replace the root by the last node Shift down

Time complexity O(lg N)

Heap

5 21 34

1

132

8

• Get 1

Heap

5 21

132

8

34

Heap

5 21

1334

8

2

Heap

34 21

135

8

2

Heap

Insert an element Add the node at the end Shift up

Time complexity O(lg N)

Heap

34 21

135

8

2

• Add 1

1

Heap

34 21

15

8

2

13

Heap

34 21

25

8

1

13

Heap

Build heap Insert each node O(N lg N) There exists a faster way!! Only need to perform shift down process from the

bottom O(N)

Heap

Find an element Not support O(N), if implement by array

Remove an element Consider that subtree O(lg N)

Hash Table

Using limited memory, storing data of unlimited range

Convert data to integers that are in a limited range by a hash function

Hash Table

Mark 5 number Each number is

between [1,10000000] Not a good idea to use

an array of size of 10000000

A[n%5]=n

0 6625800

1 65536

2 1234567

3 38

4 4

Hash Table

Insert an element O(1)

Find an element Using the same Hash function! O(1)

Delete an element Lazy Deletion O(1)

Hash Table

Collision? E.g. 56 and 111 Open Hashing Close Hashing

Hash TableOpening Hashing

0

1

2

3

4

11156

127

Hash Table

Close hashing Insert 1 If the cell is occupied,

find the next empty cell

0 6625800

1 65536

2 1234567

3 38

4 1