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Data Structure & Algorithm Basic Concepts [Part - 1]

 

Data Definition

Data Definition defines a particular data with the following characteristics.

  • Atomic − Definition should define a single concept.

  • Traceable − Definition should be able to be mapped to some data element.

  • Accurate − Definition should be unambiguous.

  • Clear and Concise − Definition should be understandable.

Data Object

Data Object represents an object having a data.

Data Type

Data type is a way to classify various types of data such as integer, string, etc. which determines the values that can be used with the corresponding type of data, the type of operations that can be performed on the corresponding type of data. There are two data types −

  • Built-in Data Type
  • Derived Data Type

Built-in Data Type

Those data types for which a language has built-in support are known as Built-in Data types. For example, most of the languages provide the following built-in data types.

  • Integers
  • Boolean (true, false)
  • Floating (Decimal numbers)
  • Character and Strings

Derived Data Type

Those data types which are implementation independent as they can be implemented in one or the other way are known as derived data types. These data types are normally built by the combination of primary or built-in data types and associated operations on them. For example −

  • List
  • Array
  • Stack
  • Queue

Basic Operations

The data in the data structures are processed by certain operations. The particular data structure chosen largely depends on the frequency of the operation that needs to be performed on the data structure.

  • Traversing
  • Searching
  • Insertion
  • Deletion
  • Sorting
  • Merging
  • Array is a container which can hold a fix number of items and these items should be of the same type. Most of the data structures make use of arrays to implement their algorithms. Following are the important terms to understand the concept of Array.

    • Element − Each item stored in an array is called an element.

    • Index − Each location of an element in an array has a numerical index, which is used to identify the element.

    Array Representation

    Arrays can be declared in various ways in different languages. For illustration, let's take C array declaration.

    Array Declaration

    Arrays can be declared in various ways in different languages. For illustration, let's take C array declaration.

    Array Representation

    As per the above illustration, following are the important points to be considered.

    • Index starts with 0.

    • Array length is 10 which means it can store 10 elements.

    • Each element can be accessed via its index. For example, we can fetch an element at index 6 as 9.

    Basic Operations

    Following are the basic operations supported by an array.

    • Traverse − print all the array elements one by one.

    • Insertion − Adds an element at the given index.

    • Deletion − Deletes an element at the given index.

    • Search − Searches an element using the given index or by the value.

    • Update − Updates an element at the given index.

    In C, when an array is initialized with size, then it assigns defaults values to its elements in following order.

    Data TypeDefault Value
    boolfalse
    char0
    int0
    float0.0
    double0.0f
    void
    wchar_t0

    Traverse Operation

    This operation is to traverse through the elements of an array.

    Example

    Following program traverses and prints the elements of an array:

    #include <stdio.h>
    main() {
       int LA[] = {1,3,5,7,8};
       int item = 10, k = 3, n = 5;
       int i = 0, j = n;   
       printf("The original array elements are :\n");
       for(i = 0; i<n; i++) {
          printf("LA[%d] = %d \n", i, LA[i]);
       }
    }

    When we compile and execute the above program, it produces the following result −

    Output

    The original array elements are :
    LA[0] = 1 
    LA[1] = 3 
    LA[2] = 5 
    LA[3] = 7 
    LA[4] = 8 
    

    Insertion Operation

    Insert operation is to insert one or more data elements into an array. Based on the requirement, a new element can be added at the beginning, end, or any given index of array.

    Here, we see a practical implementation of insertion operation, where we add data at the end of the array −

    Example

    Following is the implementation of the above algorithm −


    #include <stdio.h>
    
    main() {
       int LA[] = {1,3,5,7,8};
       int item = 10, k = 3, n = 5;
       int i = 0, j = n;
       
       printf("The original array elements are :\n");
    
       for(i = 0; i<n; i++) {
          printf("LA[%d] = %d \n", i, LA[i]);
       }
    
       n = n + 1;
    	
       while( j >= k) {
          LA[j+1] = LA[j];
          j = j - 1;
       }
    
       LA[k] = item;
    
       printf("The array elements after insertion :\n");
    
       for(i = 0; i<n; i++) {
          printf("LA[%d] = %d \n", i, LA[i]);
       }
    }

    When we compile and execute the above program, it produces the following result −

    Output

    The original array elements are :
    LA[0] = 1 
    LA[1] = 3 
    LA[2] = 5 
    LA[3] = 7 
    LA[4] = 8 
    The array elements after insertion :
    LA[0] = 1 
    LA[1] = 3 
    LA[2] = 5 
    LA[3] = 10 
    LA[4] = 7 
    LA[5] = 8 
    

    For other variations of array insertion operation click here

    Deletion Operation

    Deletion refers to removing an existing element from the array and re-organizing all elements of an array.

    Algorithm

    Consider LA is a linear array with N elements and K is a positive integer such that K<=N. Following is the algorithm to delete an element available at the Kth position of LA.

