ADT Table

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ADT Table

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Title: ADT Table

1
ADT Table

The ADT table is appropriate for problems that
must manage data by value.
Some important operations of the ADT table are
Inserting a data item containing the value x.
Delete a data item containing the value x.
Retrieve a data item containing the value x.
Various table implementations are possible for
the ADT table.
We have to analyze the possible implementations
so that we can make an intelligent choice.
Some operations are implemented more efficiently
in certain implementations.

2
An ordinary table of cities
3
ADT Table Operations

Various sets of table operations are possible.
Some of them are
Create an empty table.
Destroy a table.
Determine whether a table is empty.
Determine the number of items in the table.
Insert a new item into a table.
Delete the item with a given search key.
Retrieve the item with a given search key.
Traverse the table.
The client of the ADT table may need a subset of
these operations, or require more operations on
the table.
Are keys in the table are unique?
We will assume that keys in our table are unique.
But, some other tables allow duplicate keys.

4
Selecting an Implementation

Since an array or a linked list represents items
one after another, these implementations are
called linear.
There are four categories of linear
implementations
Unsorted, array based (an unsorted array)
Unsorted, pointer based (a simple linked list)
Sorted (by search key), array based (a sorted
array)
Sorted (by search key), pointer based (a sorted
linked list).
We have also nonlinear implementations such as
binary search tree.
Binary search tree implementation offers several
advantages over linear implementations.

5
Sorted Linear Implementations

Array-Based
Pointer-Based

6
Binary Search Tree Implementation
7
Which Implementation should be used?

Which implementation is appropriate depends on
our application.
Before we select an implementation, we should
answer following questions about our application
What operations are needed?
Our application may not need all operations.
Some operation is implemented more efficiently in
an implementation, and another operation is
implemented more efficiently in another
implementation.
How often is each operation is required?
Some applications may require many occurrences of
an operation, but other applications may not.
For example, some applications may perform many
retrievals, but not so many insertions and
deletions. On the other hand, other applications
may perform many insertions and deletions.

8
How to Select an Implementation Scenario A

Let us assume that we have an application
inserts data items into a table.
after all data items are inserted traverse this
table in no particular order.
does not perform any retrieval and deletion
operations.
Which implementation is appropriate for this
application?
Keeping the items in a sorted order does not
provide any advantage for this application.
In fact, it will be more costly for this
application.
? Unsorted implementation will be more
appropriate.
Which unsorted implementation (array-based,
pointer-based)?
Do we know the maximum size of the table?
If we know the expected size is close to the
maximum size of the table
? an array-based will be more appropriate
(because pointer-based use extra space for
pointers)
Otherwise,
? a pointer-based will be more appropriate
(because too many entries will be empty
in the array-based implementation)

9
Insertion for unsorted linear implementations
Runtime complexity of insertion in an unsorted
list O(1)

array based
(b) pointer based

10
How to Select an Implementation Scenario B

Let us assume that we have an application
performs many retrievals, but no insertions and
deletions (or so few insertions or deletions, so
that we ignore their costs).
for example, a thesaurus (to look up synonyms of
a word)
For this application, a sorted implementation is
more appropriate.
If we use a sorted array, we can use binary
search to access data.
Since the binary search is not practical with
linked lists, sorted linked list implementation
will not be appropriate.
We can also use a binary search tree (in fact, we
can use a balanced binary search tree).
If we know the tables maximum size, a sorted
array-based implementation is more appropriate
for frequent retrievals.
Otherwise, a binary search tree implementation
will be more appropriate for frequent retrievals.

11
How to Select an Implementation Scenario C

Let us assume that we have an application
performs many retrievals, insertions and
deletions.
? Sorted Array Implementation
Retrieval is efficient.
But insertion and deletion are not efficient. We
have to shift data items.
? sorted-array implementation is not appropriate.
? Sorted Linked List Implementation
Retrieval, insertion, and deletion are not
efficient (although we do not shift data items)
? sorted linked list implementation is not
appropriate.
? Binary Search Tree Implementation
Retrieval, insertion, and deletion are efficient.
? binary search tree implementation is
appropriate.

12
Insertion for sorted linear implementations

array based
(b) pointer based

13
Which Implementation?

Despite difficulties, linear implementations of a
table can be appropriate.
Linear implementations are easy to understand,
easy to implement.
Linear implementations can be appropriate for
small tables.
For large tables, if there are few deletions and
retrievals, linear implementations may be
appropriate.
In general, a binary search tree implementation
is a better choice.
Worst case O(n) for most table operations
Average case O(log2n) for most table
operations
Balanced binary search tree increases the
efficiency of the ADT table operations.

