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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.

In a more general d-Heap– A parent node can have d children

We simply refer to binary heaps as heaps

0

3 2

4 5

root

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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]

R

l r

ll lr rrrl

root

rrrllrllrlR

7654321

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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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13 14 16 19 21 19 68 65 26 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(1)

percolateDown(4) percolateDown(3)

percolateDown(2)

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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=1

h 2h-i i = 2h i=1h i/2i

– = 2h O(1) = O(2h) = O(N) Proof for i=1

h i/2i = O(1)– i/2i ≤ ∫i-1

i (x+1)/2x dx i=1

h i/2i ≤ i=1∞ i/2i ≤ ∫0

∞ (x+1)/2x dx Note: ∫u dv = uv - ∫v du, with dv = 2-x 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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