    1. Start
    2. Set J = K
    3. Repeat steps 4 and 5 while J < N
    4. Set LA[J] = LA[J + 1]
    5. Set J = J+1
    6. Set N = N-1
    7. Stop
    

    Example

    Following is the implementation of the above algorithm −

    #include <stdio.h>
    
    void main() {
       int LA[] = {1,3,5,7,8};
       int k = 3, n = 5;
       int i, j;
       
       printf("The original array elements are :\n");
    	
       for(i = 0; i<n; i++) {
          printf("LA[%d] = %d \n", i, LA[i]);
       }
        
       j = k;
    	
       while( j < n) {
          LA[j-1] = LA[j];
          j = j + 1;
       }
    	
       n = n -1;
       
       printf("The array elements after deletion :\n");
    	
       for(i = 0; i<n; i++) {
          printf("LA[%d] = %d \n", i, LA[i]);
       }
    }

    When we compile and execute the above program, it produces the following result −

    Output

    The original array elements are :
    LA[0] = 1 
    LA[1] = 3 
    LA[2] = 5 
    LA[3] = 7 
    LA[4] = 8 
    The array elements after deletion :
    LA[0] = 1 
    LA[1] = 3 
    LA[2] = 7 
    LA[3] = 8 
    

    Search Operation

    You can perform a search for an array element based on its value or its index.

    Algorithm

    Consider LA is a linear array with N elements and K is a positive integer such that K<=N. Following is the algorithm to find an element with a value of ITEM using sequential search.

    1. Start
    2. Set J = 0
    3. Repeat steps 4 and 5 while J < N
    4. IF LA[J] is equal ITEM THEN GOTO STEP 6
    5. Set J = J +1
    6. PRINT J, ITEM
    7. Stop
    

    Example

    Following is the implementation of the above algorithm −

    #include <stdio.h>
    
    void main() {
       int LA[] = {1,3,5,7,8};
       int item = 5, n = 5;
       int i = 0, j = 0;
       
       printf("The original array elements are :\n");
    	
       for(i = 0; i<n; i++) {
          printf("LA[%d] = %d \n", i, LA[i]);
       }
        
       while( j < n){
          if( LA[j] == item ) {
             break;
          }
    		
          j = j + 1;
       }
    	
       printf("Found element %d at position %d\n", item, j+1);
    }

    When we compile and execute the above program, it produces the following result −

    Output

    The original array elements are :
    LA[0] = 1 
    LA[1] = 3 
    LA[2] = 5 
    LA[3] = 7 
    LA[4] = 8 
    Found element 5 at position 3
    

    Update Operation

    Update operation refers to updating an existing element from the array at a given index.

    Algorithm

    Consider LA is a linear array with N elements and K is a positive integer such that K<=N. Following is the algorithm to update an element available at the Kth position of LA.

    1. Start
    2. Set LA[K-1] = ITEM
    3. Stop
    

    Example

    Following is the implementation of the above algorithm −


    #include <stdio.h>
    
    void main() {
       int LA[] = {1,3,5,7,8};
       int k = 3, n = 5, item = 10;
       int i, j;
       
       printf("The original array elements are :\n");
    	
       for(i = 0; i<n; i++) {
          printf("LA[%d] = %d \n", i, LA[i]);
       }
        
       LA[k-1] = item;
    
       printf("The array elements after updation :\n");
    	
       for(i = 0; i<n; i++) {
          printf("LA[%d] = %d \n", i, LA[i]);
       }
    }

    When we compile and execute the above program, it produces the following result −

    Output

    The original array elements are :
    LA[0] = 1 
    LA[1] = 3 
    LA[2] = 5 
    LA[3] = 7 
    LA[4] = 8 
    The array elements after updation :
    LA[0] = 1 
    LA[1] = 3 
    LA[2] = 10 
    LA[3] = 7 
    LA[4] = 8 
  • A linked list is a sequence of data structures, which are connected together via links.

    Linked List is a sequence of links which contains items. Each link contains a connection to another link. Linked list is the second most-used data structure after array. Following are the important terms to understand the concept of Linked List.

    • Link − Each link of a linked list can store a data called an element.

    • Next − Each link of a linked list contains a link to the next link called Next.

    • LinkedList − A Linked List contains the connection link to the first link called First.

    Linked List Representation

    Linked list can be visualized as a chain of nodes, where every node points to the next node.

    Linked List

    As per the above illustration, following are the important points to be considered.

    • Linked List contains a link element called first.

    • Each link carries a data field(s) and a link field called next.

    • Each link is linked with its next link using its next link.

    • Last link carries a link as null to mark the end of the list.

    Types of Linked List

    Following are the various types of linked list.

    • Simple Linked List − Item navigation is forward only.

    • Doubly Linked List − Items can be navigated forward and backward.

    • Circular Linked List − Last item contains link of the first element as next and the first element has a link to the last element as previous.

    Basic Operations

    Following are the basic operations supported by a list.

    • Insertion − Adds an element at the beginning of the list.

    • Deletion − Deletes an element at the beginning of the list.

    • Display − Displays the complete list.

    • Search − Searches an element using the given key.

    • Delete − Deletes an element using the given key.

    Insertion Operation

    Adding a new node in linked list is a more than one step activity. We shall learn this with diagrams here. First, create a node using the same structure and find the location where it has to be inserted.

    Linked List Insertion

    Imagine that we are inserting a node B (NewNode), between A (LeftNode) and C (RightNode). Then point B.next to C −

    NewNode.next −> RightNode;
    

    It should look like this −

    Linked List Insertion

    Now, the next node at the left should point to the new node.

