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Overview
• Priority Queue
• Priority Queue Examples
• Heap
• Heap Implementation
• Building a Heap
• Heap Operations
• Variations of Heap
Priority Queue
• A priority queue is a queue with … err … priorities!
• Special case:• Queues• Stacks
P-3Memory Address
P-2Memory Address
P-1Memory Address
PMemory Address
PMemory Address
P-1Memory Address
P-2Memory Address
P-3Memory Address
Stack Queue
Priority Queue
• Operations• Insert (push, enqueue)
• With priority
• Remove (pop, dequeue)• With priority
• isEmpty• inspectFirst (peek)
Priority Queue
• Implementation• Array• Linked list• Heap
Priority Queue
12 12 45 5 23 89
Priority Queue Examples
• A*
• Discrete Event Simulator
Priority Queue Examples
• Path finding in robotics
Priority Queue Examples
• Past path cost function• Know distance from start
• Future path cost function• Heuristics
• Priority queue• Possible route segments
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_neighboursdo 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
Priority Queue Examples • Discrete Event Simulator
• Event + time
Heap
• Tree data structure• Often a binary tree
• Ordered relationship between child and parent• Not between siblings• Heap property
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Heap
• Max / Min heap67
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Heap
• Operations• Create• Insert• Delete (max or min)• isEmpty• Extract max or min
Heap Implementation
• Usually as an array
• Binary tree
• Almost complete
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3 1 4 2
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Priority queue as a heap
• Root at index 1
• Left at index 2
• Right index 3
• For node n, children at 2n and 2n+1
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3
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7 8 12 18
1 5
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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
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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 10 19 7 8 12 18 1 5 3 6 171719 17
Building a Heap
• Build a heap
• Heapify
• HeapSort
Building a Heap
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11 10 8 4 2
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4 2
Building a Heap
• In a binary tree, where do the leaves start?
• + 1 to n
• n = 5, leaves start at 3
• n = 6, leaves start at 4
• Every leaf is a heap with one node!
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4 2 1
Building a Heap
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Building a Heap
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)
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30 125
30
Building a Heap
BuildMaxHeap(A)
size[A] = length[A]
for i = to i
MaxHeapify(A, i)
Building a Heap
• Complexity of BuildHeap
• MaxHeapify is applied to all nodes at a given level• How many nodes at each level?
• Height h = 0, n= 7, number of nodes = 4
• Height h = 1, n= 7, number of nodes = 2
Building a Heap
• Complexity at h
•O(h)• (ch)
•O =
• O(n)
Heap Operations
• Extract Max
• Insert
Heap Operations
ExtractMax(A)
max A[1]
A[1] A[A.heap_size]
A.heap_size A.heap_size – 1
MaxHeapify(A, 1)
return max
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2 1 5 3
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Heap Operations
• O(h)
• O(logn)
Heap Operations
• Insert
• Insert at the end and copy upwards till heap property is satisfied
• Worst case O(logn)
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5 7 3 2
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Heap Operations
HeapSort(A, n)
BuildMaxHeap(A)
for i A.size to 2
swap(A[A.last], A[i])
MaxHeapify(A, i -1)
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Heap Operations
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)
O(logn)
23
8 12
1 3 6 7
Variations of Heap
• Fibonacci heaps
• Binomial heaps
Variations of Heap
• Fibonacci heaps
• Set of trees
• Minimum trees
• Fibonacci numbers used in the run time analysis
5 6
1
5
5 5
5
Variations of Heap
• Fibonacci heaps have no predefined order
• However, number of children are low
• A child is root of a subtree at k
Questions?
