HOW TO COMPUTE TIME COMPLEXITY AND SPACE COMPLEXITY OF AN ALGORITHM

Summary

This video from Information Technology Skills (IPS) offers an introduction to algorithms and complexity, a cornerstone of computational complexity theory. It explains how to analyze algorithms by breaking them into operations and calculating their time and space complexity using Big O notation. Examples are provided to illustrate the process, including simple loops and nested loops, with complexities ranging from linear to quadratic and logarithmic. The tutorial encourages viewers to engage with the content and subscribe for more informative videos.

Takeaways

• 😀 Algorithm analysis is a crucial part of computational complexity theory, which offers theoretical estimations for the resources required to solve computational problems.
• 🔍 Computational complexity theory, also known as complexity theory, measures the time and space resources consumed by an algorithm during its execution.
• ⏱️ Time complexity is a measure of the time an algorithm takes to complete, often expressed using Big O notation.
• 📊 Space complexity refers to the amount of working storage an algorithm needs, which includes variables and data structures.
• 🛠️ To compute time complexity, algorithms are broken down into individual operations, and the Big O of each operation is calculated and summed up.
• 🔢 In the example provided, the time complexity of a simple summation algorithm with a for loop is O(n), where n is the number of elements in the array.
• 🔄 The space complexity of an algorithm is determined by counting the number of variables and data structures, which in the example was also O(n).
• 🔄 For nested loops, the time complexity is calculated by considering the number of iterations of the outer loop and the inner loop, resulting in O(n^2) for the given example.
• 📈 An algorithm with a loop that increments by a factor (e.g., i = i * 2) has a time complexity of O(log n), where n is the number of elements processed.
• 💬 The video encourages viewers to engage by commenting, subscribing, and liking for more tutorial videos.

Q & A

What is the main focus of the video?

The main focus of the video is an introduction to algorithm analysis and complexity, which is an important part of computational complexity theory.

What is computational complexity theory?

Computational complexity theory, also known as complexity theory, measures the amount of computing resources, such as time and space, that a particular algorithm consumes when it runs.

What is the purpose of algorithm analysis?

Algorithm analysis is used to provide a theoretical estimation for the required resources of an algorithm to solve a specific computational problem.

What are the two types of complexities discussed in the video?

The two types of complexities discussed in the video are time complexity and space complexity.

How is time complexity expressed?

Time complexity is commonly expressed using the big O notation.

What is the process of computing time complexity?

To compute time complexity, one must break the algorithm into individual operations, count the big O of each operation, add them up, remove constants, and find the highest order term.

What is considered when computing space complexity?

Space complexity considers the amount of working storage an algorithm needs, which includes variables and data structures.

How is the time complexity of a for loop with a single operation inside it calculated?

The time complexity of a for loop with a single operation inside it is calculated by considering the initialization, condition check, and increment operations, each contributing to the overall time complexity.

What is the time complexity of an algorithm with a nested loop where the inner loop iterates n times for each iteration of the outer loop?

The time complexity of such an algorithm is O(n^2), as each iteration of the outer loop results in n iterations of the inner loop.

How does the video explain the time complexity of an algorithm with an incrementing loop that increments by two?

The video explains that the time complexity of an incrementing loop that increments by two is O(n/2), which simplifies to O(n), as the loop will iterate half the number of times compared to a loop that increments by one.

What is the time complexity of an algorithm that doubles the value of a variable in each iteration?

The time complexity of an algorithm that doubles the value of a variable in each iteration is O(log2(n)), as the number of iterations required to reach n is the logarithm base 2 of n.

Outlines

00:00

📚 Introduction to Algorithm Analysis and Complexity

The video begins with an introduction to algorithm analysis, which is a crucial part of computational complexity theory. This theory provides theoretical estimations for the resources required by an algorithm to solve a computational problem. The video explains the concepts of algorithm analysis, complexity theory, time complexity, and space complexity. Time complexity is the measure of the time an algorithm takes to complete, often expressed using Big O notation. Space complexity refers to the amount of working storage an algorithm needs. The video then delves into how to compute time complexity by breaking down an algorithm into individual operations, counting the Big O of each, and finding the highest order term.