    LeftNode.next −> NewNode;
    
    Linked List Insertion

    This will put the new node in the middle of the two. The new list should look like this −

    Linked List Insertion

    Similar steps should be taken if the node is being inserted at the beginning of the list. While inserting it at the end, the second last node of the list should point to the new node and the new node will point to NULL.

    Deletion Operation

    Deletion is also a more than one step process. We shall learn with pictorial representation. First, locate the target node to be removed, by using searching algorithms.

    Linked List Deletion

    The left (previous) node of the target node now should point to the next node of the target node −

    LeftNode.next −> TargetNode.next;
    
    Linked List Deletion

    This will remove the link that was pointing to the target node. Now, using the following code, we will remove what the target node is pointing at.

    TargetNode.next −> NULL;
    
    Linked List Deletion

    We need to use the deleted node. We can keep that in memory otherwise we can simply deallocate memory and wipe off the target node completely.

    Linked List Deletion

    Reverse Operation

    This operation is a thorough one. We need to make the last node to be pointed by the head node and reverse the whole linked list.

    Linked List Reverse Operation

    First, we traverse to the end of the list. It should be pointing to NULL. Now, we shall make it point to its previous node −

    Linked List Reverse Operation

    We have to make sure that the last node is not the last node. So we'll have some temp node, which looks like the head node pointing to the last node. Now, we shall make all left side nodes point to their previous nodes one by one.

    Linked List Reverse Operation

    Except the node (first node) pointed by the head node, all nodes should point to their predecessor, making them their new successor. The first node will point to NULL.

    Linked List Reverse Operation

    We'll make the head node point to the new first node by using the temp node.

    Linked List Reverse Operation
  • Doubly Linked List is a variation of Linked list in which navigation is possible in both ways, either forward and backward easily as compared to Single Linked List. Following are the important terms to understand the concept of doubly linked list.

    • Link − Each link of a linked list can store a data called an element.

    • Next − Each link of a linked list contains a link to the next link called Next.

    • Prev − Each link of a linked list contains a link to the previous link called Prev.

    • LinkedList − A Linked List contains the connection link to the first link called First and to the last link called Last.

    Doubly Linked List Representation

    Doubly Linked List

    As per the above illustration, following are the important points to be considered.

    • Doubly Linked List contains a link element called first and last.

    • Each link carries a data field(s) and two link fields called next and prev.

    • Each link is linked with its next link using its next link.

    • Each link is linked with its previous link using its previous link.

    • The last link carries a link as null to mark the end of the list.

    Basic Operations

    Following are the basic operations supported by a list.

    • Insertion − Adds an element at the beginning of the list.

    • Deletion − Deletes an element at the beginning of the list.

    • Insert Last − Adds an element at the end of the list.

    • Delete Last − Deletes an element from the end of the list.

    • Insert After − Adds an element after an item of the list.

    • Delete − Deletes an element from the list using the key.

    • Display forward − Displays the complete list in a forward manner.

    • Display backward − Displays the complete list in a backward manner.

    Insertion Operation

    Following code demonstrates the insertion operation at the beginning of a doubly linked list.

    Example

    //insert link at the first location
    void insertFirst(int key, int data) {
    
       //create a link
       struct node *link = (struct node*) malloc(sizeof(struct node));
       link->key = key;
       link->data = data;
    	
       if(isEmpty()) {
          //make it the last link
          last = link;
       } else {
          //update first prev link
          head->prev = link;
       }
    
       //point it to old first link
       link->next = head;
    	
       //point first to new first link
       head = link;
    }

    Deletion Operation

    Following code demonstrates the deletion operation at the beginning of a doubly linked list.

    Example

    //delete first item
    struct node* deleteFirst() {
    
       //save reference to first link
       struct node *tempLink = head;
    	
       //if only one link
       if(head->next == NULL) {
          last = NULL;
       } else {
          head->next->prev = NULL;
       }
    	
       head = head->next;
    	
       //return the deleted link
       return tempLink;
    }

    Insertion at the End of an Operation

    Following code demonstrates the insertion operation at the last position of a doubly linked list.

    Example

    //insert link at the last location
    void insertLast(int key, int data) {
    
       //create a link
       struct node *link = (struct node*) malloc(sizeof(struct node));
       link->key = key;
       link->data = data;
    	
       if(isEmpty()) {
          //make it the last link
          last = link;
       } else {
          //make link a new last link
          last->next = link;     
          
          //mark old last node as prev of new link
          link->prev = last;
       }
    
       //point last to new last node
       last = link;
    }

Circular Linked List is a variation of Linked list in which the first element points to the last element and the last element points to the first element. Both Singly Linked List and Doubly Linked List can be made into a circular linked list.

Singly Linked List as Circular

In singly linked list, the next pointer of the last node points to the first node.

Singly Linked List as Circular Linked List

Doubly Linked List as Circular

In doubly linked list, the next pointer of the last node points to the first node and the previous pointer of the first node points to the last node making the circular in both directions.

Doubly Linked List as Circular Linked List

As per the above illustration, following are the important points to be considered.

  • The last link's next points to the first link of the list in both cases of singly as well as doubly linked list.

  • The first link's previous points to the last of the list in case of doubly linked list.

Basic Operations

Following are the important operations supported by a circular list.

  • insert − Inserts an element at the start of the list.

  • delete − Deletes an element from the start of the list.

  • display − Displays the list.

