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CMSC 132:
Object-Oriented Programming II
Heaps & Priority Queues
Department of Computer Science
University of Maryland, College Park
Overview
Binary trees
Full, Perfect, Complete
Heaps
Insert
getSmallest
Heap applications
Heapsort
Priority queues
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Heaps and Priority Queues: A Comprehensive Guide, Study notes of Computer Science
An in-depth exploration of heaps and priority queues, essential data structures in computer science. Topics covered include binary trees, heap properties, heap operations, heap applications, and heap implementation. Learn about the key differences between heaps and binary search trees, and discover how heaps are used in sorting algorithms and priority queues.
Typology: Study notes
Pre 2010
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CMSC 132:
Object-Oriented Programming II
Heaps & Priority Queues
Department of Computer Science
University of Maryland, College Park
Overview
Binary trees
Full, Perfect, Complete
Heaps
Insert getSmallest
Heap applications
Heapsort Priority queues
Full Binary Tree
Binary tree where all nodes have 0 or 2 children
Not Allowed
Perfect Binary Tree
A full binary tree where
All leaves at level h for tree of height h
h = 2 h = 3 h = 4
h = 1
Heaps
Two key properties
Complete binary tree Value at node Smaller than or equal to values in subtrees
Example heap
X ≤ Y X ≤ Z
Y
X
Z
Heap & Non-heap Examples
Heaps Non-heaps
Heap Properties
Heaps are balanced trees
Height = log 2 (n) = O(log(n))
Can find smallest element easily
Always at top of heap!
Can organize heap to find maximum value
Value at node larger than values in subtrees Heap can track either min or max, but not both
Heap
Key operations
Insert ( X ) getSmallest ( )
Key applications
Heapsort Priority queue
Heap Insert Example
Insert ( 8 )
- Insert toend of tree swap if parent key larger2) Compare to parent, complete3) Insert
Heap Operation – getSmallest()
Algorithm
- Get smallest node at root
- Replace root with X at end of tree
- While ( X > child ) Swap X with smallest child // X drops down tree
- Return smallest node
Complexity
swaps proportional to height of tree
O( log(n) )
Heap GetSmallest Example
getSmallest ()
with end of tree1) Replace root children, if larger swap2) Compare node to with smallest child
- Repeat swapif needed
Heap GetSmallest Example
getSmallest ()
with end of tree1) Replace root children, if larger swap2) Compare node to with smallest child
- Repeat swapif needed
Heap Implementation
Calculating node locations
Array index i starts at 0 Parent(i) = ⎣ ( i – 1 ) / 2 ⎦ LeftChild(i) = 2 × i + RightChild(i) = 2 × i +
Heap Implementation
Example
Parent(1) = ⎣ ( 1 – 1 ) / 2 ⎦ = ⎣ 0 / 2 ⎦ = 0 Parent(2) = ⎣ ( 2 – 1 ) / 2 ⎦ = ⎣ 1 / 2 ⎦ = 0 Parent(3) = ⎣ ( 3 – 1 ) / 2 ⎦ = ⎣ 2 / 2 ⎦ = 1 Parent(4) = ⎣ ( 4 – 1 ) / 2 ⎦ = ⎣ 3 / 2 ⎦ = 1 Parent(5) = ⎣ ( 5 – 1 ) / 2 ⎦ = ⎣ 4 / 2 ⎦ = 2
Heap Implementation
Example
LeftChild(0) = 2 × 0 +1 = 1 LeftChild(1) = 2 × 1 +1 = 3 LeftChild(2) = 2 × 2 +1 = 5
Heap Implementation
Example
RightChild(0) = 2 × 0 +2 = 2 RightChild(1) = 2 × 1 +2 = 4
Heapsort – Insert Values
Heapsort – Remove Values
Input
Input
- Heapsort – Insert in to Array
- 11, 5, 13, 6, - Index = - Insert
- Heapsort – Insert in to Array
- 11, 5, 13, 6, Input - Index = - Insert - Swap
- Heapsort – Remove from Array
- 11, 5, 13, 6,
- Index =
- 11, 5, 13, 6,
- Heapsort – Remove from Array
- Remove root
- Replace
- Swap w/ child
- Heapsort – Remove from Array
- 11, 5, 13, 6, Input
- Index =
- 11, 5, 13, 6, Input
- Heapsort – Remove from Array
- Remove root
- Replace
- Swap w/ child
Heap Application – Priority Queue
Queue
Linear data structure First-in First-out (FIFO) Implement as array / linked list Enqueue Dequeue
Heap Application – Priority Queue
Priority queue
Elements are assigned priority value Higher priority elements are taken out first Equal priority elements are taken out in FIFO order Implement as heap Enqueue ⇒ insert( ) Dequeue ⇒ getSmallest( ) (^) Dequeue
Enqueue