308-203A Introduction to Computing II Lecture 10: Heaps

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308-203AIntroduction to Computing IILecture
10 Heaps
Fall Session 2000
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Motivation

• Data structures supporting extraction of max
  element
• are quite useful, for example
• Priority queues - list of tasks to perform with
  priority
• (always take highest priority task)
• Event simulators, e.g. video games
• (always simulate the nearest event in the future)

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Heap
Extract-Max()
Heap
Insert(object, key)
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How could we do this?

• Sorted list but recall that Insertion-Sort
• was suboptimal
• Binary tree but binary trees can become
• unbalanced and costly
• A more clever way a special case of binary
  trees

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Arrays as Binary Trees
Take an array of n elements A1..n For an
index i e 1 .. n define Parent(i)
i/2 Left-child(i) 2i Right-child(i) 2i
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Example (as array)
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Example (as tree)
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Facts

• These trees are always balanced

• Easy to find next leaf to add
• (element n1 in the array)

• Not as flexible as trees built with pointers

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The Heap Property
For any node X with parent PARENT(X)
PARENT(X).key gt X.key
Compare to the Binary Search Tree Property
from last lecture this is a much weaker condition
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Example
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Heap-Insert
Heap-Insert(A1..n, k) A.length
An1 k j n1 while (j ? 1 and
Aj gt APARENT(j) swap(A, j,
PARENT(j)) j PARENT(j)
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Heap-Insert Example
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swap
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Heap-Insert Example
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swap
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Heap-Insert Example
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Done (15 lt 16)
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Helper routine Heapify
Given a node X, where Xs children are
heaps, guarantee the heap property for the
ensemble of X and its children
X
Left heap
Right heap
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Heapify
Heapify(A1..n, i) l Left(i) r
Right(i) if (Ai gt Al and Ai gt Ar )
return if (Al gt Ar) swap(Ai,
Al) Heapify(A, l) else swap(Ai,
Al) Heapify(A, l)
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Heapify example
HEAP PROPERTY VIOLATED
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Heapify example
Swap with max(14, 6, 10)
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Heapify example
Swap with max(8, 6, 7)
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Heapify example
Done 6 max(2, 6, 4)
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Heap-Extract-Max
Heap-Extract-Max(A1..n) returnValue
A1 A1 An heap.length --
Heapify(A1..(n-1), 1) return
returnValue
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Heap-Extract-Max example
returnValue 16
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Heap-Extract-Max example
Replace with An
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Heap-Extract-Max example
Heapify from top
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Order of Growth
All three, proportional to height of tree
O( log n ) O( log n ) O( log n )
Heap-insert Heapify Heap-Extract-Max
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Other useful operations

• Build-Heap convert an unordered array into a
  heap
• O( n )
• Heap-Sort sort by removing elements from the
  heap
• until it is empty
• O( n log n )

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Build-Heap
Build-Heap(A1..n) for i ?n/2? downto
1 Heapify(A, i)
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Heap-Sort
Heap-Sort(A1..n) result new
array1..n for i n downto 1 resulti
Heap-Extract-Max (A1..i) return
result
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Any questions?