Data Structures_ Sorting & Searching

Part – 1 - Searching
Searching : basic searching techniques, sequential search,
binary search, iterative and recursive methods,
comparison between sequential and binary search.

 Searching refers to finding for an item in any list. It is
one of the common operations in data processing.
 Searching should be efficient and quick as the list may
be large and large amount of data has to be processed.
 The programmer has to select the best technique to the
given problem. Searching is made easy when the list is
sorted.

The searching technique that is being followed in the data
structure is listed below:
 1. Linear Search or Sequential Search
 2. Binary Search

 Linear Search or Sequential Search: In this
technique of searching, the element to be found in
searching the elements to be found is searched
sequentially in the list.
 This method can be performed on a sorted or an
unsorted list (usually arrays).
 In case of a sorted list searching starts from 0th element
and continues until the element is found from the list or
the element whose value is greater than (assuming the
list is sorted in ascending order), the value being
searched is reached.

 Binary Search: It is a very fast and efficient searching
technique. It requires the list to be in sorted order.
 In this method, to search an element you can compare it
with the present element at the centre of the list. If it
matches, then the search is successful otherwise the list
is divided into two halves: one from the 0th element to
the middle element which is the centre element (first
half) another from the centre element to the last
element (which is the 2nd half) where all values are
greater than the centre element.

 Linear search is also called as sequential search
algorithm. It is the simplest searching algorithm.
 In Linear search, we simply traverse the list completely and
match each element of the list with the item whose location
is to be found. If the match is found, then the location of the
item is returned; otherwise, the algorithm returns NULL.
 It is the simplest of all the searching techniques which does
not expect the specific ordering of data.
 Sequential search is nothing but searching an element /
record in a linear way. This can be in arrays or in linked
lists.

Linear search is implemented using the following steps
 Step 1: Read the search element (key).
 Step 2: Compare the search element with the first element
in the list.
 Step 3: If both are matched, then display search element
is found and terminate the function.
 Step 4: If both are not matched, then compare search
element with the next element in the list
 Step 5: Repeat the Steps 3 & 4 until the search element is
compared with the last element in the list.
 Step 6: If the last element in the list does not match, then
display element not found and terminate the function

Algorithm: Linear Search (list [ ], n, key)
Let list [ ] be a linear array with n elements and key is an
element to be searched in the list
Step 1: Set i = 0
Step 2: while (i < n) do
If list [i] == key then
return i; //Key found at ith location
end while
Step 3: return -1; //key not found

 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 conquers.
 For this algorithm to work properly, the data collection
should be in the sorted form.

Input: Given an array A of n elements in sorted order and item is an element to
be searched.
Output: Returns the position of item element if successful and returns -1
otherwise,
Step 1: set first = 0, last = n – 1
Step 2: while (first <= last)
mid = (first + last) / 2
if (item == A[mid])
return (mid + 1);
else if (item < A[mid])
last = mid – 1
else
first = mid + 1
end while
Step 3: return -1

 Let the elements of array are –
 Let the element to search is, K = 56
 We have to use the below formula to calculate
the mid of the array –
mid = (beg + end)/2

So, in the given array -
 beg = 0
 end = 8
 mid = (0 + 8)/2 = 4. So, 4 is the mid of the array.

Now, the element to search is found.
So algorithm will return the index of
the element matched.

Basis of comparison Linear search Binary search
Worst-case scenario In a linear search, the worst- case
scenario for finding the element is
O(n).
N number of comparisons are required
for worst case.
In a binary search, the worst-case
scenario for finding the element is
O(log2n).
Log n number of comparisons are
sufficient in worst case.
Best-case scenario In a linear search, the best-case
scenario for finding the first element in
the list is O(1).
In a binary search, the best-case
scenario for finding the first element in
the list is O(1).
Dimensional array It can be implemented on both a single
and multidimensional array.
It can be implemented only on a
multidimensional array.
Code Algorithm is easy to implement, and
requires less amount of code.
Algorithm is slightly complex. It takes
more amount of code to implement.
Search

 The Recursion and Iteration both repeatedly execute
the set of instructions.
 Recursion is when a statement in a function calls
itself repeatedly. The iteration is when a
loop repeatedly executes until the controlling
condition becomes false.
 The primary difference between recursion and iteration
is that recursion is a process, always applied to a
function and iteration is applied to the set of
instructions which we want to get repeatedly
executed.

