Algorithm

SYLLABUS

Algorithms, Abstract Data Type, The Running Times Of a Program, Good Programming Practice,
The data type, Implementation of lists, Array implementation of list, Pointer implementation of
list, Doubly-link lists, Stack, Queues, Mapping.

Sets, An ADT with union, intersection and difference, Bit vector, implementation of sets, Link-list
implementation of sets, The data dictionary.

Efficiency of algorithms, Analysis of recursive programs, Solving recurrence equation, Divide and
conquer algorithms, Dynamic programming, Greedy algorithm, Prim’s algorithm,
Kruskal’s algorithm, Dijkstra’s method, Backtracking.

Basic terminology, Implementation of tree, An Array Representation of Trees, Representation of
Trees by List of Children, Binary trees, The internal sorting model, Bubble sort,
Insertion sort, Selection sort, Heap sort, Divide and conquer, Merge sort, Quick sort,
Binary search.

A Model of External Computation, External sorting, Characteristics of External Sorting, Criteria
for Developing an External Sorting Algorithm, Important Uses of External Sorting,
Merge Sort--A Digression, Top-Down Strategy, Bottom-Up Strategy, Storing Information
in Files, Hashed Files, Indexed Files

The Issues in Memory, Garbage Collection Algorithms For Equal-Sized Block, Collection in Place,
Buddy system, Distribution of blocks, Allocation blocks, Returning blocks to available
storage, Storage compaction and Compaction problem

Introduction to NP Problem, Polynomial-time, Abstract Problems, Encoding, NP-Completeness and
Reducibility, NP-Completeness, Circuit Satisfiability, NP-Complete Problems, The Vertex-cover
Problem, The Subset-sum Problem, The Hamiltonian-cycle Problem, The Traveling-salesman
Problem

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TABLE OF CONTENTS

UNIT 1
Basic of Algorithms
1.1 Algorithm
1.2 Abstract Data Type
1.3 The Running Times Of a Program
1.4 Good Programming Practice

UNIT 2
Basic Data Type
2.1 The data type “list”
2.2 Static and Dynamic Memory Allocation
2.3 Pointers
2.4 Implementation of lists
2.5 Linear Linked List
2.6 Array implementation of list
2.7 Pointer implementation of list
2.8 Doubly-link lists
2.9 Stack
2.10 Queues
2.11 Mapping

UNIT 3
Basic Operations and Sets
3.1 Sets
3.2 An ADT with union, intersection and difference
3.3 Bit vector implementation of sets
3.4 Link-list implementation of sets
3.5 The data dictionary

UNIT 4
Algorithms Analysis Techniques
4.1 Efficiency of algorithms
4.2 Analysis of recursive programs
4.3 Solving recurrence equation

UNIT 5
Algorithms Design Technique
5.1 Divide and conquer algorithms
5.2 Dynamic programming
5.3 Greedy algorithm
5.4 Minimum-cost spanning trees
5.5 Minimum Spanning Tree
5.6 Prim’s Algorithm
5.7 Kruskal’s Algorithm
5.8 Shortest Paths
5.9 Dijkastra’s Algorithm
5.10 Backtracking

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UNIT 6
Trees and Sorting
6.1 Basic terminology
6.2 Implementation of tree
6.3 An Array Representation of Trees
6.4 Representation of Trees by List of Children
6.5 Binary trees
6.6 The internal sorting model
6.7 Bubble sort
6.8 Insertion sort
6.9 Selection sort
6.10 Heap sort
6.11 Divide and conquer
6.11.1 Merge sort
6.11.2 Quick sort
6.11.3 Binary search

UNIT 7
Algorithms for External Storage
7.1 A Model of External Computation
7.2 External sorting
7.3 Characteristics of External Sorting
7.4 Criteria for Developing an External Sorting Algorithm
7.5 Important Uses of External Sorting
7.6 Merge Sort--A Digression
7.7 Top-Down Strategy
7.8 Bottom-Up Strategy
7.9 Storing Information in Files
7.10 Hashed Files
7.11 Indexed Files

UNIT 8
Memory Management
8.1 The Issues in Memory
8.2 Garbage Collection Algorithms For Equal-Sized Block
8.3 Collection in Place
8.4 Buddy system
8.5 Distribution of blocks
8.6 Allocation blocks
8.7 Returning blocks to available storage
8.8 Storage compaction and Compaction problem

UNIT 9
NP Complete Problem
9.1 Introduction
9.2 Polynomial-time
9.3 Abstract Problems
9.4 Encoding
9.5 NP-Completeness and Reducibility
9.6 NP-Completeness
9.7 Circuit Satisfiability
9.8 NP-Complete Problems
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9.9 The Vertex-cover Problem
9.10 The Subset-sum Problem
9.11 The Hamiltonian-cycle Problem
9.12 The Traveling-salesman Problem

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UNIT 1

Basic of Algorithms

1.1 Algorithm
1.2 Abstract Data Type
1.3 The Running Times Of a Program
1.4 Good Programming Practice

1.1 Introduction to Algorithms

Computer Science can be defined as the study of algorithms. This study encompasses four distinct
areas:

• Machines for executing algorithms.
• Languages for describing algorithms.
• Foundations of algorithms.
• Analysis of algorithms.

It can be easily seen that "algorithm" is a fundamental notion in computer science. So it deserves
a precise definition. The dictionary's definition, "any mechanical or recursive computational
procedure," is not entirely satisfying since these terms are not basic enough.

Definition: An algorithm is a finite set of instructions, which, if followed, accomplish a particular
task. In addition every algorithm must satisfy the following criteria:

1. Input: there are zero or more quantities which are externally supplied;
2. Output: at least one quantity is produced;
3. Definiteness: each instruction must be clear and unambiguous;
4. Finiteness: if we trace out the instructions of an algorithm, then for all cases the algorithm
will terminate after a finite number of steps;
5. Effectiveness: every instruction must be sufficiently basic that a person using only pencil
and paper can in principle carry it out. It is not enough that each operation is definite, but
it must also be feasible.

In formal computer science, one distinguishes between an algorithm, and a program. A
program does not necessarily satisfy the fourth condition. One important example of such a
program for a computer is its operating system, which never terminates (except for system
crashes) but continues in a wait loop until more jobs are entered.

An algorithm can be described in several ways. One way is to give it a graphical form of
notation such as. This form places each processing step in a "box" and uses arrows to indicate the
next step. Different shaped boxes stand for different kinds of operations.

If a computer is merely a means to an end, then the means may be an algorithm but the
end is the transformation of data. That is why we often hear a computer referred to as a data
processing machine. Raw data is input and algorithms are used to transform it into refined data.

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So, instead of saying that computer science is the study of algorithms, alternatively, we might say
that computer science is the study of data.

1.2 Abstract Data Type:

ADT is kind of data abstraction where a type's internal form is hidden behind a set of
access functions. Values of the type are created and inspected only by calls to the access
functions. This allows the implementation of the type to be changed without requiring any
changes outside the module in which it is defined. Objects and ADT are both forms of data
abstraction, but objects are not ADT. Objects use procedural abstraction (methods), not type
abstraction.
A classic example of an ADT is a stack data type for which functions might be provided to
create an empty stack, to push values onto a stack and to pop values from a stack.
ADT A type whose internal form is hidden behind a set of access functions. Objects of the
type are created and inspected only by calls to the access functions. This allows the
implementation of the type to be changed without requiring any changes outside the module in
which it is defined.
Abstract data types are central to object-oriented programming where every class is an
ADT.

1.3 The Running Time of a Program
When solving a problem we are faced frequently with a choice among algorithms. On what
basis should choose? There are two often-contradictory goals.
1. We would like an algorithm that is easy to understand code and debug.
2. We would like an algorithm that makes efficient use of the computers resources,
especially, one that runs as fast as possible.
Measuring the running time of a program: The running time of a program depends on factors
such as:
1. The input of a program.
2. The quality of a code generated by the compiler used to create the object program.
3. The nature and speed of the instructions on the machine used to execute the program.
4. The time complexity of the algorithm underlying the program.

