Priority Queues (Heaps)
 Sections 6.1 to 6.5
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The Priority Queue ADT
 DeleteMin
– log N time
 Insert
– log N time
 Other operations
– FindMin
 Constant time
– Initialize
 N time
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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)
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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
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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
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Binary Heaps
 A binary heap is a partially ordered complete binary tree
– The tree is completely filled on all levels except possibly the
lowest.
root
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0
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 In a more general d-Heap
– A parent node can have d children
 We simply refer to binary heaps as heaps
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Vector Representation of Complete Binary Tree
 Storing elements in vector in level-order
– Parent of v[k] = v[k/2]
root
– Left child of v[k] = v[2*k]
– Right child of v[k] = v[2*k + 1]
l
ll
R
r
lr
rl
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R
l
r
ll
lr
rl
rr
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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]
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Examples
Which one is a heap?
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Implementation of Priority Queue (heap)
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Insertion Example: insert(14)
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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
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Implementation of insert
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deleteMin() example
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16 19 21
19 68
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32 31
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deleteMin() Example (Cont’d)
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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
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Implementation of deleteMin()
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Implementation of deleteMin()
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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
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Constructor
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Example
percolateDown(7)
percolateDown(6)
percolateDown(5)
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percolateDown(4)
percolateDown(2)
percolateDown(3)
percolateDown(1)
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Complexity Analysis
 Consider a tree of height h with 2h-1 nodes
– Time = 1•(h) + 2•(h-1) + 4•(h-2) + ... + 2h-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)
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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)
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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()
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