1

Data Structure and Algorithm (CS 102)

Ashok K Turuk

2

m-Way Search TreeAn m-way search tree T may be an

empty tree. If T is non-empty, it satisfies the following properties:

(i) For some integer m known as the order of the tree, each node has at most m child nodes. A node may be represented as A0 , (K1, A1), (K2, A2) …. (Km-1 , Am-1 ) where Ki 1<= i <= m-1 are the keys and Ai, 0<=i<=m-1 are the pointers to the subtree of T

3

m-Way Search Tree[2] If the node has k child nodes where k<=m,

then the node can have only (k-1) keys, K1 , K2 , …… Kk-1 contained in the node such that Ki < Ki+1 and each of the keys partitions all the keys in the subtrees into k subsets

[3] For a node A0 , (K1 , A1), (K2 , A2) , …. (Km-

1 , Am-1 ) all key values in the subtree pointed to by Ai are less than the key Ki+1 , 0<=i<=m-2 and all key values in the subtree pointed to by Am-1 are greater than Km-1

4

m-Way Search Tree[4] Each of the subtree Ai , 0<=i<=m-

1 are also m-way search tree

5

m-Way Search Tree [ m=5]

18 44 76 198X X

7 12

X X

80 92 141

262

8 10148

151

172

186

X X X

X X X X XX X X

77

X X

272

286

350X X X X

6

Searching in an m-Way Search Tree

18 44 76 198X X

7 12

X X

80 92 141

262

8 10148

151

172

186

X X X

X X X X XX X X

77

X X

272

286

350X X X X

Look for 77

7

Insertion in an m-Way Search Tree

18 44 76 198X X

7 12

X X

80 92 141

262

8 10148

151

172

186

X X X

X X X X XX X X

77

X X

272

286

350X X X X

Insert 6

8

Insertion in an m-Way Search Tree

18 44 76 198X X

7 12

X X

80 92 141

262

8 10148

151

172

186

X X X

X X X XX X X

77

X X

272

286

350X X X X

Insert 6

6

X

Insert 146

146X

9

Deletion in an m-Way Search Tree

Let K be the key to be deleted from the m-way search tree.

K

Ai Aj

K : KeyAi , Aj : Pointers to subtree

10

Deletion in an m-Way Search Tree

[1] If (Ai = Aj = NULL) then delete K

[2] If (Ai NULL, Aj = NULL ) then choose the largest of the key elements K’ in the child node pointed to by Ai and replace K by K’.

[3] If (Ai = NULL, Aj NULL ) then choose the smallest of the key element K” from the subtree pointed to by Aj , delete K” and replace K by K”.

11

Deletion in an m-Way Search Tree

[4] If (Ai NULL, Aj NULL ) then choose the largest of the key elements K’ in the subtree pointed to by Ai or the smallest of the key element K” from the subtree pointed to by Aj to replace K.

12

5-Way Search Tree

18 44 76 198X X

7 12

X X

80 92 141

262

8 10148

151

172

186

X X X

X X X X XX X X

77

X X

272

286

350X X X X

Delete 151

13

5-Way Search Tree

18 44 76 198X X

7 12

X X

80 92 141

262

8 10148

172

186

X X X

X X X XX X X

77

X X

272

286

350X X X X

Delete 151

14

5-Way Search Tree

18 44 76 198X X

7 12

X X

80 92 141

262

8 10148

151

172

186

X X X

X X X X XX X X

77

X X

272

286

350X X X X

Delete 262

15

5-Way Search Tree

18 44 76 198X X

7 12

X X

80 92 141

272

8 10148

151

172

186

X X X

X X X X XX X X

77

X X

286

350X X X

Delete 262

16

5-Way Search Tree

18 44 76 198X X

7 12

X X

80 92 141

262

8 10148

151

172

186

X X X

X X X X XX X X

77

X X

272

286

350X X X X

Delete 12

17

5-Way Search Tree

18 44 76 198X X

7 10

X X

80 92 141

262

8148

151

172

186

X X X

X X X X XX X

77

X X

272

286

350X X X X

Delete 12

18

B TreesB tree is a balanced m-way search tree

A B tree of order m, if non empty is an m-way search tree in which

[i] the root has at least two child nodes and at most m child nodes

[ii] internal nodes except the root have at least m/2 child nodes and at most m child nodes