Calculating the Running Time of a Program

Some Important Formulas for finding the Running Time of a program are the following:

When finding the Running Time of a Program with that has Scary nested loops it's a good
thing to know how Summations work and why you use Summations.
We use summations because of what a Summation stands for is very similar to how a loop
is run.
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Some important properties of Summations are good to review in order to make it easier to
work with them. The following property shows one way of manipulating summations to work to
your advantage in certain questions.

Another important aspect of summations is how to manipulate the equation properly to fit the
first helpful equations given above.

Some rules for finding the Running Time that come are following:

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A good way to understand how Big Oh works is to look at the following diagram.

Expression Name
O(1) constant

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logarithmic

log squared
O(n) linear
N log n
Quadratic

cubic
exponential

Table: The Names of Common Big Oh Expressions

1.4 Good Programming Practice:

There are a substantial number of ideas we should bear in mind when designing an algorithm
and implementing it as a program. These ideas often appear platitudinous, because by and large
they can be appreciated only through their successful use in real problems, rather than by
development of theory.

1. Plan the design of a program. This strategy of sketch-then-detail tends to produce a
more organized final program that is easier to debug and maintain.
2. Encapsulate. Use procedures and ADT’s to place the code for each principal operation
and type of data in one place in the program listing. Then, if changes become
necessary, the section of code requiring change will be localized.

3. Use or modify an existing program. One of the chief inefficiencies in the
programming process is that usually a project is tackled as if it were the first program
ever written. One should first look for an existing program that does all or a part of the
task.

4. Program at the command level. A well-designed operating system will allow us to
connect a network of available programs together without writing any programs at all,
except for one list of operating system commands. To make command compos able, it
is generally necessary that each behave as a filter, a program with one input file and
one output file.

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UNIT 2

Basic Data Type

2.1 The data type “list”
2.2 Static and Dynamic Memory Allocation
2.3 Pointers
2.4 Linear Linked List
2.4.1 Array implementation of list
2.4.2 Pointer implementation of list
2.4.3 Doubly-link lists
2.5 Stack
2.6 Queues
2.7 Mapping

2.1 The Data Type “List”
Using ADT’s allows the data in a specific piece of code to be hidden from other pieces of
code that don't need and shouldn't have access to it. This is often called modular programming or
encapsulation. The idea behind the ADT is to hide the actual implementation of the code, leaving
only its abstraction visible. In order to do this, one needs to find a clear division between the
linked list code and the surrounding code. When the linked list code is removed, what remains is
its logical abstraction. This separation makes the code less dependent on any one platform. Thus,
programming using the ADT method is usually considered a necessity for cross-platform
development as it makes maintenance and management of the application code much easier.
The ADT concept is developed here via the example of how the doubly linked list
Application Programming Interface (API) was created. The doubly linked list (DLL) can also be an
independent C module and compiled into a larger program.
An abstract data type (ADT).

2.2 Static and Dynamic Memory Allocation: -
In static memory allocation memory is allocated at compile time. If we declare a stack
through an array of 100 elements (all integers) then the statement will be:
int stk[100];
This declaration would typically be used if 100 records are to be stored in memory. The
moment we make this declaration 200 bytes are reserved in memory for storing 100 integers in it.
However it may so happen that when we actually run the program we might be interested in
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storing only 60 records, even in this case 200 bytes would get reserved in memory that would
result in wastage of memory. There may be cases when we need to store more than 100 records,
in this case array would fall short in size.
We can overcome these problems by allocating memory at Run Time (instead of compile
time), this is called Dynamic Memory Allocation. It is done by using Standard Library functions
malloc() and calloc(). Syntax for allocating memory through malloc:
p = (data type*) malloc (n*size of (<datatype>));

2.3 Pointers
Pointer is a variable that gives the Address (location or link or reference) of some other
variable. As other variables Pointer also cannot be used before declaration. We can declare the
pointer variable as:
datatype *p;
This declaration specifies that p is a pointer, which will store the address of variable of
particular datatype. '*' stands for 'value of address'. Thus if we declare int *p; then it would mean,
the value at the address contained in p is an integer.
Implementation of Lists: -
In implementation of lists we shall describe data structures that can be used to represent
lists. We shall consider array, pointer and cursor implementation of list.
Linked Linear List
We saw in previous chapters how static representation of linear ordered list through Array
leads to wastage of memory and in some cases overflows. Now we don't want to assign memory to
any linear list in advance instead we want to allocate memory to elements as they are inserted in
list. This requires Dynamic Allocation of memory and it can be achieved by using malloc() or
calloc() function.
But memory assigned to elements will not be contiguous, which is a requirement for linear
ordered list, and was provided by array representation. How we could achieve this?
We have to consider a logical ordered list, i.e. elements are stored in different memory
locations but they are linked to each other and form a logical list as in Fig. This link represents
that each element

A1 A2 A3 A4 …… An
.

Figure: Logical List
has address of its logical successor element in the list. We can understand this concept through a
real life example. Suppose their is a list of 8 friends, x1, x2......x8. Each friend resides at different
locations of the city. x1 knows the address of x2, x2 knows the address of x3 and so on .... x7 has
the address of x8. If one wants to go to the house of x8 and he does not know the address he will
go to x2 and so on Fig.
The concept of linked list is like a family despaired, but still bound together. From the
above discussion it is clear that Link list is a collection of elements called nodes, each of

x1 Add.of x2 x2 Add.of x3 X3 ……. X8 NULL

Figure
Which stores two items of information:
• An element of the list
• A link or address of another node

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Link can be provided through a pointer that indicates the location of the node containing the
successor of this list element. The NULL in the last node indicates that this is the last node in the
list.

2.4 Linked List

2.4.1 Array Implementation of List: -
This type of implementation of a linked list is sometimes useful. It suffers from the usual
array problem of having to estimate the needed size of the array in advance. There are some ways
around this problem (such as using a vector instead of an array, allocating space for the array
dynamically, or even allocating space for additional arrays as needed), but they have their own
complications. We will assume here that the maximum size needed can be foretold. The array will
hold a fixed number of records (structures), each of which represents a node.
In this implementation, a linked list object contains an array of nodes, perhaps called
NodeArray, and four integers fields labelled Avail, Front, Rear, and Count. Each record in
NodeArray contains an Info field and an integer Next field. Avail, Front, Rear, and each Next
field all contain fake pointers. What we use as a fake pointer to a node is the array index of that
node. To represent the NULL pointer we can use an illegal array index, such as -1. We will not
start our list with a dummy node, though it is possible to do so by making a few small changes.
The picture of an empty list is as follows:

The idea is that Front points to the first node of the list. Since it is -1, our imitation NULL
pointer, this means that we have an empty list. The Count of zero also indicates the same thing.
Rear is -1, indicating that there is no node at the rear of the list (as the list is empty). The Avail
field is intended to point to a list of free nodes. The programmer must arrange to manage free
space within the array with this method! In out picture above, all of the nodes are free and are
linked together in a list with Avail containing zero and thus pointing at the first free node,
NodeArray[0]. Then this node has a Next value of 1, showing that it points to NodeArray[1], etc.
Finally, NodeArray[4].Next contains our imitation NULL pointer to mark the end of the list.
After items have been dynamically inserted into the list and perhaps items have been
deleted as well, we might end up with a picture such as the following.

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Can you read what is on this list? All you do is follow the (fake) pointers. Since Head contains 1,
the first node contains the string "Ann". This node contains a pointer to node 0, so the second
node contains "Sam". This node contains a pointer to node 4, so the third node contains "Mary".
Finally, this node has -1 in the Next field, marking the end of the list. So the three data items in
the list are "Ann", "Sam", and "Mary". If you need a quick way to get to the last node on the list,
you use the 4 found in the Rear field. The list of available (free) nodes contains node 2 and node
3. Note that such a list object always contains two linked lists: the list of data nodes and the list
of available (free) nodes. It is left as an exercise to the reader to implement a linked list class using
this array-based scheme.