05:08

🔍 Detailed Explanation of Time and Space Complexity Calculation

This paragraph provides a detailed explanation of how to calculate time and space complexity. It uses an example of a simple algorithm that sums elements in an array. The time complexity is calculated by breaking down the algorithm into individual operations and counting the number of times each operation is executed. The space complexity is determined by identifying the variables and data structures used by the algorithm. The video emphasizes the importance of removing constants and focusing on the highest order term when calculating complexity.

10:14

🔄 Nested Loops and Their Impact on Time Complexity

The video script discusses the impact of nested loops on time complexity. It uses an example with an outer loop and an inner loop to demonstrate how to compute the Big O notation for each operation within the loops. The example shows how the time complexity is the product of the number of iterations of the outer loop and the inner loop. The video explains that after adding the complexities of all operations, constants are removed, and the highest order term is identified to determine the overall time complexity of the algorithm.

15:37

🌐 Exploring Different Loop Increments and Their Time Complexity

The final paragraph explores how different loop increments affect time complexity. It uses examples where the loop increments by one, two, and even by powers of two. The video explains how to calculate the time complexity for each case, emphasizing the importance of understanding the relationship between the number of iterations and the complexity. It concludes with an example where the loop condition involves an exponentiation, leading to a time complexity of log base 2 of n, showcasing the diversity in algorithmic complexity analysis.

Keywords

💡Algorithm Analysis

Algorithm analysis is a systematic approach to assessing the performance of an algorithm. It is crucial for understanding how well an algorithm will run in terms of time and space complexity. In the video, algorithm analysis is introduced as an important part of computational complexity theory, which provides theoretical estimations for the resources required by an algorithm to solve a specific computational problem. The script discusses how to break down an algorithm into individual operations to compute its time complexity, which is a fundamental aspect of algorithm analysis.

💡Computational Complexity Theory

Computational complexity theory, also known as complexity theory, is a field of computer science that studies the inherent difficulty of computational problems and the computational resources required to solve them. The video script introduces this theory as a measure of the amount of computing resources, specifically time and space, that an algorithm consumes when it runs. It is foundational to understanding the efficiency of algorithms and plays a key role in the video's exploration of algorithm performance.

💡Time Complexity

Time complexity is a concept in algorithm analysis that describes the amount of time taken by an algorithm to run as a function of the length of the input. It is commonly expressed using Big O notation. In the video, time complexity is explained through the process of breaking an algorithm into individual operations, counting the Big O of each operation, and then summing them up while removing constants and finding the highest order term. This concept is illustrated with examples, such as a for loop that sums elements in an array.

💡Space Complexity

Space complexity refers to the amount of memory space required by an algorithm to solve a problem, which is another critical aspect of computational complexity. In the video, space complexity is discussed in the context of the variables and data structures used by an algorithm. The script provides an example of calculating space complexity by considering the number of variables and the size of data structures, ultimately aiming to determine the Big O notation for space usage.

💡Big O Notation

Big O notation is a mathematical notation that describes the upper bound of an algorithm's time or space complexity. It is used to classify algorithms according to how their resource use scales with input size. The video script explains how to calculate the time complexity of an algorithm using Big O notation by breaking down the algorithm, summing the complexities of individual operations, and simplifying the expression to find the highest order term. This notation is central to the video's theme of algorithm analysis.

💡Initialization

Initialization in the context of loops and algorithms refers to the process of setting up the initial state of variables before the loop begins execution. In the video, initialization is part of the explanation of how to compute time complexity, particularly when discussing the breakdown of a for loop into its constituent parts: initialization, condition check, and increment. The script uses the example of a for loop to illustrate how initialization contributes to the overall time complexity.

💡Condition

The condition in a loop is a Boolean expression that determines whether the loop should continue executing. It is a critical component in controlling the flow of an algorithm. The video script discusses how the condition of a loop is analyzed for time complexity, counting the number of times the condition is checked and executed, which is essential for determining the overall time complexity of the loop.