19

B Trees

[iii] the number of keys in each internal node is one less than the number of child nodes and these keys partition the keys in the subtrees of nodes in a manner similar to that of m-way search trees

[iv] all leaf nodes are on the same level

20

B Tree of order 5

48

31 4556 64 85

87 88 100

112X X X X X

49 51 52

X X X X

46 47

36 40 42

10 18 21

X X X X

X X X

X X X X

X X X

58 62

67 75

X X X

21

Searching a B Tree

Searching for a key in a B-tree is similar to the one on an m-way search tree.

The number of accesses depends on the height h of the B-tree

22

Insertion in a B-TreeA key is inserted according to the

following procedure[1] If the leaf node in which the key is

to be inserted is not full, then the insertion is done in the node.

A node is said to be full if it contains a maximum of (m-1) keys given the order of the B-tree to be m

23

Insertion in a B-Tree[2] If the node were to be full then insert the

key in order into the existing set of keys in the node. Split the node at its median into two nodes at the same level, pushing the median element up by one level. Accommodate the median element in the parent node if it is not full. Otherwise repeat the same procedure and this may call for rearrangement of the keys in the root node or the formation of new root itself.

5-Way Search Tree

8 96 116

2 7

X X X

104

110

37 46 55 86X X X X X X X X

137

145X X X

Insert 4, 5, 58, 6 in the order

5-Way Search Tree

8 96 116

104

110

37 46 55 86X X X X X X X X

137

145X X X

Search tree after inserting 4

2 4 7

X X X X

5-Way Search Tree

8 96 116

104

110

37 46 55 86X X X X X X X X

137

145X X X

Search tree after inserting 4, 5

2 4 5 7

X X X X X

5-Way Search Tree

8 96 116

104

110

37 46 55 86X X X X X X X X

137

145X X X

2 4 5 7

X X X X X

37,46,55,58,86

Split the node at its median into two node, pushing the median element up by one level

5-Way Search Tree

104

110X X X

137

145X X X

2 4 5 7

X X X X X

37 46

X X X

58 86

X X X

8 96 116

Insert 55 in the root

5-Way Search Tree

104

110

8 55 96 116

X X X

137

145X X X

Search tree after inserting 4, 5, 58

2 4 5 7

X X X X X

37 46

X X X

58 86

X X X

5-Way Search Tree

104

110

8 55 96 116

X X X

137

145X X X

Insert 6

2 4 5 7

X X X X X

37 46

X X X

58 86

X X X2,4,5,6,7

Split the node at its median into two node, pushing the median element up by one level

5-Way Search Tree

104

110

8 55 96 116

X X X

137

145X X X

Insert 5 at the root

37 46

X X X

58 86

X X X6 72 4

X X XX X X

5-Way Search Tree

104

110X X X

137

145X X X

Insert 5 at the root

37 46

X X X

58 86

X X X6 72 4

X X XX X X

96 116

5 8

55

5-Way Search Tree

104

110X X X

137

145X X X

Insert 5 at the root

37 46

X X X

58 86

X X X

6 7

2 4

X X X

X X X

96 1165 8

55

34

Deletion in a B-TreeIt is desirable that a key in leaf node

be removed.

When a key in an internal node to be deleted, then we promote a successor or a predecessor of the key to be deleted ,to occupy the position of the deleted key and such a key is bound to occur in a leaf node.

35

Deletion in a B-TreeRemoving a key from leaf node:If the node contain more than the

minimum number of elements, then the key can be easily removed.

If the leaf node contain just the minimum number of elements, then look for an element either from the left sibling node or right sibling node to fill the vacancy.

36

Deletion in a B-TreeIf the left sibling has more than minimum

number of keys, pull the largest key up into the parent node and move down the intervening entry from the parent node to the leaf node where key is deleted.

Otherwise, pull the smallest key of the right sibling node to the parent node and move down the intervening parent element to the leaf node.

37

Deletion in a B-TreeIf both the sibling node has minimum number of

entries, then create a new leaf node out of the two leaf nodes and the intervening element of the parent node, ensuring the total number does not exceed the maximum limit for a node.