2.4.2 Pointer Implementation of List: -
It is very common to implement a linked list using pointers and either structures or
objects for the nodes. We will choose objects for the nodes and set up the class ListNodeClass as
shown below.
class ListNodeClass
{
private:
ItemType Info;
ListNodeClass * Next;
public:
// First, the constructor:
ListNodeClass(const ItemType & Item, ListNodeClass * NextPtr = NULL):
Info(Item), Next(NextPtr)
{
};
void GetInfo(ItemType & TheInfo) const;
friend class ListClass; // very convenient to allow this
};

typedef ListNodeClass * ListNodePtr;
Note that the two data fields are private, as usual. The type ItemType would have to be set
up to handle whatever data we want to hold in the list. The Next field is a pointer to another
ListNodeClass object, since the * refers to whatever Next points to.
The most important function is the constructor. It takes a data item and a pointer as its
parameters and uses them to fill in the fields of the object being constructed. Note that the pointer
parameter has a default value of NULL. Recall that the Info(Item), Next(NextPtr) means to copy
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Item into Info and NextPtr into Next. The GetInfo function simply sends back a code of the data
from the object.
The ListNodeClass is also told that it has a friend class named ListClass. Functions of the
friend class have direct access to the private data fields (Info and Next) of ListNodeClass objects. It
is not necessary to allow this direct access, but it is easier than the alternative: using get and set
class functions every time a ListClass function wants to look at or change the private fields of a
ListNodeClass object. Some people refuse to use friend classes, however, because they give more
access than is really necessary, thus partially defeating the information hiding that class provide.
Finally, note that the above code sets up ListNodePtr as a convenient type name for a
pointer to a node. This means that we won't have to use ListNodeClass * from here on in the rest
of the code, but can instead use ListNodePtr.
Next, let's set up a class declaration for list objects. Each list will have three private data
fields: a pointer to the first node on the list, a pointer to the last node on the list, and a count of
how many data nodes are in the list.
The Front pointer will point to what is called a dummy node. This is a node containing no data. It
is easier to code the insertion and deletion operations if the first node on the list contains no data.
Otherwise, the code to insert a new first node or to delete the current first node must be different
from the code to handle the normal cases.
The following is the class declaration for ListClass. Helping functions, which are only used
by other ListClass functions are made private, so that they cannot be accessed otherwise. (This is
another instance of information hiding.) There is a comment that the three data fields are
sometimes made into protected fields. The purpose of this is that if we ever created a subclass
(using inheritance), functions of the subclass would be able to directly access these fields. With
private fields, the fields would be inherited, but not directly accessible by the subclass. Since we
do not plan to inherit a new class from ListClass, this is not a concern.
class ListClass
{
private:
ListNodePtr GetNode(const ItemType & Item,
ListNodePtr NextPtr = NULL);
void FreeNode(ListNodePtr NodePtr);
void ClearList(void);
// Next 3 are sometimes made into protected fields:
ListNodePtr Front, Rear;
int Count;
public:
// constructor:
ListClass(void);
// destructor:
~ListClass(void);
int NumItems(void) const;
bool Empty(void) const;
void InsertFront(const ItemType & Item);
void InsertRear(const ItemType & Item);
void InsertInOrder(const ItemType & Item);
ItemType RemoveFront(void);
ListNodePtr Find(const ItemType & Item) const;
};
The public functions above provide the usual linked list operations. The above class
declaration thus shows that, from an abstract data type point of view, the list operations are list
creation, list destruction, getting the number of items in the list, checking if the list is empty,
inserting an item at the front of the list, inserting an item at the rear of a list, inserting an item
into an ordered list, removing an item from the front of a list, and finding an item in the list.
Certainly other operations are possible. For example, why not have an operation to remove the
item from the rear of a list? (One practical reason not to do so is that it is not easy to adjust the
Rear pointer after deleting the last node and to get a NULL into the Next field of the new last node.
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The only way to do so appears to be traverse the list from the first to the last node, before
removing the final node, in order to get a pointer to the next to the last node!)
A ListClass object, named List is pictured below. Functions are not shown in the objects in
order to keep the diagram from becoming cluttered, but if they were, the private functions should
be shown as completely boxed in, whereas the public function should be shown in open boxes to
indicate that they are available for public use. Note that a ListClass object doesn't really contain
the data in the list. Rather, the data is contained in node objects that are outside of the List object
itself. The List object simply contains pointers to the first and last nodes of the list.

The complete code files for this example are given below. The last file contains a simple
test program to try out several linked list operations. This test program is straightforward and will
not be commented upon here. Read through these files to get the general idea, and then go on to
the notes that follow.
In the next to the last file above, you will note that the constructor initializes a ListClass
object to contain a Count of zero, and to have Front and Rear pointers that point to a dummy
node. Of course, this means that it has to create a dummy node. This is handled by the helping
function GetNode. In turn, GetNode uses new with the ListNodeClass constructor to dynamically
allocate a new node.
Since a ListClass object points to other objects outside of it, it is important to have a
destructor to clean up the space used by these ListNodeClass objects. The destructor uses the
ClearList function to reclaim the space used by the data nodes and then calls FreeNode(Front) to
get rid of the dummy node. The ListClass object itself is automatically removed.
The FreeNode function is trivial in that it just uses delete to remove the node pointed to.
The ClearList function, however, is more interesting. Let's look at it in more detail since it is a
good example of a list processing function.
/* Given: Nothing.
Task: Clear out all nodes on the list except the dummy.
The list is thus set to an empty list.
Return: Nothing directly, but the implicit ListClass object is modified.
*/
void ListClass::ClearList(void)
{
ListNodePtr Current, Temp;

Current = Front->Next;
while (Current != NULL)
{
Temp = Current;
Current = Current->Next;
FreeNode(Temp);
}

Front->Next = NULL;
Rear = Front;
Count = 0;
}
In essence the idea is to traverse the list, freeing up each data node as it is encountered.
The Current pointer is used to point to the successive nodes on the list. Note that it is initialized
to point where the dummy node's Next field points, that is, to point to the first data node. Then we
have a while loop. Inside of it, the Current pointer is copied into Temp, Current is advanced by
placing in it a copy of the Next field of the node we have been pointing at, and FreeNode is used to
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free up the node that Temp points to. Notice that we can't simply free the node that Current
points to each time around the loop. If we did that we would lose our pointer into the linked list
and have no way to advance to the next node. The loop stops when Current is advanced to pick
up the NULL value marking the end of the list. After the loop ends, the ListClass object is adjusted
so that the Front and Rear pointers are NULL and the Count is zero. In other words, the list object
is changed into the typical picture of an empty list.
Let's look at a few more of the list manipulation functions, starting with InsertFront.
/* Given: Item A data item.
Task: To insert a new node containing Item at the front of the implicit
ListClass object.
Return: Nothing directly, but the implicit object is modified.
*/
void ListClass::InsertFront(const ItemType & Item)
{
ListNodePtr NodePtr;

NodePtr = GetNode(Item, Front->Next);
Front->Next = NodePtr;

if (Count == 0)
Rear = NodePtr;
Count++;
}

It is clear that the GetNode function is being used to create a node to hold the new data item. The
second parameter to this function is used to initialize the Next field of this new node so that it
points to the first data node. At this point we have the following picture:

To finish things off, the function adjusts the Next field of the dummy node so that it points
to the new node. The Count field for the list also needs to be incremented. That finishes things for
the example in our picture.
Unfortunately, it is not always quite that simple. There is a special case lurking in the
background. What happens if we insert at the front of an empty list? Remember that an empty list
looks like this:

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The steps we previously did to insert a new node at the front of the list are still fine, but
they leave one thing out: they do not adjust the Rear pointer, currently pointing at the dummy
node, to point to the newly inserted node. That is handled by the if in the code above.
You may not have noticed it, but the ListClass functions are directly accessing the private data
fields of the ListNodeClass objects. That would not normally be allowed, but it is legal here
because we made ListClass to be a friend of ListNodeClass.
Next, let's look at the InsertInOrder function. Note that it assumes that the list is already
in order.
/* Given: Item A data item.
Task: Insert Item into a new node added to the implicit ListClass object
(assumed to already be in ascending order based on the Info
field) so as to maintain the ascending order.
Return: Nothing directly, but the implicit object is modified.
*/
void ListClass::InsertInOrder(const ItemType & Item)
{
ListNodePtr Current, Last, Temp;
bool Proceed;