Insertion Operation

Following code demonstrates the insertion operation in a circular linked list based on single linked list.

Example

insertFirst(data):
Begin
   create a new node
   node -> data := data
   if the list is empty, then
      head := node
      next of node = head
   else
      temp := head
      while next of temp is not head, do
      temp := next of temp
      done
      next of node := head
      next of temp := node
      head := node
   end if
End

Deletion Operation

Following code demonstrates the deletion operation in a circular linked list based on single linked list.

deleteFirst():
Begin
   if head is null, then
      it is Underflow and return
   else if next of head = head, then
      head := null
      deallocate head
   else
      ptr := head
      while next of ptr is not head, do
         ptr := next of ptr
      next of ptr = next of head
      deallocate head
      head := next of ptr
   end if
End

Display List Operation

Following code demonstrates the display list operation in a circular linked list.

display():
Begin
   if head is null, then
      Nothing to print and return
   else
      ptr := head
      while next of ptr is not head, do
         display data of ptr
         ptr := next of ptr
      display data of ptr
   end if
End

A stack is an Abstract Data Type (ADT), commonly used in most programming languages. It is named stack as it behaves like a real-world stack, for example – a deck of cards or a pile of plates, etc.

Stack Example

A real-world stack allows operations at one end only. For example, we can place or remove a card or plate from the top of the stack only. Likewise, Stack ADT allows all data operations at one end only. At any given time, we can only access the top element of a stack.

This feature makes it LIFO data structure. LIFO stands for Last-in-first-out. Here, the element which is placed (inserted or added) last, is accessed first. In stack terminology, insertion operation is called PUSH operation and removal operation is called POP operation.

Stack Representation

The following diagram depicts a stack and its operations −

Stack Representation

A stack can be implemented by means of Array, Structure, Pointer, and Linked List. Stack can either be a fixed size one or it may have a sense of dynamic resizing. Here, we are going to implement stack using arrays, which makes it a fixed size stack implementation.

Basic Operations

Stack operations may involve initializing the stack, using it and then de-initializing it. Apart from these basic stuffs, a stack is used for the following two primary operations −

  • push() − Pushing (storing) an element on the stack.

  • pop() − Removing (accessing) an element from the stack.

When data is PUSHed onto stack.

To use a stack efficiently, we need to check the status of stack as well. For the same purpose, the following functionality is added to stacks −

  • peek() − get the top data element of the stack, without removing it.

  • isFull() − check if stack is full.

  • isEmpty() − check if stack is empty.

At all times, we maintain a pointer to the last PUSHed data on the stack. As this pointer always represents the top of the stack, hence named top. The top pointer provides top value of the stack without actually removing it.

First we should learn about procedures to support stack functions −

peek()

Algorithm of peek() function −

begin procedure peek
   return stack[top]
end procedure

Implementation of peek() function in C programming language −

Example

int peek() {
   return stack[top];
}

isfull()

Algorithm of isfull() function −

begin procedure isfull

   if top equals to MAXSIZE
      return true
   else
      return false
   endif
   
end procedure

Implementation of isfull() function in C programming language −

Example

bool isfull() {
   if(top == MAXSIZE)
      return true;
   else
      return false;
}

isempty()

Algorithm of isempty() function −

begin procedure isempty

   if top less than 1
      return true
   else
      return false
   endif
   
end procedure

Implementation of isempty() function in C programming language is slightly different. We initialize top at -1, as the index in array starts from 0. So we check if the top is below zero or -1 to determine if the stack is empty. Here's the code −

Example

bool isempty() {
   if(top == -1)
      return true;
   else
      return false;
}

Push Operation

The process of putting a new data element onto stack is known as a Push Operation. Push operation involves a series of steps −

  • Step 1 − Checks if the stack is full.

  • Step 2 − If the stack is full, produces an error and exit.

  • Step 3 − If the stack is not full, increments top to point next empty space.

  • Step 4 − Adds data element to the stack location, where top is pointing.

  • Step 5 − Returns success.

Stack Push Operation

If the linked list is used to implement the stack, then in step 3, we need to allocate space dynamically.

Algorithm for PUSH Operation

A simple algorithm for Push operation can be derived as follows −

begin procedure push: stack, data

   if stack is full
      return null
   endif
   
   top  top + 1
   stack[top]  data

end procedure

Implementation of this algorithm in C, is very easy. See the following code −

Example

void push(int data) {
   if(!isFull()) {
      top = top + 1;   
      stack[top] = data;
   } else {
      printf("Could not insert data, Stack is full.\n");
   }
}

Pop Operation

Accessing the content while removing it from the stack, is known as a Pop Operation. In an array implementation of pop() operation, the data element is not actually removed, instead top is decremented to a lower position in the stack to point to the next value. But in linked-list implementation, pop() actually removes data element and deallocates memory space.

A Pop operation may involve the following steps −

  • Step 1 − Checks if the stack is empty.

  • Step 2 − If the stack is empty, produces an error and exit.

  • Step 3 − If the stack is not empty, accesses the data element at which top is pointing.

  • Step 4 − Decreases the value of top by 1.

  • Step 5 − Returns success.