If while borrowing the intervening element from the parent node, it leaves the number of keys in the parent node to be below the minimum number, then we propagate the process upwards ultimately resulting in a reduction of the height of B-tree

38

B-tree of Order 5110

65

86

120

226

70

81

32

44 X X X

90 95

100

X X XX X X

115

118 20

0221X X X

X X X X 300

440 550 601X X X X X

Delete 95, 226, 221, 70

39

B-tree of Order 5110

65

86

120

226

70

81

32

44 X X X

90 100

X X XX X X

115

118 20

0221X X X

X X X 300

440 550 601X X X X X

B-tree after deleting 95

40

B-tree of Order 5110

65

86

120

300

70

81

32

44 X X X

90 100

X X XX X X

115

118 20

0221X X X

X X X 300

440 550 601X X X X X

41

B-tree of Order 5110

65

86

120

300

70

81

32

44 X X X

90 100

X X XX X X

115

118 20

0221X X X

X X X 440 550 601X X X X

B-tree after deleting 95, 226

Delete 221

42

B-tree of Order 5110

65

86

120

440

70

81

32

44 X X X

90 100

X X XX X X

115

118 20

0300X X X

X X X 550 601X X X

B-tree after deleting 95, 226, 221

Delete 70

43

B-tree of Order 5110

65

86

120

440

65

81

32

44 X X X

90 100

X XX X X

115

118 20

0300X X X

X X X 550 601X X X

Delete 65

44

B-tree of Order 5

110

86

120

440

65

81

32

44 X X X

90 100

X XX X X

115

118 20

0300X X X

X X X 550 601X X X

B-tree after deleting 95, 226, 221, 70

45

Heap Suppose H is a complete binary tree

with n elements

H is called a heap or maxheap if each node N of H has the following property

Value at N is greater than or equal to the value at each of the children of N.

46

Heap

97

88 95

66 55

66 35

18 40 30 26

48

24

55

95

62 77

48

25 38

97

88

95

66

55

95

48

66

35

48

55

62

77

25

38

18

40

30

26

24

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

47

Inserting into a HeapSuppose H is a heap with N elementsSuppose an ITEM of information is given.

Insertion of ITEM into heap H is given as follows:

[1] First adjoin ITEM at the end of H so that H is still a complete tree, but necessarily a heap

[2] Let ITEM rise to its appropriate place in H so that H is finally a heap

48

Heap

97

88 95

66 55

66 35

18 40 30 26

48

24

55

95

62 77

48

25 38

97

88

95

66

55

95

48

66

35

48

55

62

77

25

38

18

40

30

26

24

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

Insert 70

70

49

Heap

97

88 95

66 55

66 35

18 40 30 26

48

24

55

95

62 77

48

25 38

97

88

95

66

55

95

48

66

35

48

55

62

77

25

38

18

40

30

26

24

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

Insert 70

70

50

Heap

97

88 95

66 55

66 35

18 40 30 26

70

24

55

95

62 77

48

25 38

97

88

95

66

55

95

48

66

35

48

55

62

77

25

38

18

40

30

26

24

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

Insert 70

48

51

Heap

97

88 95

66 70

66 35

18 40 30 26

55

24

55

95

62 77

48

25 38

97

88

95

66

55

95

48

66

35

48

55

62

77

25

38

18

40

30

26

24

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

Insert 70

48

52

Build a HeapBuild a heap from the following list

44, 30, 50, 22, 60, 55, 77, 55

53

44

44, 30, 50, 22, 60, 55, 77, 55

44

30

44

30 50

50

30 44

50

30 44

22

Complete the Rest Insertion

77

55 60

50 30

22

44 55

54

Deleting the Root of a Heap Suppose H is a heap with N elementsSuppose we want to delete the root R of HDeletion of root is accomplished as follows[1] Assign the root R to some variable ITEM[2] Replace the deleted node R by the last

node L of H so that H is still a complete tree but necessarily a heap

[3] Reheap. Let L sink to its appropriate place in H so that H is finally a heap.

55

95

85 70

55 33

15

30 65

20 15 22

22

85 70

55 33

15

30 65

20 15

85

22 70

55 33

15

30 65

20 15

85

55 70

22 33

15

30 65

20 15