Temp = GetNode(Item);
Current = Front->Next;
Last = Front;
Proceed = true;

while (Proceed && (Current != NULL))
if (Item > Current->Info)
{
Last = Current;
Current = Current->Next;
}
else
Proceed = false;

Last->Next = Temp;
Temp->Next = Current;

if (Current == NULL) // item was inserted at rear of list
Rear = Temp;
Count++;
}

Suppose that we have an ordered list containing 4, 6, 9, and that we want to insert 8
using the above function. Note that it moves a pair of pointers, Current and Last, through the list,

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with Last always pointing to the node before the one that Current points to. Current starts out by
being initialized to point to the first data node, while Last points to the dummy node. Each time
around the while loop a check is made to see if the 8 is greater than the item contained in the
node that Current points to. If so, the pointers are advanced so that each points to the next node.
If not, then the place to insert has been found, so the loop is stopped by setting the Proceed flag to
false. The picture at the point where the insertion location has been found is shown below:

All that remains to be done is to link in the new node, up the Count by 1, and check for any
special cases. It turns out that the only unique case is when the new item ends up going at the
very end of the list. In such a case the Rear pointer must be adjusted (as shown in the code
above).
The Remove Front function is simpler. However, it exits with an error message if there is
no data item to remove. If not, it sets up NodePtr to point to the first data node and extracts the
value from this node's Info field. It then adjusts the Next field of the dummy node so that it points
to the second data node, skipping around the first one. The Count is decremented and the first
data node is freed up. Not surprisingly, there is a special case. This occurs when the list only has
one data node. After removing it, the Rear field must be adjusted to point to the dummy node.

2.4.3 Doubly-Link lists: -

A doubly linked list has two pointer fields in each node, often named Left and Right. If
the linked list is pictured as a horizontal sequence of nodes, the Right pointers are used to
traverse the lists from left to right (that is, from beginning to end). However, the Left pointers can
be used to back up to the left whenever that is desired. The following is one possible picture of a
doubly linked list. Sometimes dummy nodes are added at the front and rear as well.

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One can thus traverse a doubly linked list in both the forward and backward directions.
This can make coding simpler, for example, when writing a sequential search function to find a
given node in the list, with the plan that we will use this to find the node after which to insert a
new node. With a singly linked list we needed to return both a pointer to the node found and a
pointer to the previous node. With a doubly linked list it suffices to return a pointer to the
matching node, since we can always follow its Left field to get at the previous node.

Layout of Doubly Linked List in memory

The arrows indicate to what the Prior and Next pointers point. The Current pointer can
point to any Node Struct, so it is open-ended.
In order to fulfill the first requirement, I decided to strictly adhere to the ANSI C standard,
and, with the possible exception of how one sets up one's data and uses the DLL's input/output
functions, there should be no endian (byte order) problems. The second requirement was met with
the creation of a top-level structure. There is only one of these structures per linked list. It keeps
track of the node pointers, the size of the applications data in bytes, how many nodes are in the
list, whether or not the list has been modified since it was created or loaded into memory, where
searching starts from, and what direction a search proceeds in. Figure 1 illustrates how the top-
level structure is integrated into the DLL.
typedef struct list
{
Node *head;
Node *tail;
Node *current;
Node *saved;
size_t infosize;
unsigned long listsize;
DLL_Boolean modified;
DLL_SrchOrigin search_origin;
DLL_SrchDir search_dir;
} List;

This and the next typedef structure remain hidden from the application program. The
node pointers mentioned above are defined in the next structure, which includes the pointers to
the application's data and the pointers to the next and prior nodes. One of these structures is
created for each node in the list.

typedef struct node
{
Info *info;
struct node *next;
struct node *prior;
19

} Node;
The last definition is a dummy typedef of the user's data. It is defined as type void so that
the DLL's functions will be able to handle any of C's or an application's data types.
typedef void Info;
As you can see, if the two structures mentioned above are hidden from the application, all
of the ugly innards of how the linked list operates will by default be hidden from the application.
Thus, we have an abstract data type.

2.5 STACK: -

A stack is an abstract data structure that consists of an ordered collection of items with
two basic operations called push and pop. The push operation appends a single new item into the
stack collection. The pop operation removes a single item off the stack collection, but in the
reverse order that the item was added using the push operation.
We can model a stack data structure in terms of a pipe that has one end open and the
other closed. The push operation places items into the pipe from the open end. The pop
operation removes an item only from the open end. Assuming the items do not juggle around in
the pipe, the items placed into the pipe can only be removed in the order that they are placed into
the pipe. Although, in theory the stack is like an infinite length pipe that can hold an endless
number of items, in practice, due to the finite memory available on a computer system, the stack
has a maximum size. This is called the maximum stack size. In theory, one never needs to
track the number of elements currently in the stack collection, but in practice this is called the
current stack size. The operation of performing a pop on an empty stack is given a special error
called a stack underflow. Since in practice, a stack can only contain a finite number of elements,
an error called a stack overflow occurs when the stack is full and a push operation is performed.
Stack will work on LIFO application.

Adding into stack

procedure add(item : items)
{add item to the global stack stack;
top is the current top of stack and n is its maximum size}
begin
if top = n then stackfull;
top := top+1;
stack(top) := item;
end: {of add}

Deletion in stack

procedure delete(var item : items)
{remove top element from the stack stack and put it in the item}
begin
if top = 0 then stackempty;
item := stack(top);
top := top-1;
end; {of delete}

These two procedures are so simple that they perhaps need no more explanation. Procedure delete actually
combines the functions TOP and DELETE, stackfull and stackempty are procedures which are left unspecified since
they will depend upon the particular application. Often a stackfull condition will signal that more storage needs to be
allocated and the program re-run. Stackempty is often a meaningful condition.

20

2.6 QUEUE: -

A Queue is an ordered list of items that have two basic operations: adding new items to the
queue, and removing items of the queue. The operation of adding new items on the queue occurs
only at one end of the queue called the rear(or back). The operation of removing items of the
queue occurs at the other end called the front. The following consists of a queue that has five
integers:
front: 23 33 -23 90 2 : rear
The operations of adding 2 new items,5 and 4, to the queue occurs in the back and the
queue looks like so.
front: 23 33 -23 90 2 5 4 :rear
The operations of removing three items from the queue would yield the queue
front: 90 2 5 4 : rear
The first item on the queue, the item that will be removed with the next remove operation,
is called the front of the queue. The last item is called
the rear of the queue. The number of items in the queue is called the queue size. If the number of
items in the queue is zero, an attempt at a remove
operation produces a queue underflow error. In theory, there does not exist a queue overflow.
But in practice, implementations of a queue have a maximum queue size.

Addition into a queue

procedure addq (item : items)
{add item to the queue q}
begin
if rear=n then queuefull
else begin
rear :=rear+1;
q[rear]:=item;
end;
end;{of addq}

Deletion in a queue

procedure deleteq (var item : items)
{delete from the front of q and put into item}

21

begin
if front = rear then queueempty
else begin
front := front+1
item := q[front];
end;
end; {of deleteq}

2.7 Mapping

A mapping or associative store is a function from elements of one type, called the domain
type to elements of another type, called the range type. We express the fact that the mapping M
associates element r of range type range type with element d of domain type domain type by M(d)=
r.
Certain mapping such as square(i)= i2 can be implemented easily by giving an arithmetic
expression or other simple means for calculating M(d) from d . However for many mapping there is
no apparent way to describe M(d) other than to store for each d the value of M(d).

22

UNIT 3

Basic Operations and Sets

3.1 Sets
3.3 Bit vector implementation of sets
3.4 Link-list implementation of sets
3.5 The data dictionary

3.1 Sets
A set is a collection of member (or elements) each member of a set either is itself a set or
primitive element called an atom. All member of a set are different, which means no set can
contain two copies of the same elements.
When used as tool in algorithm and data structure design, atoms usually are integers,
characters or strings and all elements in any one set are usually of the same type. We shall often
assume that a relation, usually denoted and read “less than” or “process”, linearly orders atoms. A
linear order < on a set S satisfies two properties:
1. For any a and b in S, exactly one a< b, a=b or b< a is true.
2. For all a, b and c in S, if a<b and b<c then a<c (transitivity).

