Download heap property

GRIFFITH
COLLEGE
DUBLIN
Data Structures, Algorithms &
Complexity
The Heap Data Structure
Lecture 7
1
Introduction
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The binary heap data structure is a data structure that can be
viewed as complete binary tree
A binary tree is a tree in which each node can have a left-child and
a right-child only
A complete binary tree is a tree which is completely filled at all
levels except possibly the lowest level, which is filled from left to
right
Because of its nature a heap can very easily be implemented using
an array
An array A that represents a heap is an array with two attributes
 length[A], the number of elements in the array
 heap-size[A], the number of heap elements stored in the array
Lecture 7
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Heap as Self-Referencing Structure
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Node has attributes, value, leftchild, rightchild and parent
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Lecture 7
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Heap as an Array
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Viewed as a binary tree and as an array
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The root of the tree is stored at A[0], its left-child at A[1], its
right child at A[2] etc.
Lecture 7
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Accessing the Heap Values
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Given the index i of a node, the indices of its parent Parent(i),
left-child Left(i) and right child Right(i) can be computed simply
Parent(i)
return i/2
Left(i)
return 2i
Right(i)
return 2i+1
 Heaps must also satisfy the heap property for every node, i,
other than the root.
A[Parent(i)]  A[i]
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Therefore, the largest element in a heap is stored at the root,
and the subtrees rooted at a node contain smaller values than
does the node itself.
Lecture 7
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Heap Operations
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The height of a node in a tree is the number of
edges on the longest simple downward path from
the node to a leaf.
The height of an n-element heap based on a
binary tree is (lg n)
The basic operations on heaps run in time at most
proportional to the height of the tree and thus
take O(lg n) time.
Lecture 7
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Maintaining the Heap Property
 Heapify is an important subroutine for manipulating heaps.
 Its inputs are an array A and an index i into the array.
 When Heapify is called, it is assumed that the binary trees
rooted at Left(i) and Right(i) are heaps, but that A[i] may be
smaller than its children, thus violating the heap property.
 The function of Heapify is to let the value at A[i] ‘float down’ in
the heap so that the subtree rooted at index i becomes a heap.
 The action required from Heapify is as follows:
Lecture 7
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Example of Heapify
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Lecture 7
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The Heapify Algorithm
Heapify(A, i)
left = Left(i)
right = Right(i)
if left  heap_size[A] and A[left] > A[i] then
largest = left
else
largest = i
endif
if right  heap-size[A] and A[right] > A[largest] then
largest = right
if largest <> i then
exchange A[i] and A[largest]
Heapify(A, largest)
endif
endalg
Lecture 7
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The Heapify Algorithm
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At each step, the index of the largest of the elements A[i],
A[Left(i)], and A[Right(i)] is stored in the variable largest.
If A[i] is largest, then the subtree rooted at node i is a heap and
the procedure ends.
Otherwise, one of the two children has the largest element, and
A[i] is swapped with A[largest], which causes node i and its
children to satisfy the heap property.
The node largest, however, now has the original value A[i], and
thus the subtree rooted at largest may violate the heap
property.
Therefore, Heapify must be called recursively on that subtree.
Lecture 7
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Analysis of Heapify
 The running time of Heapify on a subtree of size n rooted at a
given node i is the (1) time to fix up the relationships among
the elements A[i], A[Left(i)] and A[Right(i)], plus the time to run
Heapify on a subtree rooted at one of the children of node i.
 The subtrees can have size of at most 2n/3, and the running
time of Heapify can therefore be described by the recurrence,
T(n)  T(2n/3) + (1)
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and the solution to this works out as,
T(n) = O(log n)
Lecture 7
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Building a Heap
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We can use the Heapify procedure in a bottom up
manner to convert an array A[1..n], where n =
length[A], into a heap.
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Now, since the subarray A[(n/2 + 1)..n] are all
leaves of the tree, each is a 1-element heap.
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The procedure Build-Heap goes through the
remaining nodes and runs Heapify on each.
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The order of processing guarantees that the subtrees
rooted at children of node i are heaps before Heapify
is run at that node.
Lecture 7
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Build Heap Algorithm
Build-Heap(A)
heap-size[A] = lenght[A]
for i = length[A]/2 downto 1
Heapify(A, i)
endfor
endalg
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Each call to Heapify costs O(lg n) time, and there are O(n) such
calls.