14
The average-case order of the ADT table
operations for various implementations
15
Sorted Array-Based Implementation

// Header file TableA.h for the ADT table.
// Sorted array-based implementation.
// Assumption A table contains at most one item
with a
// given search key at any time.
include "KeyedItem.h" // definition of KeyedItem
and KeyType
include "TableException.h"
const int MAX_TABLE maximum-size-of-table
typedef KeyedItem TableItemType
typedef void (FunctionType)(TableItemType
anItem)
class Table
public
Table() // default constructor
// copy constructor and destructor are
supplied by the compiler

16
Sorted Array-Based Implementation (cont.)

// Table operations
virtual bool tableIsEmpty() const
// Determines whether a table is empty.
virtual int tableLength() const
// Determines the length of a table.
virtual void tableInsert(const TableItemType
newItem)throw (TableException)
// Inserts an item into a table in its proper
sorted
// order according to the item's search key.
virtual void tableDelete(KeyType searchKey)throw
(TableException)
// Deletes an item with a given search key from a
table.
virtual void tableRetrieve(KeyType searchKey,
TableItemType tableItem) const throw
(TableException)
// Retrieves an item with a given search key from
a table.
virtual void traverseTable(FunctionType visit)
// Traverses a table in sorted search-key order,
calling
// function visit() once for each item.

17
Sorted Array-Based Implementation (cont.)

protected
void setSize(int newSize)
// Sets the private data member size to
newSize.
void setItem(const TableItemType newItem, int
index)
// Sets itemsindex to newItem.
int position(KeyType searchKey) const
// Finds the position of a table item or its
insertion point.
private
TableItemType itemsMAX_TABLE // table items
int size // table size
int keyIndex(int first, int last, KeyType
searchKey) const
// Searches a particular portion of the
private array
// items for a given search key by using a
binary search.
// end Table class

18
tableInsert

void TabletableInsert(const TableItemType
newItem)
// Note Insertion is unsuccessful if the table
is full,
// that is, if the table already contains
MAX_TABLE items.
// Calls position.
if (size MAX_TABLE)
throw TableException("TableException Table
full")
// there is room to insert
// locate the position where newItem belongs
int spot position(newItem.getKey())
// shift up to make room for the new item
for (int index size-1 index gt spot
--index)
itemsindex1 itemsindex
// make the insertion
itemsspot newItem
size
// end tableInsert

19
tableDelete

void TabletableDelete(KeyType searchKey)
// Calls position.
// locate the position where searchKey
exists/belongs
int spot position(searchKey)
// is searchKey present in the table?
if ((spot gt size) (itemsspot.getKey() !
searchKey))
// searchKey not in table
throw TableException(
"TableException Item not found on
delete")
else // searchKey in table
--size // delete the item
// shift down to fill the gap
for (int index spot index lt size
index)
itemsindex itemsindex1
// end if
// end tableDelete

20
tableRetrieve

void TabletableRetrieve(KeyType searchKey,
TableItemType tableItem) const
// Calls position.
// locate the position where searchKey
exists/belongs
int spot position(searchKey)
// is searchKey present in table?
if ((spot gt size) (itemsspot.getKey() !
searchKey))
// searchKey not in table
throw TableException(
"TableException Item not found on
retrieve")
else
tableItem itemsspot // item present
retrieve it
// end tableRetrieve

21
traverseTable

void TabletraverseTable(FunctionType visit)
for (int index 0 index lt size index)
visit(itemsindex)
// end traverseTable

22
Binary Search Tree Implementation TableB.h

include "BST.h" // binary search tree operations
include "TableException.h"
typedef TreeItemType TableItemType
class Table
public
Table() // default constructor
// copy constructor and destructor are
supplied by the compiler
// Table operations
virtual bool tableIsEmpty() const
virtual int tableLength() const
virtual void tableInsert(const TableItemType
newItem) throw(TableException)
virtual void tableDelete(KeyType searchKey)
throw(TableException)
virtual void tableRetrieve(KeyType searchKey,
TableItemType tableItem) const
throw(TableException)
virtual void traverseTable(FunctionType
visit)
protected
void setSize(int newSize)
private
BinarySearchTree bst // binary search tree
that contains the tables items