A set is an unordered collection of particular objects. These objects are referred to as elements.
An example is shown as:
A = {1, 2, 3, 4, 5, 9, 10}
Where. A is the set and the elements are identified within the braces. Generally a set is denoted
by a capital letter while lower case letters represents the elements. For a large series with a
pattern, the set could be designated as:

N = {1, 2, 3, 4, 5, … }

The use of the symbol ∈ in set theory means that object is an element of a set. For example, a ∈ S
means that a is an element within the set S. More than one element can be identified such as a, b
∈ S which indicates that both a and b are elements of the set S. Continuing in this vein, ∉ means
that an element is not in a set. For example, a ∉ S states that a is not an element within the set S.

A subset is a set where every element belongs to another set. The symbols ⊂ and ⊃ are used to
designate subsets, depending upon the syntax. If all of the elements of V are also elements of T
then V is a subset of T and this can be written as V ⊂ T , which can be interpreted as saying V is
contained in T, or T ⊃ V, which can be used to imply that T contains V. For example,
T = {all letters in the English alphabet}
V = {all vowels}

then V⊂T

If there exists one element that exists in V that is not in T, then R is not a subset and this
is designated as V ⊄ T.

23

If two sets, S and T, contain exactly the same identical elements, then they are equal and
S = T. It is also true that S ⊂ T and T ⊂ S. If on the other hand, S ⊂ T and S ≠ T then S is called a
proper subset of T. This means that there is at least one element of T, which is not an element of
S.

It is important to recognize the syntax utilized in set theory. For example, if the set S = {a,
b, c}, then a ∈ S is the correct form that means a is an element of the set S. But, using {a} ∈ S is
incorrect because the braces mean a set. S can consist of subsets that contain only one element.
For example, {a} can be used to designate a subset of S containing only a. Then, it is proper to
state that {a} ⊂ S which means that the set containing {a} is contained within S.

It is also possible to represent a set of elements in one set from another set, which has
certain properties. This is written as:

S = {x ∈ T | x has the property p}

or, if the set T is clear to the user, this can be shortened to the form:

S = {x | x has the property p}

The following example shows the set S contains all of those elements from the set of natural
numbers N, where x + 2 = 7.

S = {x ∈ N | x + 2 = 7}

Since S contains only one element, 5, this is called a unit set. Another example shows that the
definition of a particular set can be done in different fashions. In this first definition, the set E will
contain all even natural numbers:

E = {x ∈ N | x is even}
hence,
E = {2, 4, 6, 8, … }

Another way of defining this set is as follows:

E = {2n | n ∈ N }

By convention, uses of certain letters usually define the element of a set. For example, the
null set, ∅, is an empty set, which contains no elements. ∅ is a subset of all sets. The universal
set, U, is the set consisting of all the elements that are of interest for a problem.
N is generally used for natural values (N = {1, 2, 3, 4, ...}, sometimes 0 is considered a
natural number), and Z is used for integers (Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}), and R represents all
real numbers.

The union of sets defines another set, which consists of elements from two or more sets. It
can be shown mathematically as A ∪ B whose elements are {x | x ∈ A or x ∈ B}. This operation is
often called a Boolean OR operation since the element must be in either set A or set B to be
included in the newly formed set. For example, if

A = {0, 2, 3, 5, 7, 9, 11}
and
B = {1, 2, 4, 6, 7, 9, 11}
then
C = A ∪ B = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11}

24

Venn Diagram showing the union of two
sets.

A figure called a Venn diagram is often used to depict the different relationships in set theory. If
the two sets have at least one common element then they are called conjoint. If, on the other
hand, there are no common elements in the two sets then they are said to be disjoint. For
example, E = {all male students at Ferris} and F = {all female students at Ferris}, the sets are
disjoint because a student cannot be a member of both groups. In the example above, the sets A
and B are conjoint since there is at least one common element.

The intersection of two sets is denoted as C = A ∩ B where the elements of this new set are
defined by {x | x ∈ A and x ∈ B}. This is sometimes called a Boolean AND operation since an
element must be in both sets to be included in the new set. If the sets are disjoint then A ∩ B = 0.
In the example above, where:

A = {0, 2, 3, 5, 7, 8, 9, 11}
and
B = {1, 2, 4, 6, 7, 9, 11}
then
C = A ∩ B = {2, 7, 9, 11}

Venn Diagram showing the intersection of
two sets.

The complement of a set is defined by the elements that are a part of the universal set but not in
the subset. If A is a subset then the complement is shown as Ac or comp (A). As an example, if
U = {all male and female students at FSU}
then the subset
A = {all male students at FSU}
then
25

comp (A) = {all female students at FSU}

Venn Diagram showing the A minus B set
operation.

Another useful set of operations is AB which means A minus B. The new set will contain all of
those elements of A that are not in B or {x | x ∈ A and x ∉ B}. Using the examples of A and B from
above,

C = AB = {0, 3, 5, 8}

Sets also follow the distributive law. Thus,

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
and
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

While not a proof, this property is depicted in using Venn diagrams.

26

A B A B A B

C C C

a

A B A B A B

C C C

b
Graphical "proof" of the distributive law.

3.3 A Bit –Vector Implementation of Sets
The best implementation of a SET ADT depends on the operation to be performed and on
the size of the set. When all sets in our domain of discourse are subsets of a small “universal set”
whose elements are the integers 1,2,3…………….N for some fixed N, then we can use a bit-vector
(boolean array) implementation. A set is represented by a bit vector in which the I th bit is true if I
is an element of the set. The major advantage of this representation is that MEMBER, INSERT and
DELETE operation can be performed in constant time by directly addressing the appropriate bit.
UNION, INTERSECTION and DIFFERENCE can be performed in time proportional to the size of
the universal set.

3.4 Link-list implementation of sets: -
It should also be evident that linked lists can represent sets, where the items in the list are the
elements of the set. Unlike the bit-vector representation, the list representation uses space
proportional to the size of the set represented, not the size of the universal set.

When we have operations like INTERSECTION on sets represented by linked lists, we have several
options. If the universal set is linearly ordered, then we can represent a set by a stored list. That
is, we assume all set members are comparable by a relation “<” and the members of a set appear
on a list in the order e1, e2, e3………………..en, where e1<e2<e3<………………en. The advantage of a

27

stored list is that we do not need to search the entire list to determine whether an element is on
the list.

3.5 The data dictionary: -

“A data dictionary is used to describe all aspects of a computer based information system: -
in particular its data and its processes, i.e. data about data.”

“To DOCUMENT for all posterity the data elements, structures, flows, stores, processes, and
external entities in the system to be designed.”

Specific arrangements of data attributed (elements) that define the organization of a single
instance of a data flow. Data structures belong to a particular data store. Think about this. For
example, you may desire to collect information about an instance of an "employee." Employees
pave payroll information; perhaps work address information, and human resources information.
Because the purpose of this information is to serve several systems or data flows, you might want
to store the whole of the employee information in several data structures.

Data Elements: -

The descriptive property or characteristic of an entity. In database terms, this is a "attribute" or a
"field." In Microsoft and other vendor terms, this is a "property." Therefore, the data element is
part of a data structure, which is part of a data store. It has a source, and there may be data
entry rules you wish to enforce on this element.

Data Flows:

Represent an input of data to a process or the output of data (or information) from a process. A
data flow is also used to represent the creation, deletion, or updating of data in a file or database
(called a "data sore" in the Data Flow Diagram (DFD.) Note that a data flow is not a process, but
the passing of data only. These are represented in the DFD. Each data flow is a part of a DFD
Explosion of a particular process.

