1. Definition Applications Implementations Heap Comparison
Priority Queues
Definition
Applications
Implementations
Heap
Comparison
4/6/2019
CS 303 – Priority Queues
Lecture 12

2. Priority Queues Specialization of SET Operations
Insert(x,A) A  A U {x}
DeleteMin(A,x) x  Min(A)
MakeNull(A) A  A – {Min(A)}
cf: QUEUE – difference is that Insert can occur anywhere
4/6/2019
CS 303 – Priority Queues
Lecture 12

3. Priority Queue Applications
Scheduling
Simulation
Other “event driven” applications
move clock to next event and execute – DeleteMin
spawn new event in the future – Insert
Sorting(!)
A  NULL
for i  1 to n do Insert(x[i],A)
for i  1 to n do DeleteMin(A,x[i])
4/6/2019
CS 303 – Priority Queues
Lecture 12

4. Priority Queue Implementations
Unsorted List Sorted List
Insert: easy O(1) Insert: hard O(n/2)
DeleteMin: hard O(n/2) DeleteMin: easy O(1)
Unsorted Array Sorted Array
Insert: easy O(1) Insert: hard O(log n + n/2)
DeleteMin: hard O(n/2) DeleteMin: easy O(1) [how?]
Why keep the entire list sorted, just to find min?
Partial Order is a better idea! a
a < b
a < c
b ? c b c
4/6/2019
CS 303 – Priority Queues
Lecture 12

5. Heap Partially Ordered Binary Tree HeapShape Balanced
Missing nodes are at leaf level and to the far right
DeleteMin
x  a
Need to replace root
1) find rightmost leaf = r
2) swap r with the hole at the root
3) push r down until
r < b, r < c a O b c b c r b c
if (b<c) & (r<b) SWAP(r,b) and continue down...
4/6/2019
CS 303 – Priority Queues
Lecture 12

6. Heap ... Insert into Heap 1) add x as the new rightmost leaf
2) push x UP until p < x
(or x == root)
Q/ do you need to push p down?
A/ no!
Why? Because p was already
smaller than everything in both
subtrees below p. When it becomes the root of one subtree,
it is still smaller than both children x p s x
if (x < p) SWAP(x,p) and continue up...
4/6/2019
CS 303 – Priority Queues
Lecture 12

7. Array Implementation of Heap (a classic hack!)1 a b c d e f g h i j a c b d e f g h i j
RightMostLeaf
No pointers (they are implicit)
Easy to find/create/remove RightMostLeaf
Insert & Delete are O(log n) (but Member is O(n), so beware)
4/6/2019
CS 303 – Priority Queues
Lecture 12

8. Implementation Comparison Chart
Binary Search Tree Hash Table Heap
(average case only!)
Insert O(log n) O(n/b) O(log n)
Delete O(log n) O(n/b) [ O(n) ]
DeleteMin O(log n) O(n) O(log n)
Member O(log n) O(n/b) [ O(n) ]
Don’t use Heap if you need Delete or Member!
Don’t use Hash Table if you need DeleteMin!
If you use a BST – beware of worst-case behavior!
4/6/2019
CS 303 – Priority Queues
Lecture 12