 Recursion uses selection structure.
 Infinite recursion occurs if the recursion step does not
reduce the problem in a manner that converges on some
condition (base case) and Infinite recursion can crash the
system.
 Recursion terminates when a base case is recognized.
 Recursion is usually slower than iteration due to the
overhead of maintaining the stack.
 Recursion uses more memory than iteration.
 Recursion makes the code smaller.
int factorial(int n) {
if (n==0) {
return 1; }
else
{
return n*factorial(n-1);
}
}

 Iteration uses repetition structure.
 An infinite loop occurs with iteration if the loop
condition test never becomes false and Infinite looping
uses CPU cycles repeatedly.
 An iteration terminates when the loop condition fails.
 An iteration does not use the stack so it's faster than
recursion.
 Iteration consumes less memory.
 Iteration makes the code longer.
 for(int i = 1; i <= 5; i++)
 {
 Printf(‘The value of i is %d’,i);
 }

 Topics to be discussed
 SORTING
◦ Definition
◦ Different types
◦ Bubble sort
◦ Selection sort
◦ Insertion sort
◦ Merge sort
◦ Quick sort
◦ Heap sort

 Sorting refers to ordering data in an increasing or decreasing fashion according to some linear
relationship among the data items.
 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.
 Types :
◦ Bubble sort (or) Exchange Sort.
◦ Selection sort.
◦ Insertion sort (or) Linear sort.
◦ Merge Sort (or) External sort.
◦ Quick sort (or) Partition exchange sort.
◦ Heap sort

 Bubble Sort is the simplest
sorting algorithm that works by
repeatedly swapping the adjacent
elements if they are in wrong
order.

 SWAPPING using
temporary variable
temp =a;
a=b;
B=temp; Output:
enter the two numbers
20
45
before swap: a=20,
b=45
after swap: a=45,b=20
• SWAPPING using without
temporary variable
• a=a+b;
• b=a-b;
• a=a-b;

In the algorithm given below, suppose arr is an array
of n elements.
The assumed swap function in the algorithm will swap the values
of given array elements.
begin BubbleSort(arr)
for all array elements
if arr[i] > arr[i+1]
swap(arr[i], arr[i+1])
end if
end for
return arr
end BubbleSort

 #include<stdio.h>
 void main()
 {
 int n,i,j,temp,a[100];
 clrscr();
 printf("nEnter the size of array
:");
 scanf("%d",&n);
 printf("n Enter the elementsn");
 for(i=0;i<n;i++)
 {
 scanf("%d",&a[i]);
 }
 printf("Entered array isn");
 for(i=0;i<n;i++)
 {
 printf("%dt",a[i]);
 }

 for(i=0;i<n-1;i++)
 {
 for(j=0;j<n-i-1;j++)
 {

if(a[j]>=a[j+1])
 {

temp=a[j];

a[j]=a[j+1];

a[j+1]=temp;
 }
 }
 }
 printf("nThe sorted arrayn");
 for(i=0;i<n;i++)
 {
 }
 getch();
 }

 It's a simple algorithm that can be implemented on a
computer.
 Efficient way to check if a list is already in order.
 Doesn't use too much memory.

 It's an efficient way to sort a list.
 Due to being efficient , the bubble sort algorithm is
pretty slow for very large lists of items.

 Selection sort is a simple sorting algorithm. This sorting algorithm is
comparison-based algorithm in which the list is divided into two parts, the
sorted part at the left end and the unsorted part at the right end.
 Initially, the sorted part is empty and the unsorted part is the entire list.
 The smallest element is selected from the unsorted array and swapped with the
leftmost element, and that element becomes a part of the sorted array. This
process continues moving unsorted array boundary by one element to the
right.
 This algorithm is not suitable for large data sets as its best, average and worst
case complexities are of Ο(n2), where n is the number of items.

 Selection sort is a sorting algorithm that
selects the smallest element from an
unsorted list in each iteration and places that
element at the beginning of the unsorted list.