Stack Pop Operation

Algorithm for Pop Operation

A simple algorithm for Pop operation can be derived as follows −

begin procedure pop: stack

   if stack is empty
      return null
   endif
   
   data  stack[top]
   top  top - 1
   return data

end procedure

Implementation of this algorithm in C, is as follows −

Example

int pop(int data) {

   if(!isempty()) {
      data = stack[top];
      top = top - 1;   
      return data;
   } else {
      printf("Could not retrieve data, Stack is empty.\n");
   }
}

The way to write arithmetic expression is known as a notation. An arithmetic expression can be written in three different but equivalent notations, i.e., without changing the essence or output of an expression. These notations are −

  • Infix Notation
  • Prefix (Polish) Notation
  • Postfix (Reverse-Polish) Notation

These notations are named as how they use operator in expression. We shall learn the same here in this chapter.

Infix Notation

We write expression in infix notation, e.g. a - b + c, where operators are used in-between operands. It is easy for us humans to read, write, and speak in infix notation but the same does not go well with computing devices. An algorithm to process infix notation could be difficult and costly in terms of time and space consumption.

Prefix Notation

In this notation, operator is prefixed to operands, i.e. operator is written ahead of operands. For example, +ab. This is equivalent to its infix notation a + b. Prefix notation is also known as Polish Notation.

Postfix Notation

This notation style is known as Reversed Polish Notation. In this notation style, the operator is postfixed to the operands i.e., the operator is written after the operands. For example, ab+. This is equivalent to its infix notation a + b.

The following table briefly tries to show the difference in all three notations −

Sr.No.Infix NotationPrefix NotationPostfix Notation
1a + b+ a ba b +
2(a + b) ∗ c∗ + a b ca b + c ∗
3a ∗ (b + c)∗ a + b ca b c + ∗
4a / b + c / d+ / a b / c da b / c d / +
5(a + b) ∗ (c + d)∗ + a b + c da b + c d + ∗
6((a + b) ∗ c) - d- ∗ + a b c da b + c ∗ d -

Parsing Expressions

As we have discussed, it is not a very efficient way to design an algorithm or program to parse infix notations. Instead, these infix notations are first converted into either postfix or prefix notations and then computed.

To parse any arithmetic expression, we need to take care of operator precedence and associativity also.

Precedence

When an operand is in between two different operators, which operator will take the operand first, is decided by the precedence of an operator over others. For example −

Operator Precendence

As multiplication operation has precedence over addition, b * c will be evaluated first. A table of operator precedence is provided later.

Associativity

Associativity describes the rule where operators with the same precedence appear in an expression. For example, in expression a + b − c, both + and – have the same precedence, then which part of the expression will be evaluated first, is determined by associativity of those operators. Here, both + and − are left associative, so the expression will be evaluated as (a + b) − c.

Precedence and associativity determines the order of evaluation of an expression. Following is an operator precedence and associativity table (highest to lowest) −

Sr.No.OperatorPrecedenceAssociativity
1Exponentiation ^HighestRight Associative
2Multiplication ( ∗ ) & Division ( / )Second HighestLeft Associative
3Addition ( + ) & Subtraction ( − )LowestLeft Associative

The above table shows the default behavior of operators. At any point of time in expression evaluation, the order can be altered by using parenthesis. For example −

In a + b*c, the expression part b*c will be evaluated first, with multiplication as precedence over addition. We here use parenthesis for a + b to be evaluated first, like (a + b)*c.

Postfix Evaluation Algorithm

We shall now look at the algorithm on how to evaluate postfix notation −

Step 1 − scan the expression from left to right 
Step 2 − if it is an operand push it to stack 
Step 3 − if it is an operator pull operand from stack and perform operation 
Step 4 − store the output of step 3, back to stack 
Step 5 − scan the expression until all operands are consumed 
Step 6 − pop the stack and perform operation

Queue is an abstract data structure, somewhat similar to Stacks. Unlike stacks, a queue is open at both its ends. One end is always used to insert data (enqueue) and the other is used to remove data (dequeue). Queue follows First-In-First-Out methodology, i.e., the data item stored first will be accessed first.

Queue Example

A real-world example of queue can be a single-lane one-way road, where the vehicle enters first, exits first. More real-world examples can be seen as queues at the ticket windows and bus-stops.

Queue Representation

As we now understand that in queue, we access both ends for different reasons. The following diagram given below tries to explain queue representation as data structure −

Queue Example

As in stacks, a queue can also be implemented using Arrays, Linked-lists, Pointers and Structures. For the sake of simplicity, we shall implement queues using one-dimensional array.

Basic Operations

Queue operations may involve initializing or defining the queue, utilizing it, and then completely erasing it from the memory. Here we shall try to understand the basic operations associated with queues −

  • enqueue() − add (store) an item to the queue.

  • dequeue() − remove (access) an item from the queue.

Few more functions are required to make the above-mentioned queue operation efficient. These are −

  • peek() − Gets the element at the front of the queue without removing it.

  • isfull() − Checks if the queue is full.

  • isempty() − Checks if the queue is empty.

In queue, we always dequeue (or access) data, pointed by front pointer and while enqueing (or storing) data in the queue we take help of rear pointer.