Thus, the running time is at most O(n lg n)
A more complex analysis, however, gives us a tighter upper
bound of O(n).
Hence, we can build a heap from an unordered array in linear
time.
Lecture 7
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Example of Build Heap
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Lecture 7
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Priority Queue
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Many applications require that we process records with keys in
order, but not necessarily in full sorted order
Items in a priority queue are not processed strictly in order of
entry into the queue
Items can be placed on the queue at any time, but processing
always takes the item with the largest key
The priority queue is a generalised queue Abstract Data
Structure (ADT)
In fact, the priority queue is a proper generalisation of the stack
and the queue ADTs, as we can implement either with priority
queues, using appropriate priorities
Lecture 7
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Definition
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A priority queue is a data structure of items with keys that
support two basic operations : insert a new item, and delete the
item with the largest key.
Applications of priority queues include:
 Simulation Systems, where events need to be processed
chronologically.
 Job scheduling in computer systems.
The Heap Data Structure is an efficient structure for
implementing a simple priority queue
In practice, priority queues are more complex than the simple
definition above.
Lecture 7
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Priority Queue
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One of the reasons that priority queue implementations are so
useful is their flexibility
For that reason any implementation will need to support some
of the following operations
 Construct a priority queue from N given items
 Insert a new item
 Delete the maximum item
 Change the priority of an arbitrary specified item
 Delete an arbitrary specified item
We take “maximum” to mean “any record with the largest key
value”.
Lecture 7
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Heap Extract
 As with many data structures, we also need to add standard
initialise, test if empty and perhaps destroy and copy operations
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Lets look at insert and delete maximum.
Heap-Extract-Max(A)
if heap-size[A] < 1 then
error “heap underflow”
endif
max = A[1]
A[1] = A[heap-size[A]]
heap-size[A] = heap-size[A] – 1
Heapify(A, 1)
return max
endalg
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The running time of Heap-Extract-Max is O(lg n), since it
performs only a constant amount of work on top of the O(lg n)
time for Heapify
Lecture 7
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Heap Insert
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Heap-Insert inserts a node into heap A. Expand by adding a
new leaf to the tree. Traverse a path from this leaf toward the
root to find a place for the new element.
Heap-Insert(A, key)
heap-size[A] = heap-size[A] + 1
i = heap-size[A]
while i > 1 and A[Parent(i)] < key do
A[i] = A[Parent(i)]
i = Parent(i)
endwhile
A[i] = key
endalg
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Running time of on an n-element heap is O(lg n) since the path
traced from the new leaf to the root has length O(lg n)
This is an example of a non-recursive sift up procedure.
Lecture 7
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Example Heap Insert (15)
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(a) is the heap before insert, (b) add a leaf (c) Copy from leaf to root until place
for new item found(d) Add new item
Lecture 7
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Heap Delete
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To construct a priority queue it is simply necessary to call BuildHeap for the array
To delete an arbitrarily specified item from the queue follow the
design for Heap-Extract-Max
Heap-Delete(A, i)
if heap-size[A] < 1 then
error “heap underflow”
endif
item = A[i]
A[i] = A[heap-size[A]]
heap-size[A] = heap-size[A] – 1
Heapify(A, i)
return item
endalg
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Which will run in at worst O(lg n) time
Lecture 7
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Changing Priorities
 What if we want to change a node priority?
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Implement based on the existing functions?
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‘Delete old value’ and ‘Insert new value’
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Each of these will run in O(lg n) time, which means that the
procedure will also run in O(lg n) time.
The heap provides an efficient implementation for a priority
queue using the operations outlined, but is not so efficient in
other cases
For example, an operation which may sometimes be required is
the join operation to merge two priority queues
There are other implementations (e.g. doubly linked list) which
support this operation more efficiently
Lecture 7
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Summary
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The binary heap data structure is a data structure
that can be viewed as complete binary tree
Because of its nature a heap can very easily be
implemented using an array
Heaps must also satisfy the heap property for every
node, i, other than the root.
A[Parent(i)]  A[i]
The heapify algorithm can be used to create a heap,
and maintain a heap
Lecture 7
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