23
Binary Search Tree Implementation TableB.cpp

include "TableB.h" // header file
void TabletableInsert(const TableItemType
newItem)
try
bst.searchTreeInsert(newItem)
size
// end try
catch (TreeException e)
throw TableException(
"TableException Cannot insert item")
// end catch
// end tableInsert

24
tableDelete

void TabletableDelete(KeyType searchKey)
try
bst.searchTreeDelete(searchKey)
// end try
catch (TreeException e)
throw TableException(
"TableException Item not found on
delete")
// end catch
// end tableDelete

25
tableRetrieve traverseTable

void TabletableRetrieve(KeyType searchKey,
TableItemType tableItem)
const
try
bst.searchTreeRetrieve(searchKey,
tableItem)
// end try
catch (TreeException e)
throw TableException(
"TableException Item not found on
retrieve")
// end catch
// end tableRetrieve
void TabletraverseTable(FunctionType visit)
bst.inorderTraverse(visit)
// end traverseTable
// End of implementation file.

26
The ADT Priority Queue

Priority queue is a variation of the table.
Each data item in a priority queue has a priority
value.
We insert an item with a priority value into its
proper position in the priority queue.
Deletion in the priority queue is not same as the
deletion in the table. We delete operation
deletes the item with the highest priority.
Using a priority queue we prioritize a list of
tasks
Job scheduling

27
ADT Priority Queue Operations

createPriorityQueue() create an empty priority
queue.
destroyPriorityQueue destroys a priority
queue.
isEmpty determines whether a priority queue is
empty or not.
insert Inserts a new item (with a priority
value) into a priority queue.
delete retrieves the item in a priority queue
with the highest priority value, and deletes that
item from the priority queue.

28
Some implementations of the ADT priority queue
(a) array based (b) pointer based (c) binary
search tree
29
Implementations Analysis

None of these implementations of the priority
queue is not efficient enough.
Array-Based
Insertion will be O(n)
Pointer-Based
Insertion will be O(n)
BST Implementation
Insertion is O(log2n) in average, but O(n) in the
worst case.
We need a balanced BST so that we can get better
performance ( O(logn) in the worst case) ? Heap

30
Heaps

Definition A heap is a complete binary tree such
that
It is empty, or
Its root contains a search key greater than or
equal to the search key in each of its children,
and each of its children is also a heap.
In this definition, since the root contains the
item with the largest search key, heap in this
definition is also known as maxheap.
On the other hand, a heap which places the
smallest search key in its root is know as
minheap.
We will talk about maxheap as heap in the rest of
our discussions.

31
Difference between Heap and BST

A heap is NOT a binary search tree.
Differences between heap and BST
While we can see a binary search tree as sorted,
but a heap is ordered in much weaker sense.
The order of the heap (although it is not sorted)
is sufficient for the efficient performance of
the priority queue operations.
While binary search trees come in many different
shapes, heaps are always complete binary trees.

32
Heap Examples
33
An Array-Based Implementation of a Heap
An array and an integer counter are the data
members for an array-based implementation of a
heap.
34
Major Heap Operations

Two major heap operations are insertion and
deletion.
Insertion
Inserts a new item into a heap.
After the insertion, the heap must satisfy heap
properties.
Deletion
Retrieves and deletes the root of the heap.
After the deletion, the heap must satisfy heap
properties.

35
Heap Delete First Step
36
Heap Delete Second Step
37
Heap Delete Last Step
The last step of heapDelete transforms the
semiheap into a heap.
Last Step
First Step
Second Step
38
Heap Delete
Recursive calls to heapRebuild
39
Heap Delete Analysis

Since the height of a complete binary tree with n
nodes is always
? log2(n1)?
? heapDelete is O(log2n)

40
Heap Insert

A new item is inserted at the bottom of the
tree, and
it trickles up to its proper place

41
Heap Insert Analysis

Since the height of a complete binary tree with n
nodes is always
? log2(n1)?
? heapInsert is O(log2n)

42
Heap.h

const int MAX_HEAP maximum-size-of-heap
include "KeyedItem.h" // definition of KeyedItem
typedef KeyedItem HeapItemType
class Heap
public
Heap() // default constructor
// copy constructor and destructor are
supplied by the compiler
// Heap operations
virtual bool heapIsEmpty() const
// Determines whether a heap is empty.
virtual void heapInsert(const HeapItemType
newItem) throw(HeapException)
// Inserts an item into a heap.
virtual void heapDelete(HeapItemType rootItem)
throw(HeapException)
// Retrieves and deletes the item in the root of
a heap.
// This item has the largest search key in the
heap.
protected
void heapRebuild(int root)
// Converts the semiheap rooted at index root
into a heap.
private