Let's first learn about supportive functions of a queue −

peek()

This function helps to see the data at the front of the queue. The algorithm of peek() function is as follows −

Algorithm

begin procedure peek
   return queue[front]
end procedure

Implementation of peek() function in C programming language −

Example

int peek() {
   return queue[front];
}

isfull()

As we are using single dimension array to implement queue, we just check for the rear pointer to reach at MAXSIZE to determine that the queue is full. In case we maintain the queue in a circular linked-list, the algorithm will differ. Algorithm of isfull() function −

Algorithm

begin procedure isfull

   if rear equals to MAXSIZE
      return true
   else
      return false
   endif
   
end procedure

Implementation of isfull() function in C programming language −

Example

bool isfull() {
   if(rear == MAXSIZE - 1)
      return true;
   else
      return false;
}

isempty()

Algorithm of isempty() function −

Algorithm

begin procedure isempty

   if front is less than MIN  OR front is greater than rear
      return true
   else
      return false
   endif
   
end procedure

If the value of front is less than MIN or 0, it tells that the queue is not yet initialized, hence empty.

Here's the C programming code −

Example

bool isempty() {
   if(front < 0 || front > rear) 
      return true;
   else
      return false;
}

Enqueue Operation

Queues maintain two data pointers, front and rear. Therefore, its operations are comparatively difficult to implement than that of stacks.

The following steps should be taken to enqueue (insert) data into a queue −

  • Step 1 − Check if the queue is full.

  • Step 2 − If the queue is full, produce overflow error and exit.

  • Step 3 − If the queue is not full, increment rear pointer to point the next empty space.

  • Step 4 − Add data element to the queue location, where the rear is pointing.

  • Step 5 − return success.

Insert Operation

Sometimes, we also check to see if a queue is initialized or not, to handle any unforeseen situations.

Algorithm for enqueue operation

procedure enqueue(data)      
   
   if queue is full
      return overflow
   endif
   
   rear  rear + 1
   queue[rear]  data
   return true
   
end procedure

Implementation of enqueue() in C programming language −

Example

int enqueue(int data)      
   if(isfull())
      return 0;
   
   rear = rear + 1;
   queue[rear] = data;
   
   return 1;
end procedure

Dequeue Operation

Accessing data from the queue is a process of two tasks − access the data where front is pointing and remove the data after access. The following steps are taken to perform dequeue operation −

  • Step 1 − Check if the queue is empty.

  • Step 2 − If the queue is empty, produce underflow error and exit.

  • Step 3 − If the queue is not empty, access the data where front is pointing.

  • Step 4 − Increment front pointer to point to the next available data element.

  • Step 5 − Return success.

Remove Operation

Algorithm for dequeue operation

procedure dequeue
   
   if queue is empty
      return underflow
   end if

   data = queue[front]
   front  front + 1
   return true

end procedure

Implementation of dequeue() in C programming language −

Example

int dequeue() {
   if(isempty())
      return 0;

   int data = queue[front];
   front = front + 1;

   return data;
}

Linear search is a very simple search algorithm. In this type of search, a sequential search is made over all items one by one. Every item is checked and if a match is found then that particular item is returned, otherwise the search continues till the end of the data collection.

Linear Search Animation

Algorithm

Linear Search ( Array A, Value x)

Step 1: Set i to 1
Step 2: if i > n then go to step 7
Step 3: if A[i] = x then go to step 6
Step 4: Set i to i + 1
Step 5: Go to Step 2
Step 6: Print Element x Found at index i and go to step 8
Step 7: Print element not found
Step 8: Exit

Pseudocode

procedure linear_search (list, value)

   for each item in the list
      if match item == value
         return the item's location
      end if
   end for

end procedure

Binary search is a fast search algorithm with run-time complexity of Ο(log n). This search algorithm works on the principle of divide and conquer. For this algorithm to work properly, the data collection should be in the sorted form.

Binary search looks for a particular item by comparing the middle most item of the collection. If a match occurs, then the index of item is returned. If the middle item is greater than the item, then the item is searched in the sub-array to the left of the middle item. Otherwise, the item is searched for in the sub-array to the right of the middle item. This process continues on the sub-array as well until the size of the subarray reduces to zero.

How Binary Search Works?

For a binary search to work, it is mandatory for the target array to be sorted. We shall learn the process of binary search with a pictorial example. The following is our sorted array and let us assume that we need to search the location of value 31 using binary search.

Binary search

First, we shall determine half of the array by using this formula −

mid = low + (high - low) / 2

Here it is, 0 + (9 - 0 ) / 2 = 4 (integer value of 4.5). So, 4 is the mid of the array.

Binary search

Now we compare the value stored at location 4, with the value being searched, i.e. 31. We find that the value at location 4 is 27, which is not a match. As the value is greater than 27 and we have a sorted array, so we also know that the target value must be in the upper portion of the array.

Binary search

We change our low to mid + 1 and find the new mid value again.

low = mid + 1
mid = low + (high - low) / 2

Our new mid is 7 now. We compare the value stored at location 7 with our target value 31.

Binary search

The value stored at location 7 is not a match, rather it is more than what we are looking for. So, the value must be in the lower part from this location.

Binary search

Hence, we calculate the mid again. This time it is 5.

Binary search

We compare the value stored at location 5 with our target value. We find that it is a match.

Binary search

We conclude that the target value 31 is stored at location 5.

Binary search halves the searchable items and thus reduces the count of comparisons to be made to very less numbers.