 1) Set the first element as minimum
Select first element as minimum
 2) Compare minimum with the second element. If the second element is smaller
than minimum, assign the second element as minimum.
Compare minimum with the third element. Again, if the third element is smaller,
then assign minimum to the third element otherwise do nothing. The process goes
on until the last element.
Compare minimum with the remaining elements

 3) After each iteration, minimum is placed in
the front of the unsorted list.
Swap the first with minimum

 4) For each iteration, indexing starts from the first unsorted element. Step 1 to 3 are
repeated until all the elements are placed at their correct positions.
The first iteration

 void main()
 {
 int
n,i,j,small,loc,temp,a[100];
 clrscr();
 printf("nEnter the size of
array :");
 scanf("%d",&n);
 printf("n Enter the
elementsn");
 for(i=0;i<n;i++)
 {
 }
 printf("Entered array isn");
 for(i=0;i<n;i++)
 {
 }

 for(i=0;i<=n;i++)
 { small=a[i];
 loc=i;
 for(j=i+1;j<n;j++)
 {

if(a[j]<small)
 {

small=a[j];
 loc=j;
 }
 }
 a[loc]=a[i];
 a[i]=small;
 }
 printf("nThe sorted array
isn");
 for(i=0;i<n;i++)
 {
 }
 getch();
 }

 Complexity = O(n2)
 Also, we can analyze the complexity by simply observing the number of loops. There are 2
loops so the complexity is n*n = n2.
 Time Complexities:
 Worst Case Complexity: O(n2)
If we want to sort in ascending order and the array is in descending order then, the worst case
occurs.
 Best Case Complexity: O(n2)
It occurs when the array is already sorted
 Average Case Complexity: O(n2)
It occurs when the elements of the array are in jumbled order (neither ascending nor
descending).

 Quick sort is a highly efficient sorting algorithm.
 A large array is partitioned into two arrays
◦ One is smaller than specified value(pivot)
◦ One is larger than specified value(pivot)
 Quite efficient for large-sized data sets
 Based on the rule of Divide and Conquer.
 Quick sort is the quickest comparison-based sorting algorithm.
 Quick sort picks an element as pivot and partitions the array around the picked
pivot.

 Step 1: Choose the highest index value as pivot.
Step 2: Take two variables to point left and right of the list excluding pivot.
Step 3: Left points to the low index.
Step 4: Right points to the high index.
Step 5: While value at left < (Less than) pivot move right.
Step 6: While value at right > (Greater than) pivot move left.
Step 7: If both Step 5 and Step 6 does not match, swap left and right.
Step 8: If left = (Less than or Equal to) right, the point where they met is new pivot.

 Let the elements of array are –
 In the given array, we consider the leftmost element as pivot. So, in this case, a[left] = 24,
a[right] = 27 and a[pivot] = 24.
 Since, pivot is at left, so algorithm starts from right and move towards left.
 Now, a[pivot] < a[right], so algorithm moves forward one position towards
left, i.e. -

 Now, a[left] = 24, a[right] = 19, and a[pivot] = 24.
 Because, a[pivot] > a[right], so, algorithm will swap a[pivot] with a[right], and pivot moves
to right, as –
 Now, a[left] = 19, a[right] = 24, and a[pivot] = 24. Since, pivot is at right, so algorithm starts
from left and moves to right.

 As a[pivot] > a[left], so algorithm moves one position to right as –
 Now, a[left] = 9, a[right] = 24, and a[pivot] = 24. As a[pivot] > a[left], so algorithm moves
one position to right as -

 Now, a[left] = 29, a[right] = 24, and a[pivot] = 24. As a[pivot] < a[left], so, swap a[pivot] and
a[left], now pivot is at left, i.e. –
 Since, pivot is at left, so algorithm starts from right, and move to left. Now, a[left] = 24,
a[right] = 29, and a[pivot] = 24. As a[pivot] < a[right], so algorithm moves one position to
left, as -

 Now, a[pivot] = 24, a[left] = 24, and a[right] = 14. As a[pivot] > a[right], so, swap a[pivot]
and a[right], now pivot is at right, i.e. –
 Now, a[pivot] = 24, a[left] = 14, and a[right] = 24. Pivot is at right, so the algorithm starts
from left and move to right.