When we use a set in the design of an algorithm, we may not need powerful operations like union
and intersection. Often, we only need to keep a set of “current” objects, with periodic insertions
and deletions from the set. Also, from time to time we may need to know whether a particular
element is in the set. A set ADT with the operations INSERT, DELETE and MEMBER has been
given the name dictionary. We shall also include MAKENULL as a dictionary operation to initialize
whether data structure is used in the implementation.

28

UNIT 4

Algorithms Analysis Techniques

4.2 Analysis of recursive programs
4.3 Solving recurrence equation

1. Algorithms (Algorithm is a well-defined sequence of steps)
2. Computational resources: time and space (which leads to solving a certain problem. Steps
should be:
a. Unambiguous
b. There should be finitely many of them)
3. Best, worst and average case performance
4. How to compare algorithms: machine-independent measure of efficiency
5. Growth rate.
6. Complexity measure O ( ).

Best, worst and average case

• Best performance: the item we search for is in the first position; examines one position.
• Worst performance: item not in the array or in the last position; examines all positions.
• Average performance (given that the item is in the array): examines half of the array.

Rate of Growth

We don't know how long the steps actually take; we only know it is some constant time. We can
just lump all constants together and forget about them.

What we are left with is the fact that the time in sequential search grows linearly with the input,
while in binary search it grows logarithmically much slower.

Big Oh complexity measure

Big O notation gives an asymptotic upper bound on the actual function, which describes
time/memory usage of the algorithm.

The complexity of an algorithm is O(f(N)) if there exists a constant factor K and an input size N0
such that the actual usage of time/memory by the algorithm on inputs greater than N0 is always
less than K f(N).

Big O

"Big O" refers to a way of rating the efficiency of an algorithm. It is only a rough estimate of the
actual running time of the algorithm, but it will give you an idea of the performance relative to the
size of the input data.

29

An algorithm is said to be of order O(expression), or simply of order expression (where
expression is some function of n, like n2 and n is the size of the data) if there exist numbers p, q
and r so that the running time always lies below between p.expression+q for n > r. Generally
expression is made as simple and as small as possible.

For example, the following piece of code executes the innermost code(n2 - n) / 2, and the whole
expression for the running time will be a quadratic. Since this lies below n2 for all n > 100, the
algorithm is said to have order O(n2).

for i := n downto 2 do
begin
h := 1;
for j := 2 to i do
if list[j] > list[h] then h := j;
temp := list[h];
list[h] := list[i];
list[i] := list[temp];
end;

(this is the algorithm for Selection Sort)

Choosing an algorithm

Choosing an algorithm isn't a case of simply picking the fastest one and implementing it.
Especially in a competition, you will also have to balance this against the time it will take you to
code the algorithm. In fact you should use the simplest algorithm that will run in the time
provided, since you generally get no more points for a faster but more complex algorithm. In fact it
is sometimes worth implementing an algorithm that you know will not be fast enough to get
100%, since 90% for a slow, correct algorithm is better than 0% for a fast but broken one.

When choosing an algorithm you should also plan what type of data structures you intend
to use. The speed of some algorithms depends on what types of data structures they are combined
with.

Analysis of recursive programs
The techniques for solving different equation are sometimes subtle and bear considerable
resemblance to the methods for solving differential equation. Consider the sorting program, there
the procedure merge sort takes a list of length n as input and returns a sorted list as its output.
The procedure merge(L1, L2) takes as input two sorted list L1 and L2, scans them each, element
by element, from the front. At each step, the larger of the two front elements is deleted from its list
and emitted as output. The result is a single sorted list containing the elements of L1 and L2. The
time taken by merge on lists of length n/2 is O(n).
function mergesort(L : LIST, n : integer) : LIST;
{ L is a list of length n. A sorted version of is
returned. We assume n is a power of 2. }
var
L1,L2: LIST
begin
if n=1 then
return(L);
else begin
Break L into two halves L1 and L2, each of length n/2;
return(merge(mergesort(L1,n/2), mergesort(L2,n/2)));

30

end
end; { mergesort }

Let T(n) be the worst case running time of procedure mergesort, we can write a recurrence
( or difference ) equation the upper bounds T(n), as follows

T(n) <= c1 if n=1
Or
T(n) < = 2T(n/2) + c2 n if n>1
C1 and C2 are constant.

Solving recurrence equation

There are three different approaches we might take to solving a recurrence equation.
1. Guess a solution f(n) and use the recurrence to show that T(n) <= f(n). Some times we
guess only the form of f(n), leaving some parameters unspecified (e.g guess f(n)=an 2 for
some a) and deduce suitable values for the parameters as we try to prove T(n) <= f(n) for all
n.
2. Use the recurrence itself to substitute for any T(m), m < n, on the right until all terms T(m)
for m > 1 have been replaced by formulas involving only T(1) . Since T(1) is always a
constant, we have a formula for T(n) in terms of n and constants. This formula is what we
have referred to as a “closed form” for T(n).

31

UNIT 5

Algorithms Design Technique

5.1 Divide and conquer algorithms
5.2 Dynamic programming
5.3 Greedy algorithm
5.4 Minimum-cost spanning trees
5.5 Minimum Spanning Tree
5.8 Shortest Paths
5.9 Dijkastra’s Algorithm
5.10 Backtracking

5.1 Divide-and-Conquer algorithms
This is a method of designing algorithms that (informally) proceeds as follows:
Given an instance of the problem to be solved, split this into several, smaller, sub-instances (of
the same problem) independently solve each of the sub-instances and then combine the sub-
instance solutions so as to yield a solution for the original instance. This description raises the
question:
By what methods are the sub-instances to be independently solved?
The answer to this question is central to the concept of Divide-&-Conquer algorithm and is
a key factor in gauging their efficiency.
Consider the following: We have an algorithm, alpha say, which is known to solve all
problem instances of size n in at most c n^2 steps (where c is some constant). We then discover
an algorithm, beta say, which solves the same problem by:
• Dividing an instance into 3 sub-instances of size n/2.
• Solves these 3 sub-instances.
• Combines the three sub-solutions taking d n steps to do this.
Suppose our original algorithm, alpha, is used to carry out the `solves these sub-instances' step 2.
Let
T(alpha)( n ) = Running time of alpha
T(beta)( n ) = Running time of beta
Then,
T(alpha)( n ) = c n^2 (by definition of alpha)
But

T(beta)( n ) = 3 T(alpha)( n/2 ) + d n

= (3/4)(cn^2) + dn
So if dn < (cn^2)/4 (i.e. d < cn/4) then beta is faster than alpha
In particular for all large enough n, (n > 4d/c = Constant), beta is faster than alpha.
This realisation of beta improves upon alpha by just a constant factor. But if the problem
size, n, is large enough then

n > 4d/c
n/2 > 4d/c
...
n/2^i > 4d/c

32

Which suggests that using beta instead of alpha for the `solves these' stage repeatedly until the
sub-sub-sub..sub-instances are of size n0 < = (4d/c) will yield a still faster algorithm.
So consider the following new algorithm for instances of size n

Procedure gamma (n : problem size ) is
Begin
if n <= n^-0 then
Solve problem using Algorithm alpha;
else
Split into 3 sub-instances of size n/2;
Use gamma to solve each sub-instance;
Combine the 3 sub-solutions;
end if;
end gamma;
Let T(gamma)(n) denote the running time of this algorithm.

cn^2 if n < = n0
T(gamma)(n) =
3T(gamma)( n/2 )+dn otherwise

We shall show how relations of this form can be estimated later in the course. With these methods
it can be shown that
T(gamma)( n ) = O( n^{log3} ) (=O(n^{1.59..})
This is an asymptotic improvement upon algorithms alpha and beta.
The improvement that results from applying algorithm gamma is due to the fact that it
maximise the savings achieved beta.
The (relatively) inefficient method alpha is applied only to "small" problem sizes.
The precise form of a divide-and-conquer algorithm is characterised by:
• The threshold input size, n0, below which the problem size is not sub-divided.
• The size of sub-instances into which an instance is split.
• The number of such sub-instances.
• The algorithm used to combine sub-solutions.
It is more usual to consider the ratio of initial problem size to sub-instance size. The threshold in
(I) is sometimes called the (recursive) base value. In summary, the generic form of a divide-and-
conquer algorithm is:
Procedure D-and-C (n : input size) is
Begin
if n < = n0 then
Solve problem without further
sub-division;
else
Split into r sub-instances
each of size n/k;
for each of the r sub-instances do
D-and-C (n/k);
Combine the r resulting
sub-solutions to produce
the solution to the original problem;
end if;
end D-and-C;

33

5.2 Dynamic Programming

The term ``Dynamic Programming" (DP) refers to a collection of algorithms that can be
used to compute optimal policies given a perfect model of the environment as a Markov decision
process (MDP). Classical DP algorithms are of limited utility in reinforcement learning both
because of their assumption of a perfect model and because of their great computational expense,
but they are still very important theoretically. DP provides an essential foundation for the
understanding of the methods presented in the rest of this book. In fact, all of these methods can
be viewed as attempts to achieve much the same effect as DP, only with less computation and
without assuming a perfect model of the environment.

