Binary HeapsText• Read Weiss, §21.1 - 21.4

Binary Heap • One-array representation of a tree• Complete trees

Building a Binary Heap• Insert• Delete

Motivation

• Development of a data structure which allows efficient inserts and efficient deletes of the minimum value (minheap) or maximum value (maxheap)

Implementation

• One-array implementation of a binary tree

• Root of tree is at element 1 of the array

• If a node is at element i of the array, then its children are at elements 2*i and 2*i+1

• If a node is at element i of the array, then its parent is at element floor(i/2)=└ i/2┘

Implementation

29 45 35 31 21

4

125

26 25 14 15

i 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

array __ 4 5 12 26 25 14 15 29 45 35 31 21 __ __ __

currentsize = 12

Implementatation

• Heap must be a complete tree

• all leaves are on the lowest two levels

• nodes are added on the lowest level, from left to right

• nodes are removed from the lowest level, from right to left

Inserting a Value

29 45 35 31 21

4

125

26 25 14 15

i 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

array __ 4 5 12 26 25 14 15 29 45 35 31 21 3 __ __

currentsize = 13Insert 3

3 insert here to keep tree complete

Inserting a Value

29 45 35 31 21

4

125

26 25 14 15

Insert 3

save new value in a temporary location: tmp 3

Inserting a Value

29 45 35 31 21

4

125

26 25 14 15

Insert 3

14

tmp 3

copy 14 down because 14 > 3

Inserting a Value

29 45 35 31 21

4

125

26 25 12 15

Insert 3

14

tmp 3

copy 12 down because 12 > 3

Inserting a Value

29 45 35 31 21

4

45

26 25 12 15

Insert 3

14

tmp 3

copy 4 down because 4 > 3

Inserting a Value

29 45 35 31 21

3

45

26 25 12 15

Insert 3

14

insert 3

Heap After Insert

29 45 35 31 21

3

45

26 25 12 15

14

Deleting a Value (note new tree!)

29 45 35 31 21

3

75

26 25 12 15

14

Delete 3

Deleting a Value

29 45 35 31 21

75

26 25 12 15

14

Delete 3

3save root value … tmp

Deleting a Value

29 45 35 31 21

75

26 25 12 15

14

Delete 3

3tmp

X

copy value of last node in complete tree into temporary location; decrement currentsize

14

Deleting a Value

29 45 35 31 21

75

26 25 12 15

Delete 3

3tmp

push down root …

compare children select smaller

14

Deleting a Value

29 45 35 31 21

5

7

26 25 12 15

Delete 3

3tmp

push down root …

copy smaller value into parent

14

Deleting a Value

29 45 35 31 21

5

7

26 25 12 15

Delete 3

3tmp

push down root … 14

compare children select smaller (25)

Deleting a Value

29 45 35 31 21

5

714

26 25 12 15

Delete 3

3tmp

push down root …

copy 14 into parent because 14 < smaller child

Deleting a Value

29 45 35 31 21

5

714

26 25 12 15

Delete 3

3return