1. Priority Queues (Heaps)
Sections 6.1 to 6.5

2. The Priority Queue ADT DeleteMin log N time Insert Other operations
FindMin
Constant time
Initialize
N time 2 2

3. Applications of Priority Queues
Any event/job management that assign priority to events/jobs
In Operating Systems
Scheduling jobs
In Simulators
Scheduling the next event (smallest event time)

4. Priority Queue Implementation
Implemented as adaptor class around
Linked lists
O(N) worst-case time on either insert() or deleteMin()
Binary Search Trees
O(log(N)) average time on insert() and delete()
Overkill: all elements are sorted
However, we only need the minimum element
Heaps
This is what we’ll study and use to implement Priority Queues
O(logN) worst case for both insertion and delete operations

5. Partially Ordered Trees
A partially ordered tree (POT) is a tree T such that:
There is an order relation <= defined for the vertices of T
For any vertex p and any child c of p, p <= c
Consequences:
The smallest element in a POT is the root
No conclusion can be drawn about the order of children

6. Binary Heaps A binary heap is a partially ordered complete binary tree
The tree is completely filled on all levels except possibly the lowest.
In a more general d-Heap
A parent node can have d children
We simply refer to binary heaps as heaps
root 3 2 4 5

7. Vector Representation of Complete Binary Tree
Storing elements in vector in level-order
Parent of v[k] = v[k/2]
Left child of v[k] = v[2*k]
Right child of v[k] = v[2*k + 1]
root R l r ll lr rl rr 7 6 5 4 3 2 1 R l r ll lr rl rr

8. Heap example Parent of v[k] = v[k/2] Left child of v[k] = v[2*k]
Right child of v[k] = v[2*k + 1]

9. Examples
Which one is a heap?

10. Implementation of Priority Queue (heap)

11. Insertion Example: insert(14)

12. Basic Heap Operations: insert(x)
Maintain the complete binary tree property and fix any problem with the partially ordered tree property
Create a leaf at the end
Repeat
Locate parent
if POT not satisfied
Swap with parent
else
Stop
Insert x into its final location

13. Implementation of insert

14. deleteMin() example 31 13 14 16 19 21 68 65 26 32 31

15. deleteMin() Example (Cont’d)31 31 31

16. Basic Heap Operations: deleteMin()
Replace root with the last leaf ( last element in the array representation
This maintains the complete binary tree property but may violate the partially ordered tree property
Repeat
Find the smaller child of the “hole”
If POT not satisfied
Swap hole and smaller child
else
Stop

17. Implementation of deleteMin()

18. Implementation of deleteMin()

19. Constructor Construct heap from a collection of item How to?
Naïve methods
Insert() each element
Worst-case time: O(N(logN))
We show an approach taking O(N) worst-case
Basic idea
First insert all elements into the tree without worrying about POT
Then, adjust the tree to satisfy POT, starting from the bottom

20. Constructor

21. Example
percolateDown(7)
percolateDown(6)
percolateDown(5)

22. percolateDown(4)
percolateDown(3)
percolateDown(2)
percolateDown(1)

23. Complexity Analysis Consider a tree of height h with 2h-1 nodes
Time = 1•(h) + 2•(h-1) + 4•(h-2) h-1•1
= i=1h 2h-i i = 2h i=1h i/2i
= 2h O(1) = O(2h) = O(N)
Proof for i=1h i/2i = O(1)
i/2i ≤ ∫i-1i (x+1)/2x dx
i=1h i/2i ≤ i=1∞ i/2i ≤ ∫0∞ (x+1)/2x dx
Note: ∫u dv = uv - ∫v du, with dv = 2x and u = x and ∫2-x dx = -2-x/ln 2
∫0∞ (x+1)/2x dx = -x 2-x/ln 2|0∞ + 1/ln 2 ∫0∞ 2-x dx + ∫0∞ 2-x dx
= -2-x/ln2 2|0∞ - 2-x/ln 2|0∞ = (1 + 1/ln 2)/ln 2 = O(1) 23 23

24. Alternate Proof Prove i=1h i/2i = O(1)
i=0∞ xi = 1/(1-x) when |x| < 1
Differentiating both sides with respect to x we get
i=0∞ i xi-1 = 1/(1-x)2
So, i=0∞ i xi = x/(1-x)2
Substituting x = 1/2 above gives
i=0∞ i 2-i = 0.5/(1-0.5)2 = 2 = O(1) 24 24

25. C++ STL Priority Queues
priority_queue class template
Implements deleteMax instead of deleteMin in default
MaxHeap instead of MinHeap
Template
Item type
container type (default vector)
comparator (default less)
Associative queue operations
Void push(t)
void pop()
T& top()
void clear()
bool empty()