Starting with this chapter, we usually assume that the environment is a finite MDP. That is, we

assume that its state and action sets, and , for , are finite, and that its dynamics

are given by a set of transition probabilities, , and

expected immediate rewards, , for all ,

, and ( is plus a terminal state if the problem is episodic). Although DP
ideas can be applied to problems with continuous state and action spaces, exact solutions are
possible only in special cases. A common way of obtaining approximate solutions for continuous
state and action tasks is to quantize the state and action spaces and then apply finite-state DP
methods.

The key idea of DP, and of reinforcement learning generally, is the use of value functions
to organize and structure the search for good policies. As discussed there, we can easily obtain

optimal policies once we have found the optimal value functions, or , which satisfy the
Bellman optimality equations:

or

for all , , and . As we shall see, DP algorithms are obtained by turning
Bellman equations such as these into assignment statements, that is, into update rules for
improving approximations of the desired value functions.

34

5.3 Greedy Algorithm
An algorithm is a step-by-step recipe for solving a problem. A greedy algorithm might also
be called a "single-minded" algorithm or an algorithm that gobbles up all of its favorites first. The
idea behind a greedy algorithm is to perform a single procedure in the recipe over and over again
until it can't be done any more and see what kind of results it will produce. It may not completely
solve the problem, or, if it produces a solution, it may not be the very best one, but it is one way of
approaching the problem and sometimes yields very good (or even the best possible) results.

5.4 Minimum-Cost spanning trees

Let G=(V,E) be an undirected connected graph. A sub-graph t = (V,E1) of G is a spanning
tree of G if and only if t is a tree.

Above figure shows the complete graph on four nodes together with three of its spanning
tree.
Spanning trees have many applications. For example, they can be used to obtain an
independent set of circuit equations for an electric network. First, a spanning tree for the electric
network is obtained. Let B be the set of network edges not in the spanning tree. Adding an edge
from B to the spanning tree creates a cycle. Kirchoff’s second law is used on each cycle to obtain a
circuit equation.
Another application of spanning trees arises from the property that a spanning tree is a
minimal sub-graph G’ of G such that V(G’) = V(G) and G’ is connected. A minimal sub-graph with
n vertices must have at least n-1 edges and all connected graphs with n-1 edges are trees. If the
nodes of G represent cities and the edges represent possible communication links connecting two
cities, then the minimum number of links needed to connect the n cities is n-1. The spanning
trees of G represent all feasible choice.
In practical situations, the edges have weights assigned to them. These weights may
represent the cost of construction, the length of the link, and so on. Given such a weighted graph,
one would then wish to select cities to have minimum total cost or minimum total length. In either
case the links selected have to form a tree. If this is not so, then the selection of links contains a
cycle. Removal of any one of the links on this cycle results in a link selection of less const
connecting all cities. We are therefore interested in finding a spanning tree of G. with minimum
cost since the identification of a minimum-cost spanning tree involves the selection of a subset of
the edges; this problem fits the subset paradigm.

5.5 Minimum Spanning Tree: -
Let G = (V,E) be an undirected connected graph. A subgraph t = (V, E 1) of g is a spanning tree of g
if t is a tree.

Example: shows the complete graph on four nodes together with three of its spanning trees.

35

Spanning trees have many applications. For example, they can be used to obtain an independent set of circuit equations
for an electric network. In practical situations, the edges have weights assigned to them. These weights may represent
the cost of construction, the length of the link, and so on. Given such a weighted graph, one would then wish to select
cities to have minimum total cost or minimum total length. In either case the links selected have to form a tree. We are
therefore interested in finding a spanning tree g with minimum cost. A graph and one of its minimum spanning tree
involves the selection of a subset of the edges, this problem fits the subset paradigm.

1 1
10 28 10

6 2 6 2
14 16 25 14 16
25 3 7 3
7 5
24 18 12 12
5 4 22 4
22
Figure (a): Graph Fig. (b): Minimum spanning tree
We present two algorithms for finding a minimum spanning tree of a weighted graph: Prim’s
algorithm and Kruskal’s algorithm.

A greedy method to obtain a minimum spanning tree is the to build the tree edge by edge. The
next edge to include is chosen according to some optimization criterion. The simplest such
criterion is to choose an edge that results in a minimum increase in the sum of the costs of the
edges so far included. There are two possible ways to interpret this criterion. In the first, the set of
edges so far selected from a tree. Thus if A is the set of edges selected so far, then A forms a tree.
The next edge (u,v) to be included in A is a minimum cost edge not in A with property that A ∪
{(u,v)} is also a tree. The following example shows this selection criterion results in a minimum
spanning tree. The corresponding algorithm is known as Prim’s algorithm.

Example shows the working of Prim’s method on the graph . The spanning tree obtained is shown
in figure (b) and has a cost of 99.

36

1 1
10 10

6 2 6 2
7 25 7
3 3
5 5
5 4

Fig. (a) Fig. (b)

1 1
10 10

6 2 6 2
7 25 7
25 3 3
5 5
4 4 12
22 22
Fig. (c) Fig. (d)

1 1
10 10

6 2 16 6 2 16
7 25 7 14
25 3 3
5 12 5
4 4 12
22 22

Fig. (e) Fig. (f)

Having seen how Prim’s method works, let us obtain a n algorithm to find a minimum cost
spanning tree using this method. The algorithm will start with a tree that include only a minimum
cost edge of g. Then, edges are added to this tree one by one. The next edge (i, j) to be added is
such that I is a vertex already included in the tree, j is a vertex not yet included, and the cost of (i,
j), cost [i, j], is minimum among all edges (k, l) such that vertex k is in the tree and vertex e is not
in the tree. To determine this edge (i, j) efficiently, we associate with each vertex j not yet included
37

in the tree a value near [j]. The value near [j] [near (j)] is minimum among all choices for near [j].
We define near [j] = 0 for all vertices j that are already in the tree. The next edge to include is
defined by the vertex j such that near [j] # 0 (j not already in the tree) and cost [j] [near (j)] is
minimum.
Prim (E, cost, n, t)
//E is the set of edges in g. cost [n] [n] is
// the cost adjacmcy matrix of an n vertex
//graph such that cost [i,j] is
//either a positive real number or is it
//no edge (i, j) exists.
//A minimum spanning tree is computed
//and stored as a set of edges in
//The array t[n–1] [2]. (The final cost is
//returned.
{
Let (k, l) be an edge of minimum cost in E;
Min cost = cost [k] [l];
t [1] [1] = k; k[1] [2] = l;
for (i = 1; i < = n : i + l)
if (cost [i] [l] < cost [i] [k])
near [i] = l;
else
near [i] = k;
near [k] = near [l] = 0;
for (i = 2; i < = n–1; i + l)
{ //Find n–2 additional edges for t.
Let j be an index such that near [j] ! = 0 and
Cost [j] near [(j)] is minimum;
T [i] [1] = j; t [i] [2] = near [j];
min cost = mincost + cost [j] [near [j]];
near [j] = 0;
for (k = 1; k < = n; k + l) //update near
if (near [k] ! = 0 && cost [k] [near [k] > cost [k] [j])
near [k] = j;
}
return (min cost);

38

}
The time required by algorithm prim is 0 (n2), where n is the number of vertices in the
graph g.

