1. Dr. Andrew Wallace PhD BEng(hons) EurIng andrew.wallace@cs.umu.se
Priority Queue / Heap
Dr. Andrew Wallace PhD BEng(hons) EurIng

2. Overview Priority Queue Priority Queue Examples Heap
Heap Implementation
Building a Heap
Heap Operations
Variations of Heap

3. Priority Queue A priority queue is a queue with … err … priorities!
Special case:
Queues
Stacks
Memory Address
Memory Address P
P-3
Memory Address
Memory Address
P-1
P-2
Memory Address
Memory Address
P-2
P-1
Memory Address
Memory Address
P-3 P
Queue
Stack

4. Priority Queue Operations Insert (push, enqueue) Remove (pop, dequeue)
With priority
Remove (pop, dequeue)
isEmpty
inspectFirst (peek)

5. Priority Queue
Implementation
Array
Linked list
Heap

6. Priority Queue 12 12 45 5 23 89

7. Priority Queue Examples
Discrete Event Simulator

8. Priority Queue Examples
Path finding in robotics

9. Priority Queue Examples
Past path cost function
Know distance from start
Future path cost function
Heuristics
Priority queue
Possible route segments

10. Priority Queue Examples
A*(G, start) create priority queue F create came_from create cost_so_far F.put(start) came_from[start] ← NULL cost_so_far ← 0 While not F.empty() n ← F.get() if n = goal return for next ← G.number_of_neighbours do new_cost ← cost_so_far[n] + G.cost(n, next) if next not in cost_so_far or new_cost < cost_so_far[next] cost_so_far[next] = new_cost priority = new_cost + Heuristic(goal, next) F.put(next, priority) came_from[next] ← current Return came_from, cost_so_far

11. Priority Queue Examples
Discrete Event Simulator
Event + time

12. Heap Tree data structure Ordered relationship between child and parent67
Tree data structure
Often a binary tree
Ordered relationship between child and parent
Not between siblings
Heap property 13 8 3 1

13. Heap 67 1
Max / Min heap 13 8 13 8 3 1 30
100

14. Heap Operations Create Insert Delete (max or min) isEmpty
Extract max or min

15. Heap Implementation Usually as an array Binary tree Almost complete 6713 8 67 13 8 9 10 6 5 3 1 4 2 9 10 6 5 3 1 4 2

16. Priority queue as a heap20
Root at index 1
Left at index 2
Right index 3
For node n, children at 2n and 2n+1 10 19 7 8 12 18 1 5 3 6 20 10 19 7 8 12 18 1 5 3 6

17. Priority queue as a heap
Insert
Add at end
New leaf on the tree
Fix the priorities
Compare with parent
If smaller, swap
Continue until priorities are fixed 20 10 19 7 8 17 12 18 1 5 3 6 12 17

18. Priority queue as a heap
RemoveMax
Return data at index 1
Copy end of array to index 1
Compare and swap to restore priorities 20 17 19 10 17 19 7 8 12 18 1 5 3 6 17

19. Building a Heap
Build a heap
Heapify
HeapSort

20. Building a Heap 11 10 8 2 4 10 8 11 4 2 11 10 8 4 2

21. Building a Heap In a binary tree, where do the leaves start?11
In a binary tree, where do the leaves start?
𝒏 𝟐 to n
n = 5, leaves start at 3
n = 6, leaves start at 4
Every leaf is a heap with one node! 10 8 4 2 1

22. Building a Heap 1 3 6 9 12 67
211
211 1 12 3
211 1 6 67 9 3 12 67 1
211 6

23. Building a Heap 30 5
MaxHeapify(A, i) l ← 2i r ← 2i+1 if l <= A.size and A[l] > A[i] then largest ← l else largest ← i if r <= A.size and A[r] > A[largest] then largest = r if largest not = i then swap(A[i], A[largest]) MaxHeapify(A, largest) 5 30 12

24. Building a Heap
BuildMaxHeap(A) size[A] = length[A] for i = length[A] to i MaxHeapify(A, i)

25. Building a Heap 𝒏 𝟐 𝒉+𝟏 Complexity of BuildHeap
MaxHeapify is applied to all nodes at a given level
How many nodes at each level?
𝒏 𝟐 𝒉+𝟏
Height h = 0, n= 7, number of nodes = 4
Height h = 1, n= 7, number of nodes = 2

26. Building a Heap 𝒏 𝟐 𝒉+𝟏 O(h) 𝒉=𝟎 𝒍𝒐𝒈𝒏 𝒏 𝟐 𝒉+𝟏 (ch) 𝒄𝒏 𝟐 𝒉=𝟎 𝒍𝒐𝒈𝒏 𝒉 𝟐𝒉
Complexity at h
𝒏 𝟐 𝒉+𝟏 O(h)
𝒉=𝟎 𝒍𝒐𝒈𝒏 𝒏 𝟐 𝒉+𝟏 (ch)
𝒄𝒏 𝟐 𝒉=𝟎 𝒍𝒐𝒈𝒏 𝒉 𝟐𝒉
O 𝒄𝒏 𝟐 𝒉=𝟎 ∞ 𝒉 𝟐𝒉 =
O(n)

27. Heap Operations
Extract Max
Insert

28. Heap Operations 3 10 8
ExtractMax(A) max ← A[1] A[1] ← A[A.heap_size] A.heap_size ← A.heap_size – 1 MaxHeapify(A, 1) return max 8 3 6 2 1 5 3 10

29. Heap Operations
O(h)
O(logn)

30. Heap Operations Insert10 8 6
Insert
Insert at the end and copy upwards till heap property is satisfied
Worst case O(logn) 5 7 3 2 12

31. Heap Operations 1 8 2 5 1 2 10 1
HeapSort(A, n) BuildMaxHeap(A) for i ← A.size to 2 swap(A[A.last], A[i]) MaxHeapify(A, i -1) 8 2 1 2 1 5 5 8 1 10 2 1 2 5 8 10

32. Heap Operations O(logn)
Heap Increase Key(A, i, key) A[i] ← key while i > 1 and A.parent(i) < A[i] do swap(A[i], A.parent(i)) i ← parent(i) 23 8 12 1 3 6 7

33. Variations of Heap
Fibonacci heaps
Binomial heaps

34. Variations of Heap Fibonacci heaps Set of trees Minimum trees
Fibonacci numbers used in the run time analysis 1 5 5 5 6 5 5

35. Variations of Heap Fibonacci heaps have no predefined order
However, number of children are low
A child is root of a subtree at k

36. Questions?