Pseudocode

The pseudocode of binary search algorithms should look like this −

Procedure binary_search
   A  sorted array
   n  size of array
   x  value to be searched

   Set lowerBound = 1
   Set upperBound = n 

   while x not found
      if upperBound < lowerBound 
         EXIT: x does not exists.
   
      set midPoint = lowerBound + ( upperBound - lowerBound ) / 2
      
      if A[midPoint] < x
         set lowerBound = midPoint + 1
         
      if A[midPoint] > x
         set upperBound = midPoint - 1 

      if A[midPoint] = x 
         EXIT: x found at location midPoint
   end while
   
end procedure

Interpolation search is an improved variant of binary search. This search algorithm works on the probing position of the required value. For this algorithm to work properly, the data collection should be in a sorted form and equally distributed.

Binary search has a huge advantage of time complexity over linear search. Linear search has worst-case complexity of Ο(n) whereas binary search has Ο(log n).

There are cases where the location of target data may be known in advance. For example, in case of a telephone directory, if we want to search the telephone number of Morphius. Here, linear search and even binary search will seem slow as we can directly jump to memory space where the names start from 'M' are stored.

Positioning in Binary Search

In binary search, if the desired data is not found then the rest of the list is divided in two parts, lower and higher. The search is carried out in either of them.

BST Step OneBST Step TwoBST Step ThreeBST Step Four

Even when the data is sorted, binary search does not take advantage to probe the position of the desired data.

Position Probing in Interpolation Search

Interpolation search finds a particular item by computing the probe position. Initially, the probe position is the position of the middle most item of the collection.

Interpolation Step OneInterpolation Step Two

If a match occurs, then the index of the item is returned. To split the list into two parts, we use the following method −

mid = Lo + ((Hi - Lo) / (A[Hi] - A[Lo])) * (X - A[Lo])

where −
   A    = list
   Lo   = Lowest index of the list
   Hi   = Highest index of the list
   A[n] = Value stored at index n in the list

If the middle item is greater than the item, then the probe position is again calculated in the sub-array to the right of the middle item. Otherwise, the item is searched in the subarray to the left of the middle item. This process continues on the sub-array as well until the size of subarray reduces to zero.

Runtime complexity of interpolation search algorithm is ÎŸ(log (log n)) as compared to ÎŸ(log n) of BST in favorable situations.

Algorithm

As it is an improvisation of the existing BST algorithm, we are mentioning the steps to search the 'target' data value index, using position probing −

Step 1 − Start searching data from middle of the list.
Step 2 − If it is a match, return the index of the item, and exit.
Step 3 − If it is not a match, probe position.
Step 4 − Divide the list using probing formula and find the new midle.
Step 5 − If data is greater than middle, search in higher sub-list.
Step 6 − If data is smaller than middle, search in lower sub-list.
Step 7 − Repeat until match.

Pseudocode

A  Array list
N  Size of A
X  Target Value

Procedure Interpolation_Search()

   Set Lo    0
   Set Mid  -1
   Set Hi    N-1

   While X does not match
   
      if Lo equals to Hi OR A[Lo] equals to A[Hi]
         EXIT: Failure, Target not found
      end if
      
      Set Mid = Lo + ((Hi - Lo) / (A[Hi] - A[Lo])) * (X - A[Lo]) 

      if A[Mid] = X
         EXIT: Success, Target found at Mid
      else 
         if A[Mid] < X
            Set Lo to Mid+1
         else if A[Mid] > X
            Set Hi to Mid-1
         end if
      end if
   End While

End Procedure

Hash Table is a data structure which stores data in an associative manner. In a hash table, data is stored in an array format, where each data value has its own unique index value. Access of data becomes very fast if we know the index of the desired data.

Thus, it becomes a data structure in which insertion and search operations are very fast irrespective of the size of the data. Hash Table uses an array as a storage medium and uses hash technique to generate an index where an element is to be inserted or is to be located from.

Hashing

Hashing is a technique to convert a range of key values into a range of indexes of an array. We're going to use modulo operator to get a range of key values. Consider an example of hash table of size 20, and the following items are to be stored. Item are in the (key,value) format.

Hash Function
  • (1,20)
  • (2,70)
  • (42,80)
  • (4,25)
  • (12,44)
  • (14,32)
  • (17,11)
  • (13,78)
  • (37,98)
Sr.No.KeyHashArray Index
111 % 20 = 11
222 % 20 = 22
34242 % 20 = 22
444 % 20 = 44
51212 % 20 = 1212
61414 % 20 = 1414
71717 % 20 = 1717
81313 % 20 = 1313
93737 % 20 = 1717

Linear Probing

As we can see, it may happen that the hashing technique is used to create an already used index of the array. In such a case, we can search the next empty location in the array by looking into the next cell until we find an empty cell. This technique is called linear probing.

Sr.No.KeyHashArray IndexAfter Linear Probing, Array Index
111 % 20 = 111
222 % 20 = 222
34242 % 20 = 223
444 % 20 = 444
51212 % 20 = 121212
61414 % 20 = 141414
71717 % 20 = 171717
81313 % 20 = 131313
93737 % 20 = 171718

Basic Operations

Following are the basic primary operations of a hash table.

  • Search − Searches an element in a hash table.

  • Insert − inserts an element in a hash table.

  • delete − Deletes an element from a hash table.