💡Increment

Increment in a loop refers to the operation that modifies the loop variable to bring the loop closer to its termination condition. In the video, the increment is one of the parts of a for loop that is analyzed for time complexity. The script explains how the number of increments is counted and contributes to the overall time complexity, with examples showing how increments are considered in the context of different loop structures.

💡Constant

In the context of Big O notation and algorithm analysis, a constant refers to a value that does not change with the size of the input and thus does not affect the algorithm's scalability. The video script mentions that when calculating time complexity, constants are removed from the Big O expression because they do not influence the algorithm's performance as the input size grows. This concept is important for simplifying the time complexity expression to its most significant term.

💡Highest Order Term

The highest order term in an algorithm's time or space complexity expression is the term that dominates the growth of the function as the input size increases. In the video, the script explains that when calculating time complexity, one should find the highest order term after summing the complexities of individual operations and removing constants. This term is crucial for understanding the scalability and efficiency of an algorithm, as it represents the primary factor affecting performance.

Highlights

Introduction to algorithm analysis, an important part of computational complexity theory.

Complexity theory provides theoretical estimation for the required resources of an algorithm.

Definition of algorithm analysis and its role in measuring computing resources.

Explanation of time complexity as the amount of time an algorithm takes to complete.

Space complexity defined as the amount of working storage an algorithm needs.

Methodology for computing time complexity by breaking an algorithm into individual operations.

Use of big O notation to express the time complexity of an algorithm.

Example of computing time complexity for a simple summation algorithm.

Breaking down a for loop into initialization, condition check, and increment steps.

Counting the big O of each operation and summing them to find the overall time complexity.

Explanation of how to remove constants and find the highest order term for time complexity.

Computing space complexity by identifying variables and data structures used in an algorithm.

Example of a nested loop algorithm and its time complexity computation.

Concept of adding big O notations of individual operations to get the overall time complexity.

Example of an algorithm with a time complexity of O(n) due to a single loop.

Explanation of how to compute time complexity for an algorithm with a loop that increments by two.

Example of an algorithm with a time complexity of O(log n) due to exponential growth in conditions.

Encouragement for viewers to comment, like, and subscribe for more tutorial videos.

Transcripts

[Music]

hello guys welcome again to ips

information technology skills

in this video we're going to have an

introduction to algorithm

and complexity so let's start algorithm

analysis

an important part of computational

complexity theory which provides

theoretical estimation for the required

resources

of an algorithm to solve a specific

computational

problem okay so for say algorithm

analysis

we're going to analyze algorithm the

next

we have the word complexity theory it is

also called as computational complexity

it measures the amount of computing

resources which is the time and space

that a particular algorithm consume

when it runs i say complexity duty

these are the variables or data

structures

is the time complexity it is the amount

of time the algorithm is

completed

it is commonly expressed using the big o

notation

then we also have the space complexity

the amount of working storage and

algorithm need

these are the data structures

algorithm so how do we compute the time

complexity first we need to break the

algorithm or fractions into individual

operations so later on makikita natin

company natin eber break

is an algorithm then after breaking the

algorithm into individual operations

that's the time now or the time

complexity

each operation

considers the next add up the big

o of each operation together remove the

constant

then find the highest order term so

let's have the first

example let's say meron tayong ari with

five elements

and melon algorithm here we have

sum equals to zero meron tayong for loop

and inside the for loop is

sum equals sum plus e or x

index

the first instruction computation of

time complexity is to break

the algorithm into individual operations

so dito

consider not in each line of code is an

individual operation so

it's a line of code another line and

third line

una cello obnong so after breaking the

algorithm into individual operations

let's count the

big o or the time complexity each line

so

for the sum equals to zero the time

complexity is

one so

execute that is the time complexity

for the for loop

so we're going to break this into three

parts

the initialization the condition and the

increment for the initialization means

[Music]

is

again the condition execute increment

again

as i get the condition execute increment

again

so you and paladito guys this is the

number of elements inside an

array condition

if the value of i is equal to 5.