 Now, a[pivot] = 24, a[left] = 24, and a[right] = 24. So, pivot, left and right are pointing the
same element. It represents the termination of procedure.
 Element 24, which is the pivot element is placed at its exact position.
 Elements that are right side of element 24 are greater than it, and the elements that are left
side of element 24 are smaller than it.
 Now, in a similar manner, quick sort algorithm is separately applied to the left and right sub-
arrays. After sorting gets done, the array will be -

 QUICK SORT
 void main()
 {
 int i,n,a[50];
 void quicksort(int a[],int,int);
 clrscr();
 printf("nEnter the size of the array :
");
 scanf("%d",&n);
 printf("Enter the elements of the
arraynt");
 for(i=0;i<n;i++)
 {
 }
 printf("nThe entered unsorted
arrayn");
 for(i=0;i<n;i++)
 {
 printf("t%d",a[i]);
 }
 quicksort(a,0,n-1);
 printf("nSorted Arrayn");
 for(i=0;i<n;i++)
 {
 printf("t%d",a[i]);
 }
 getch();
 }
 void quicksort(int a[],int low,int high)
 {
 int pivot,down,up,key,temp;
 if(low<high)
 {
 down=low;
 pivot=low;
 up=high;
 key=a[pivot];
 while(down<up)
 {

while((a[down]<=key)&&(down<=high))
 down++;

while((a[up]>key)&&(up>=low))
 up--;
 if(down<up)
 {
 temp=a[up];
 a[up]=a[down];
 a[down]=temp;
 }
 }
 temp=a[up];
 a[up]=a[pivot];
 a[pivot]=temp;
 quicksort(a,low,up-1);
 quicksort(a,up+1,high);
 }
 return;
 }

Time Complexity
Best Case O(n*logn)
Average Case O(n*logn)
Worst Case O(n
2
)

Space Complexity O(n*logn)
Stable NO

 Merge sort is similar to the quick sort algorithm as it uses the divide and conquer approach to
sort the elements.
 It is one of the most popular and efficient sorting algorithm.
 It divides the given list into two equal halves, calls itself for the two halves and then merges
the two sorted halves.
 The sub-lists are divided again and again into halves until the list cannot be divided further.
Then we combine the pair of one element lists into two-element lists, sorting them in the
process. The sorted two-element pairs is merged into the four-element lists, and so on until
we get the sorted list.

 Step 1 − if it is only one element in the list it is already sorted, return.
 Step 2 − divide the list recursively into two halves until it can no more be divided.
 Step 3 − merge the smaller lists into new list in sorted order.


MERGE_SORT(arr, beg, end)
if beg < end
set mid = (beg + end)/2
MERGE_SORT(arr, beg, mid)
MERGE_SORT(arr, mid + 1, end)
MERGE (arr, beg, mid, end)
end of if
END MERGE_SORT

Example of Merge sort : unsorted array
 Step 1: Merge sort first divides the whole array iteratively into equal halves unless the atomic
values are achieved.

 Step 2: This does not change the sequence of appearance of items in the original. Now divide
these two arrays into halves.

 Step 3: Further divide these arrays and we achieve atomic value which can no more be
divided.


 Step 4: First compare the element for each list and then combine them into another list in a
sorted manner. We see that 14 and 33 are in sorted positions. We compare 27 and 10 and in
the target list of 2 values we put 10 first, followed by 27. We change the order of 19 and 35
whereas 42 and 44 are placed sequentially.
 Step 5: In the next iteration of the combining phase, compare lists of two data values, and
merge them into a list of found data values placing all in a sorted order.
 Step 6: After the final merging.
 Merge sort keeps on dividing the list into equal halves until it can no more be divided. By
definition, if it is only one element in the list, it is sorted. Then, merge sort combines the
smaller sorted lists keeping the new list sorted too.