DataItem

Define a data item having some data and key, based on which the search is to be conducted in a hash table.

struct DataItem {
   int data;
   int key;
};

Hash Method

Define a hashing method to compute the hash code of the key of the data item.

int hashCode(int key){
   return key % SIZE;
}

Search Operation

Whenever an element is to be searched, compute the hash code of the key passed and locate the element using that hash code as index in the array. Use linear probing to get the element ahead if the element is not found at the computed hash code.

Example

struct DataItem *search(int key) {
   //get the hash
   int hashIndex = hashCode(key);
	
   //move in array until an empty
   while(hashArray[hashIndex] != NULL) {
	
      if(hashArray[hashIndex]->key == key)
         return hashArray[hashIndex];
			
      //go to next cell
      ++hashIndex;
		
      //wrap around the table
      hashIndex %= SIZE;
   }

   return NULL;        
}

Insert Operation

Whenever an element is to be inserted, compute the hash code of the key passed and locate the index using that hash code as an index in the array. Use linear probing for empty location, if an element is found at the computed hash code.

Example

void insert(int key,int data) {
   struct DataItem *item = (struct DataItem*) malloc(sizeof(struct DataItem));
   item->data = data;  
   item->key = key;     

   //get the hash 
   int hashIndex = hashCode(key);

   //move in array until an empty or deleted cell
   while(hashArray[hashIndex] != NULL && hashArray[hashIndex]->key != -1) {
      //go to next cell
      ++hashIndex;
		
      //wrap around the table
      hashIndex %= SIZE;
   }
	
   hashArray[hashIndex] = item;        
}

Delete Operation

Whenever an element is to be deleted, compute the hash code of the key passed and locate the index using that hash code as an index in the array. Use linear probing to get the element ahead if an element is not found at the computed hash code. When found, store a dummy item there to keep the performance of the hash table intact.

Example

struct DataItem* delete(struct DataItem* item) {
   int key = item->key;

   //get the hash 
   int hashIndex = hashCode(key);

   //move in array until an empty 
   while(hashArray[hashIndex] !=NULL) {
	
      if(hashArray[hashIndex]->key == key) {
         struct DataItem* temp = hashArray[hashIndex]; 
			
         //assign a dummy item at deleted position
         hashArray[hashIndex] = dummyItem; 
         return temp;
      } 
		
      //go to next cell
      ++hashIndex;
		
      //wrap around the table
      hashIndex %= SIZE;
   }  
	
   return NULL;        
}

Sorting refers to arranging data in a particular format. Sorting algorithm specifies the way to arrange data in a particular order. Most common orders are in numerical or lexicographical order.

The importance of sorting lies in the fact that data searching can be optimized to a very high level, if data is stored in a sorted manner. Sorting is also used to represent data in more readable formats. Following are some of the examples of sorting in real-life scenarios −

  • Telephone Directory − The telephone directory stores the telephone numbers of people sorted by their names, so that the names can be searched easily.

  • Dictionary − The dictionary stores words in an alphabetical order so that searching of any word becomes easy.

In-place Sorting and Not-in-place Sorting

Sorting algorithms may require some extra space for comparison and temporary storage of few data elements. These algorithms do not require any extra space and sorting is said to happen in-place, or for example, within the array itself. This is called in-place sorting. Bubble sort is an example of in-place sorting.

However, in some sorting algorithms, the program requires space which is more than or equal to the elements being sorted. Sorting which uses equal or more space is called not-in-place sorting. Merge-sort is an example of not-in-place sorting.

Stable and Not Stable Sorting

If a sorting algorithm, after sorting the contents, does not change the sequence of similar content in which they appear, it is called stable sorting.

Stable Sorting

If a sorting algorithm, after sorting the contents, changes the sequence of similar content in which they appear, it is called unstable sorting.

Unstable Sorting

Stability of an algorithm matters when we wish to maintain the sequence of original elements, like in a tuple for example.

Adaptive and Non-Adaptive Sorting Algorithm

A sorting algorithm is said to be adaptive, if it takes advantage of already 'sorted' elements in the list that is to be sorted. That is, while sorting if the source list has some element already sorted, adaptive algorithms will take this into account and will try not to re-order them.

A non-adaptive algorithm is one which does not take into account the elements which are already sorted. They try to force every single element to be re-ordered to confirm their sortedness.

Important Terms

Some terms are generally coined while discussing sorting techniques, here is a brief introduction to them −

Increasing Order

A sequence of values is said to be in increasing order, if the successive element is greater than the previous one. For example, 1, 3, 4, 6, 8, 9 are in increasing order, as every next element is greater than the previous element.

Decreasing Order

A sequence of values is said to be in decreasing order, if the successive element is less than the current one. For example, 9, 8, 6, 4, 3, 1 are in decreasing order, as every next element is less than the previous element.

Non-Increasing Order

A sequence of values is said to be in non-increasing order, if the successive element is less than or equal to its previous element in the sequence. This order occurs when the sequence contains duplicate values. For example, 9, 8, 6, 3, 3, 1 are in non-increasing order, as every next element is less than or equal to (in case of 3) but not greater than any previous element.

Non-Decreasing Order

A sequence of values is said to be in non-decreasing order, if the successive element is greater than or equal to its previous element in the sequence. This order occurs when the sequence contains duplicate values. For example, 1, 3, 3, 6, 8, 9 are in non-decreasing order, as every next element is greater than or equal to (in case of 3) but not less than the previous one.

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