Dsa – data structure and algorithms searching

Data Structure and
Algorithms-Searching
Techniques
Prepared By,
S.Sajini
AP/CSE

DSA – Data Structure
and Algorithms
What is DS?
A data structure is
collection of data elements
that are arranged in one
name, and which defines a
particular way of storing
and organizing data in a
computer so that it can be
used efficiently.

Searching Algorithms
Linear Search

Data Search − Consider an inventory of 1 million(106) items of a
store. If the application is to search an item, it has to
search an item in 1 million(106) items every time slowing
down the search. As data grows, search will become slower.
This is why searching algorithms are important
Need for Searching Algorithms

1. Linear search or sequential search is a method for finding a
particular value in a list that checks each element in sequence
until the desired element is found or the list is exhausted.
2. The list need not be ordered.
3. It is very simple and works as follows: We keep on comparing
each element with the element to search until the desired
element is found or list ends.
Linear search

This is the array we are going to perform linear search
Let’s search for the number 3. We start at the beginning
and check the first element in the array.
Example

Is the first value 3? No, not it. Is it the next
element?
Is the second value 3?
Not there either. The next element?

Is the third value 3?
Not there either. Next?
Is the fourth value 3? Yes!
We found it!!! Now you understand the idea of
linear searching; we go through each element,
in order, until we find the correct value.

Linear search - Pseudocode:
# Input: Array D, integer key
# Output: first index of key in D, or -1 if not found
For i = 0 to last index of D:
if D[i] equals key:
return i
return -1

Time Complexity
In the linear search problem, the best case occurs when Target is
present at the first location. The number of operations in the
best case is constant (not dependent on n). So time complexity in
the best case would be O(1)
Best case Analysis

Time Complexity
For Linear Search, the worst case happens when the element to be
searched is not present in the array or it is present at the end of
the array.
In this scenario the search() compares it with all the elements of
array one by one. Therefore, the worst case time complexity of
linear search would be O(n).
Worst case Analysis

Time Complexity
The key is equally likely to be in any position in the array
If the key is in the first array position: 1 comparison
If the key is in the second array position: 2 comparisons ...
If the key is in the ith postion: i comparisons ...
So average all these possibilities:
(1+2+3+...+n)/n = [n(n+1)/2] /n = (n+1)/2 comparisons.
The average number of comparisons is (n+1)/2 = O(n).
Average case Analysis

Time Complexity
In average case analysis, we take all possible inputs and calculate
computing time for all of the inputs. Sum all the calculated values and
divide the sum by total number of inputs.
We must know distribution of cases. For the linear search problem, let us
assume that all cases are uniformly distributed (including the case of
target not being present in array).
Average case Analysis

Space Complexity
This type of search requires only a single unit of memory to store the
element being searched. This is not relevant to the size of the input
Array.
Hence, the Space Complexity of Linear Search is O(1).

Searching Algorithms
Binary Search

1. The binary search algorithm begins by comparing the target value to the
value of the middle element of the sorted array.
2. If the target value is equal to the middle element's value, then the
position is returned and the search is finished.
Binary search

3. If the target value is less than the middle element's value, then the
search continues on the lower half of the array; or if the target value is
greater than the middle element's value, then the search continues on the
upper half of the array.
4. This process continues, eliminating half of the elements, and comparing
the target value to the value of the middle element of the remaining
elements - until the target value is either or until the entire array has
been searched.
Binary search

This is the array we are going to perform binary search
Let’s search for the number 19.
Example
In the first step we start with the entire sequence and
consider its middle element 8

As the sequence is sorted and 8 is less than the key 19,
we conclude that 19 can occur only in the second half of
the sequence, not including the middle element 8.

We then repeat the same with the subsequence
consisting of the second half. Its middle element is
21, which is greater than the key 19.

We thus next consider the first half of this subsequence,
whose middle element is 11. In the last step the subsequence
consists of one element, and this element is both the middle
element and is equal to the key 19.

Thus we have discovered 19 in the sequence. Observe
that only the 4 elements (8, 21, 11 and 19) indexed by
the middle point index mid in the figure are compared
against the key 19.
If the same search is performed in linear search, it
will take 8 comparisons.

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 = (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

Time Complexity
n
n/2 n/2
n/4 n/4
n/8 n/8
Array
Sub -Array
.
.
.
Therefore the last sub array will be n/n or
n/2k
K = number of comparisons
made

n/n = n/2k
n/2k = 1
n = 2k
Taking log on both the sides
Log n = log 2k
Log n = k log 2
log n = k
Time Complexity is O(log n)

Space Complexity
This type of search requires only a single unit of memory to store the
element being searched. This is not relevant to the size of the input
Array.
Hence, the Space Complexity of Binary Search is O(1).

Dsa – data structure and algorithms searching

Dsa – data structure and algorithms searching

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