There is a second possible interpretation of the optimization criteria mentioned earlier in
which the edges of the graph are considered in non-decreasing order of cost. This interpretation is
that the set t of edges so far selected for the spanning tree be such that it is possible to complete t
into a tree. Thus t may not be a tree at all stages in the algorithm. In fact it will generally only be
a forest since the set of edges t can be completed into a tree if there are no cycles in t. This
method is due to kruskal.

Example: Consider the graph of figure (a). We begin with no edges selected figure(a) shows
the current graph with no edges selected Edge (1,6) is the first edge considered. It is included in
the spanning tree being built. This yields the graph of figure (b). Next the edge (3,4) is selected and
included in the tree (fig. (c)). The next edge to be considered is (2,7). Its inclusion in the tree being
built does not create a cycle, so we get the graph of figure. Edge (2,3) is considered next and
included in the tree figure (e). Of the edges not yet considered (7,4) has the least cost. It is
considered next. Its inclusion in the tree results in a cycle, so this edge is discarded. Edge (5,4) is
the next edge to be added in the tree being built. This result in the configuration of figure (f). The
next edge to be considered is the edge (7,5). It is discarded as its inclusion creates a cycle. Finally
edge (6,5) is considered an included in the tree built. This completes the spanning tree. The
resulting tree has cost 99.

1 1
10

6 2 6 2
7 7
3 3
5 5
4 4

Fig. (a) Fig. (b)

1 1
10 10

6 2 6 2

39

7 14
3 7 3
5 12 5
4 4 12

Fig. (c) Fig. (d)

1 1
10 10
2
6 16 6 2 16
7 14 14
3 7 3
5 12 12
4 4
5 22

Fig. (e) Fig. (f)

For clarity, kruskal’s method is written out more formally in following algorithm.
1. t = 0;
2. while [(it has less than n–1 edges) R& (E! = 0)]
3. {
4. Choose an edge (u,v) from E of lowest cost;
5. Delete (q, w) from E;
6. If (u, w) does not create a cycle in it)
add (v,w) to t;
7. else
Discard (v, w);
8. }
Initially E is the set of all edges in g. The only functions we wish to perform on this set are
(1) determine an edge with minimum cost (line 4) and (2) delete this edge (line 5). Both these
functions can be performed efficiently if the edges in E are maintained as a sorted sequential list.
It is not essential to sort all the edges so long as the next edge for line 4 can be determined easily.
If the edges are maintained as a min heap, then the next edge to consider can be obtained in 0
(long |E|) line. The construction of heap it self take O (|E|) time. To be able to perform step 6
efficiently, the vertices in g should be grouped together in such a way that one can easily
determine whether the vertices v and w are already connected by the earlier selection of edges. If
40

they are, then the edge (v,w) is to be added to t. One possible grouping is to place all vertices in
the same connected component by t into a set. For example, when the edge (2,6) is to be
considered, the sets are {1,2}, {2,4,6}, and {5}. Vertices 2 and 6 are in different sets so these sets
are combined to give {1,2,3,4,6} and {5}. The next edge to be considered is (1,4). Since vertices 1
and 4 are in the same set, the edge is rejected. The edge (3,5) connects vertices in different sets
and results in the final spanning tree.

5.8 Shortest Paths
Graphs can be used to represent the highway structure of a state on country with vertices
representing cities and edges representing sections of highway. The edge can them be assigned
weights which may be either the distance along that section of highway. A motoriot wishing to
drive from city A to B would be interested in answers to the following question:

• Is there a path from A to B?

• If there is more than one path from A to B, which is the shortest path?

The problems defined by these questions are special cases of the path problems we study in
this section. The length of a path is now defined to be the sum of the weights of the edges or that
path. The starting vertex of the path is referred to as the source, and the last vertex the
destination. The graphs are digraphs to allow for one-way structs. In the problem we consider we
are given a directed graph g = (V,E), a weighting function cost for the edges of g, and a source
vertex V0. The problem is to determine the shortest path from V0 to all the remaining vertices of
g. It is assumed that all the weight are positive.

5.9 Dijkstra’s Algorithm
This algorithm determines the lengths of the shortest paths from v0 to all other vertices in g.
Dijkstra’s (v, cost, dist, n)
//dist [j], 1< = j < = n, is set to the legnth
//of the hortest path from vertex v to
//vertex j in a diagraph g with n
//vertices dist [v] set to zero. G is
//represented by its cost adjacency matrix
//cost [n] [n].
{
for (i = 1; i < = n; i &&)
{ //intializes
S [i] = false; dist [i] = cost [v][i]
}
S [v] = true; dist [v] = 0.0; ||put v in S.
{
//Determines n–1 paths from v.

41

Choose u from among these vertices not in S such that digit [u] is minimum;
S[u] = true; ||put u in S
For (each is adjacent to u with S[w] = false)
//update distances
dist [w] = dist [u] + cost [u] [w]
}
}

Example: Consider the eight vertex diagraph with cost adjacency matrix as in figure. The values of
dist and the vertices selected at each iteration of for loop of line 12 is previous algorithm, for
finding all the shortest paths from Boston are shown in figure. To begin with, S contains only
Boston. In the first iteration of for loop (that is num = 2), the city u that is not in S and whose dist
[4] is minimum is identified to be New York. In the next iteration of the for loop, the city that
enters S is Miami since it has the smallest dist [ ] value from among all the nodes not in S. None
of the dist [ ] values are altered. The algorithm when only seven of the eight vertices are in S. By
the definition of dist, the distance of the last vertex, in this case Los Angeles, is correct as the
shortest path from Boston to Los Angeles can go through only the remaining six vertices.

Boston
1500 5
Chicago

4 250
1200 1000

San Francisco 6 New York
800
2 3
300 Denver 1400 900
1000

1 8 1000
Los Angeles New Orleans
1700 7 Miami

Figure (a)

1 2 3 4 5 6 7 8

1 0
2 300 0
3 100 800 0
4 1200 0
5 1500 0 250
6 1000 0 900 1400
7 0 1000
8 1700 0
42

Figure (b)

Ilevation S Vertex Distance

Selcted LA SF DEN CHI BOST NY WA NO

[1] [2] [3] [4] [5] [6] [7] [8]

Intial — — +00 +00 +00 1500 0 250 +00 +00

1 {5} 6 +00 +00 +00 1250 0 250 1150 1650

2 {5,6} 7 +00 +00 +00 1250 0 250 1150 1650

3 {5,6,7} 4 +00 +00 2450 1250 0 250 1150 1650

4 {5,6,7,4} 8 3350 +00 2450 1250 0 250 1150 1650

5 {5,6,7,4,8} 3 350 3250 2450 1250 0 250 1150 1650

6 {5,6,7,4,8,3} 2 3350 3250 2450 1250 0 250 1150 1650

{5,6,7,4,8,3,2}

Figure

5.10 Backtracking
Backtracking is a systematic way to go through all the possible configurations of a space.
These configurations may be all possible arrangements of objects (permutations) or all possible
ways of building a collection of them (subsets). Other applications may demand enumerating all
spanning trees of a graph, all paths between two vertices, or all possible ways to partition the
vertices into color classes.
What these problems have in common is that we must generate each one of the possible
configurations exactly once. Avoiding both repetitions and missing configurations means that we
must define a systematic generation order among the possible configurations. In combinatorial
search, we represent our configurations by a vector , where each element is
selected from an ordered set of possible candidates for position i. As shown below, this
representation is general enough to encode most any type of combinatorial object naturally.
The search procedure works by growing solutions one element at a time. At each step in
the search, we will have constructed a partial solution with elements fixed for the first k elements
of the vector, where . From this partial solution , we will construct the set of
possible candidates for the (k+1)st position. We will then try to extend the partial solution by
adding the next element from . So long as the extension yields a longer partial solution, we
continue to try to extend it.
However, at some point, might be empty, meaning that there is no legal way to extend
the current partial solution. If so, we must backtrack, and replace , the last item in the solution
value, with the next candidate in . It is this backtracking step that gives the procedure its name:

43

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