Binary Heaps Text Binary Heap Building a Binary Heap 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 4 12 5 26 25 14 15 29 45 35 31 21 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 4 12 5 26 25 14 15 29 45 35 31 21 3 Insert 3 insert here to keep tree complete 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 = 13 Insert 3

Inserting a Value 4 12 5 26 25 14 15 29 45 35 31 21 3 Insert 3 save new value in a temporary location: tmp  3 Insert 3

Inserting a Value 4 12 5 26 25 14 15 29 45 35 31 21 14 3 Insert 3  copy 14 down because 14 > 3 3 tmp  Insert 3

Inserting a Value 4 12 5 26 25 12 15 29 45 35 31 21 14 3 Insert 3 copy 12 down because 12 > 3 3 tmp  Insert 3

Inserting a Value 4 4 5 26 25 12 15 29 45 35 31 21 14 3 Insert 3 copy 4 down because 4 > 3 3 tmp  Insert 3

Inserting a Value 3 4 5 26 25 12 15 29 45 35 31 21 14 Insert 3

Heap After Insert 3 4 5 26 25 12 15 29 45 35 31 21 14

Deleting a Value (note new tree!) 3 7 5 26 25 12 15 29 45 35 31 21 14 Delete 3

Deleting a Value 7 5 26 25 12 15 29 45 35 31 21 14 3 Delete 3 save root value … tmp  3 Delete 3

X Deleting a Value 14 7 5 26 25 12 15 29 45 35 31 21 14 3 Delete 3 copy value of last node in complete tree into temporary location; decrement currentsize 14 7 5 26 25 12 15 X 29 45 35 31 21 14 tmp  3 Delete 3

compare children select smaller Deleting a Value 14 push down root … compare children select smaller 7 5 26 25 12 15 29 45 35 31 21 tmp  3 Delete 3

copy smaller value into parent Deleting a Value 14 5 push down root … copy smaller value into parent 7 26 25 12 15 29 45 35 31 21 tmp  3 Delete 3

compare children select smaller (25) Deleting a Value 14 5 push down root … 7 26 25 12 15 29 45 35 31 21 compare children select smaller (25) tmp  3 Delete 3

copy 14 into parent because 14 < smaller child Deleting a Value 5 push down root … 7 14 26 25 12 15 29 45 35 31 21 copy 14 into parent because 14 < smaller child tmp  3 Delete 3

Deleting a Value 5 7 14 26 25 12 15 29 45 35 31 21 return 3 Delete 3