1. CS 2511
Fall 2014
Binary Heaps

2. Disadvantage of Linear Priority Queues
Two Basic Types of Implementation:
Array
Both adding and removing takes O(n) time due to data movements
Linked List
Adding takes O(n) time due to searching

3. Representing a Heap 2 3 4 8 7 9 14 16 10 1 2 3 4 5 6 7 8 9 10 11 12
Array Index 2 4 3 8 7 9 14 16 10
Array Value

4. Adding to a Heap (Step 1) 2 Want to add 1 3 4 8 7 9 14 16 10 1 1 2 3 41 2 3 4 5 6 7 8 9 10 11 12
Array Index 2 4 3 8 7 9 14 16 10 1
Array Value

5. Adding to a Heap (Step 2) 2 Want to add 1 3 4 8 1 9 14 16 10 7 1 2 3 41 2 3 4 5 6 7 8 9 10 11 12
Array Index 2 4 3 8 1 9 14 16 10 7
Array Value

6. Adding to a Heap (Step 3) 2 Want to add 1 3 1 8 4 9 14 16 10 7 1 2 3 41 2 3 4 5 6 7 8 9 10 11 12
Array Index 2 1 3 8 4 9 14 16 10 7
Array Value

7. Adding to a Heap (Step 4) 1 Want to add 1 3 2 8 4 9 14 16 10 7 1 2 3 41 2 3 4 5 6 7 8 9 10 11 12
Array Index 1 2 3 8 4 9 14 16 10 7
Array Value

8. Removing from a Heap (Step 1)7 1
Want to remove 1 3 2 8 4 9 14 16 10 7 1 2 3 4 5 6 7 8 9 10 11 12
Array Index 7 1 2 3 8 4 9 14 16 10 7
Array Value

9. Removing from a Heap (Step 2)
Want to remove 1 3 7 8 4 9 14 16 10 1 2 3 4 5 6 7 8 9 10 11 12
Array Index 2 7 3 8 4 9 14 16 10
Array Value

10. Removing from a Heap (Step 3)2
Want to remove 1 3 4 8 7 9 14 16 10 1 2 3 4 5 6 7 8 9 10 11 12
Array Index 2 4 3 8 7 9 14 16 10
Array Value

11. Relationship between Maximum Size and Height
Height (h)
Maximum Size (n) 1 3 2 7 15 4 31 … h
2h+1-1

12. Efficiency of Binary Heaps
Number of Operations for Add or Remove limited by its height
Maximum height is log2(n+1) – 1, where n is the number of nodes
O(log(n))
n = 2h + 1 − 1
n + 1 = 2h + 1
log2(n + 1) = log2(2h + 1)
log2(n + 1) = (h + 1) * log2(2)
log2(n + 1) = (h + 1) * 1
log2(n + 1) = h + 1
h = log2(n + 1) – 1
Reminder: log2(n + 1) = log(n+1) / log(2)
True height is ceil(log2(n + 1) – 1)