 #include<conio.h>
 void partition(int [20],int,int);
 void merge(int [20],int,int,int);
 void main()
 {
 int i,a[20],n;
 clrscr();
 printf("Enter the number of elementsn");
 scanf("%d",&n);
 printf("Enter the elementsn");
 for(i=0;i<n;i++)
 printf("Orginal listn");
 for(i=0;i<n;i++)
 partition(a,0,n-1);
 printf("nSorted listn");
 for(i=0;i<n;i++)
 getch();
 }
 void partition(int a[20],int lb,int ub)
 {
 int mid;
 if(lb<ub)
 {
 mid=(lb+ub)/2;
 partition(a,0,mid);
 partition(a,mid+1,ub);
 merge(a,lb,mid,ub);
 }
 }
 void merge(int a[20],int lb,int mid,int ub)
 {
 int c[20],i,j,k;
 i=lb;
 k=lb;
 j=mid+1;
 while((i<=mid)&&(j<=ub))
 {
 if(a[i]<a[j])
 {
 c[k]=a[i];
 k++;
 i++;
 }
 else
 {
 c[k]=a[j];
 k++;
 j++;
 }
 }
 while(i<=mid)
 {
 c[k]=a[i];
 k++;
 i++;
 }
 while(j<=ub)
 {
 c[k]=a[j];
 k++;
 j++;
 }
 for(i=lb;i<k;i++)
 a[i]=c[i];
 }

 Best Case Complexity - It occurs when there is no sorting required, i.e. the array is already
sorted. The best-case time complexity of merge sort is O(n*logn).
 Average Case Complexity - It occurs when the array elements are in jumbled order that is
not properly ascending and not properly descending. The average case time complexity of
merge sort is O(n*logn).
 Worst Case Complexity - It occurs when the array elements are required to be sorted in
reverse order. That means suppose you have to sort the array elements in ascending order, but
its elements are in descending order. The worst-case time complexity of merge sort
is O(n*logn).
Case Time Complexity
Best Case O(n*logn)
Average Case O(n*logn)
Worst Case O(n*logn)

 Merge sort can be used with linked lists without taking
up any more space.
 A merge sort algorithm is used to count the number of
inversions in the list.
 Merge sort is employed in external sorting.

 For small datasets, merge sort is slower than other
sorting algorithms.
 Even if the array is sorted, the merge sort goes through
the entire process.

 The space complexity of merge sort is O(n). It
is because, in merge sort, an extra variable is
required for swapping.
Space Complexity O(n)
Stable YES

 It is an in-place comparison-based sorting algorithm. A sub-list is maintained which is
always sorted. For example, the lower part of an array is maintained to be sorted.
 An element which is to be inserted in this sorted sub-list, has to find its appropriate place and
then it has to be inserted there. Hence the name, insertion sort.
 The array is searched sequentially and unsorted items are moved and inserted into the sorted
sub-list (in the same array).
 This algorithm is not suitable for large data sets as its average and worst case complexity are
of Ο(n2), where n is the number of items.

Advantages:
 Simple implementation
 Efficient for small data sets
 Adaptive, i.e., it is appropriate for data sets that are
already substantially sorted.

 The simple steps of achieving the insertion sort are listed as follows -
 Step 1 - If the element is the first element, assume that it is already sorted.
Return 1.
 Step2 - Pick the next element, and store it separately in a key.
 Step3 - Now, compare the key with all elements in the sorted array.
 Step 4 - If the element in the sorted array is smaller than the current
element, then move to the next element. Else, shift greater elements in the
array towards the right.
 Step 5 - Insert the value.
 Step 6 - Repeat until the array is sorted.

 Example ( Working of Insertion)

Insertion sort compares the first two elements.

It finds that both 14 and 33 are already in ascending order. For now, 14 is in sorted sub-list.

 Insertion sort moves ahead and compares 33 with 27.

 Found that 33 is not in the correct position.

 It swaps 33 with 27. It also checks with all the elements of sorted sub-list. Here that the sorted
sub-list has only one element 14, and 27 is greater than 14. Hence, the sorted sub-list remains
sorted after swapping.

 But now 14 and 27 in the sorted sub-list. Next, it compares 33 with 10.

 These values are not in a sorted order.