43
Heap.cpp

//
// Implementation file Heap.cpp for the ADT heap.
//
include "Heap.h" // header file for class Heap
HeapHeap() size(0)
// end default constructor
bool HeapheapIsEmpty() const
return bool(size 0)
// end heapIsEmpty

44
heapInsert

void HeapheapInsert(const HeapItemType
newItem)
// Method Inserts the new item after the last
item in the heap and
// trickles it up to its proper position.
// The heap is full when it contains MAX_HEAP
items.
if (size gt MAX_HEAP)
throw HeapException("HeapException Heap
full")
// place the new item at the end of the heap
itemssize newItem
// trickle new item up to its proper position
int place size
int parent (place - 1)/2
while ( (parent gt 0)
(itemsplace.getKey() gt
itemsparent.getKey()) )
// swap itemsplace and itemsparent
HeapItemType temp itemsparent
itemsparent itemsplace
itemsplace temp
place parent
parent (place - 1)/2

45
heapDelete

void HeapheapDelete(HeapItemType rootItem)
// Method Swaps the last item in the heap with
the root
// and trickles it down to its proper position.
if (heapIsEmpty())
throw HeapException("HeapException Heap
empty")
else
rootItem items0
items0 items--size
heapRebuild(0)
// end if
// end heapDelete

46
heapRebuild

void HeapheapRebuild(int root)
// if the root is not a leaf and the root's
search key
// is less than the larger of the search keys
in the root's children
int child 2 root 1 // index of root's
left child, if any
if ( child lt size )
// root is not a leaf, so it has a left
child at child
int rightChild child 1 // index of
right child, if any
// if root has a right child, find larger
child
if ( (rightChild lt size)
(itemsrightChild.getKey() gt
itemschild.getKey()) )
child rightChild // index of larger
child
// if the root's value is smaller than the
// value in the larger child, swap values
if ( itemsroot.getKey() lt
itemschild.getKey() )
HeapItemType temp itemsroot
itemsroot itemschild
itemschild temp
// transform the new subtree into a heap
heapRebuild(child)

47
Heap Implementation of Priority Queue

Since the heap operations and the priority queue
operations are same, the heap implementation of
the priority queue is straightforward.
Both insertion and deletion operations of the
priority queue will be O(log2n), when we use the
heap.

48
PQ.h

// Header file PQ.h for the ADT priority queue.
// Heap implementation.
include "Heap.h" // ADT heap operations
typedef HeapItemType PQueueItemType
class PriorityQueue
public
// default constructor, copy constructor, and
// destructor are supplied by the compiler
// priority-queue operations
virtual bool pqIsEmpty() const
virtual void pqInsert(const PQueueItemType
newItem)
throw (PQueueException)
virtual void pqDelete(PQueueItemType
PQueueItemType)
throw (PQueueException)
private
Heap h
// end PriorityQueue class

49
PQ.cpp

// Implementation file PQ.cpp for the ADT
priority queue.
// A heap represents the priority queue.
include "PQ.h" // header file for priority queue
bool PriorityQueuepqIsEmpty() const
return h.heapIsEmpty()
// end pqIsEmpty
void PriorityQueuepqInsert(const
PQueueItemType newItem)
try
h.heapInsert(newItem)
// end try
catch (HeapException e)
throw PQueueException(
"PQueueException Priority queue full")
// end catch
// end pqInsert

50
pqDelete

void PriorityQueuepqDelete(PQueueItemType
priorityItem)
try
h.heapDelete(priorityItem)
// end try
catch (HeapException e)
throw PQueueException(
"PQueueException Priority queue
empty")
// end catch
// end pqDelete

51
Heapsort

We can use a heap to sort an array
Create a heap from the given initial array with n
items.
Swap the root of the heap with the last element
in the heap.
Now, we have a semiheap with n-1 items, and a
sorted array with one item.
Using heapRebuild convert this semiheap into a
heap. Now we will have a heap with n-1 items.
Repeat the steps 2-4 as long as the number of
items in the heap is more than 1.

52
Heapsort (cont.)