In the context of data structures and algorithms, "complexity" refers to the analysis of the efficiency and performance of data structures.

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Data Structure & Algorithm
Topic: Complexity
by:
Dr. Shweta Saraswat

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Introduction
• We love technology and how it has made our
lives efficient – food, taxis, courses, friends, and
whatnot are accessible with just one tap. Still,
there are a few annoying things, like the
notification saying storage is full and more than 2
seconds of buffering. Just intolerable!
• The main aim behind technological
advancements is to increase efficiency, which is
done by reducing the time and space complexity
or reducing the memory needed to execute
commands. Thus professionals work on
producing time-efficient algorithms that use less
memory.

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What is Time Complexity?
• Time complexity is the amount of time taken to run an
algorithm. It is the measure of the number of
elementary operations performed by an algorithm and
an estimate of the time required for that operation. It
also depends on external factors such as the compiler,
the processor’s speed, etc.
• If we ask you how much time you can add the first five
natural numbers? (you start counting 1+2+3+4+5).
Assume that, it took you 3 seconds. But how will you
calculate this for the computer? We cannot! And thus,
computer scientists have come up with an approach to
calculate a rough estimate of the time taken to execute

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What is space complexity?
• Space complexity is the amount of memory
space an algorithm/program uses during its entire
execution. It measures the number of variables
created to store values, including both the inputs
and the outputs.
• In simple terms, it is a rough estimation of how
much storage your code will take in RAM.
• Anyone dreaming of getting into a product-based
industry should not just be able to write code but
write efficient code which takes the least time and
memory to execute. So, let’s begin to establish a
solid foundation for this concept.

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• In simple words, time complexity is the amount of
time taken to run an algorithm whereas, space
complexity is the amount of memory used by an
algorithm. Time and space complexity are
measured using Big O notation which shows the
upper bound of the growth of the function.

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How to calculate Time complexity?
• Frequency count method:
• Let’s look at the algorithms to find the sum of
n numbers. There are two ways to get the
results.
• Code 1:- Frequency in front of code 1, 1, 1,
(n+1), n, 1
• Sum = 5+2n
• Code 2:- Frequency 1,1,1,1,1,1,1,1,1
• Sum= 9

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Frequency count method
• According to the frequency count method, we estimate
the time by counting the number of times each
statement executes and adding them all together.
• After this, we remove all the constants and keep only the
highest-order term. This gives the time complexity of
that algorithm.
• For program 1- Sum = 5 + 2n
• After removing the constants and keeping only the
highest order term we get,
• The time complexity for the first program is O(n).
• We show complexity using O(), called Big O notation. It
describes the complexity of the code using algebraic
terms.

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Frequency count method
• For program 2- Sum = 9
• After removing the constants and keeping only
the highest order term we get,
• The time complexity for the first program is O(1).
• To develop efficient software, we choose the
method with less time complexity. Thus for the
above example, we prefer a second method with
less time complexity of O(1).

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How to calculate Space Complexity?
• The space needed by an algorithm is the sum of
the fixed space and the variable space required.
Different data types take different memory
spaces as shown in the table.
• Consider an example of the sum of the first N
numbers.
• Here input value ‘n’ is a constant of type integer
and which will take the space of 4 bytes. Similarly
‘i’ and ‘sum’. Thus a total space of 12 bytes.
• Now removing the constants and keeping the
highest power term we get, Space complexity
=O(1).

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How to calculate Space Complexity?
• Consider another example of adding values to an
Array.
• Here fixed variables are ‘sum’ and ‘i’ of the integer
type.
• There’s also a temporary or extra space used by the
algorithm while ‘return’ is being executed. This
temporary space is called auxiliary space and is
calculated as a fixed space.
• Thus, the fixed part is 3 variables× 4 bytes each= 12
bytes.
• The size of the array is variable with integer type
each, thus taking the space of 4xN.
• Therefore, Total space = 4N+ 12
• Removing the constants and keeping the highest
power term we get Space complexity = O(N).

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• Together time and space complexity define
the effectiveness of an algorithm/ program. In
most cases, there is more than one algorithm
for a particular operation. It is always best to
use an algorithm with less complexity.