 So swapping take place,

 Swapping makes 27 and 10 unsorted and swap it

 Swapping makes 14 and 10 unsorted and swap it

 Finally, the sorted element in the insertion sort will be as,

 #include <stdio.h>

 void insert(int a[], int n) /* function
to sort an aay with insertion sort */
 {
 int i, j, temp;
 for (i = 1; i < n; i++) {
 temp = a[i];
 j = i - 1;

 while(j>=0 && temp <= a[j])
/* Move the elements greater than t
emp to one position ahead from the
ir current position*/
 {
 a[j+1] = a[j];
 j = j-1;
 }
 a[j+1] = temp;
 }
 }

 void printArr(int a[], int n) /* function to
print the array */
 {
 int i;
 for (i = 0; i < n; i++)
 printf("%d ", a[i]);
 }

 int main()
 {
 int a[] = { 12, 31, 25, 8, 32, 17 };
 int n = sizeof(a) / sizeof(a[0]);
 printf("Before sorting array elements a
re - n");
 printArr(a, n);
 insert(a, n);
 printf("nAfter sorting array elements
are - n");
 printArr(a, n);

 return 0;
 }

 Best Case Complexity - It occurs when there is no sorting required, i.e. the array is already sorted.
The best-case time complexity of insertion sort is O(n).
 Average Case Complexity - It occurs when the array elements are in jumbled order that is not
properly ascending and not properly descending. The average case time complexity of insertion sort
is O(n2).
 Worst Case Complexity - It occurs when the array elements are required to be sorted in reverse
order. That means suppose you have to sort the array elements in ascending order, but its elements
are in descending order. The worst-case time complexity of insertion sort is O(n2).
Best Case O(n)
Average Case O(n2)
Worst Case O(n2)

 the space complexity of insertion sort is O(1). It is
because, in insertion sort, an extra variable is required
for swapping.
Space Complexity O(1)
Stable YES

 Heap sort to be discussed in 5th unit

 Heap - A heap is a complete binary tree, and the binary tree is a tree in which the node can
have the utmost two children. A complete binary tree is a binary tree in which all the levels
except the last level, i.e., leaf node, should be completely filled, and all the nodes should be
left-justified.
Heap sort basically recursively performs two main operations -
 Build a heap H, using the elements of array.
 Repeatedly delete the root element of the heap formed in 1st phase.
 Heap Sort - javatpoint

 HeapSort(arr)
 BuildMaxHeap(arr)
 for i = length(arr) to 2
 swap arr[1] with arr[i]
 heap_size[arr] = heap_size[arr] ? 1
 MaxHeapify(arr,1)
 End

 BuildMaxHeap(arr)
 heap_size(arr) = length(arr)
 for i = length(arr)/2 to 1
 MaxHeapify(arr,i)
 End

 MaxHeapify(arr,i)
 L = left(i)
 R = right(i)
 if L ? heap_size[arr] and arr[L] > arr[i]
 largest = L
 else
 largest = i
 if R ? heap_size[arr] and arr[R] > arr[largest]
 largest = R
 if largest != i
 swap arr[i] with arr[largest]
 MaxHeapify(arr,largest)
 End

 n heap sort, basically, there are two phases involved in the sorting of elements. By using the
heap sort algorithm, they are as follows -
 The first step includes the creation of a heap by adjusting the elements of the array.
 After the creation of heap, now remove the root element of the heap repeatedly by shifting it
to the end of the array, and then store the heap structure with the remaining elements.

 First, we have to construct a heap from the
given array and convert it into max heap.

 After converting the given heap into max heap, the array elements are –
 Next, we have to delete the root element (89) from the max heap. To delete this node, we
have to swap it with the last node, i.e. (11). After deleting the root element, we again have to
heapify it to convert it into max heap.

 After swapping the array element 89 with 11, and
converting the heap into max-heap, the elements
of array are –
 In the next step, again, we have to delete the
root element (81) from the max heap. To delete
this node, we have to swap it with the last node,
i.e. (54). After deleting the root element, we
again have to heapify it to convert it into max
heap.

 After swapping the array element 81 with 54 and
of array are –
 In the next step, we have to delete the root
element (76) from the max heap again. To delete
this node, we have to swap it with the last node,
i.e. (9). After deleting the root element, we again
have to heapify it to convert it into max heap.

of array are –
 In the next step, again we have to delete the root
element (54) from the max heap. To delete this
node, we have to swap it with the last node,
heap.

of array are –
heap.

of array are –

 After swapping the array
element 11 with 9, the elements of array are –
 Now, heap has only one element left. After
deleting it, heap will be empty.
 After completion of sorting, the array
elements are –
 Now, the array is completely sorted.

Data Structures_ Sorting & Searching

Data Structures_ Sorting & Searching

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