[Music]

or the number of elements inside the

array is five

so we believe nothing to one two three

four

five six so five plus one

six so n plus one what if in the united

alumni

element in the area is ten okay so

11 times now condition so

better and plus one

and then increments

one two three four five five times

the increment so increment

is n

five times increment what if the number

of elements again is ten

so ten times chance increments so next

we have this

sum equals sum plus e rx

index of i individual operation

time complexity is

okay so as long as the condition is

false

so one times n pretty nothing

one two three

we need to remove the constant a name

constantly

the time complexity of this algorithm is

big o of n now computer not

and space complexity for the space

complexity ito among

variables and data structures and a

gametes

algorithm so the space complexity we

have

a rx dito

and if we're going to compute your space

complexity

belonging

value or the number of elements inside

the eyes so pretty much

[Music]

n plus three then

constant

and so pretty much that the space

complexity of

this algorithm is big o of n

let's have another example guys

the first loop is x equals to zero x

less than n x plus plus young n guys

let's say

number of elements or any number now for

the condition

and inside the first loop meron another

lo tapos equals to sum plus a

r index of i inside no inner loop

so follow instruction at all we need to

break the algorithm or functions in the

individual

operation and compute the big o of each

operation so they don't consider that in

that each line of code or

algorithm is individual

so for the sum the time complexity is

1. may execute

and for the for loop the first loop nut

and we have x equals to zero so the time

complexity is one

x less than n so the time complexity is

n plus one and plus n

so if we're going to observe

first example okay

decrement by one similarly time

this is for the initialization and plus

one this is for the condition

[Music]

end so must have nothing that the time

complexity of

this loop is n

time complexity

operations

the time complexity of the inner loop is

one plus n plus one plus n

or simply

this is the time complexity

on time complexity

time complexity

now let's try to add all only big o

of h operation d does n times n predict

that is n square same with this part

that is n square so not in parameters

we have two n square plus n plus

one then after adding

we're going to remove the constant

is n square plus n then find the

highest order term in algebra the

highest order term is

um square so we can say that the time

complexity

of this algorithm is n square big o

of n is square okay so that is the

highest order term

now let's try to compute namandi space

complexity

here we have a r r x again marantayang

sum

now we're going to add we have n plus

four then tangalyn constant

so we can say that the space complexity

of this algorithm is

big o of n let's have another example

example

number three so if you're going to

observe

that the time complexity increments

by one is n what if the loop will

increment by

two so so let's say

five element initialization

is

five five now asking of

conditions increment one

two three four four times

like increment pero your number of

one two three four

five so five times now xiaomi has no

condition but the number of element

is ten so ten divided by two that is

five

so whether that is a beginner and divide

by two

so if the time complexity known loop 9

is and divided by 2 sim silang

time complexity number operations inside

the equilibrium of the

loop so putting that things have become

that the time complexity in the non-sum

is and divide by two now we're going to

add this

in

[Music]

subtraction sim sila non veena

computation

detonator example number four net n

let's see

we have the loop na i equals to one i

less than n

then i equals to i times two

multiplication

nandito guys now to compute these guys

because if we're going to initialize

that to zero any number multiplied by

zero

will be equal

then as again the condition tapos

incremental indian

2 times 2 the magicking 4 as the

condition 4 times 2 forgiving 8 as again

the condition

8 times 2 making 16 as again the

condition

16 times 2 magicking 32 as again the

condition so

process as long as the condition is true

or hang on

so continue one

exponent so data's a one but then that

is a beginner

two raised to zero two raised to zero

that is equal to one

then sub two but then two raised to one

so four padding two raised to two so

eight

to raise the 3 2 raised to 4 4 is the 5

and so on and gang 2 raised to x

or let's say you want to raise the x not

n is the

n value so dipping the nut

n equals to base two raised to an

exponent pretty nothing young it

right into log base two of n

so we can say that this algorithm has a

time complexity of

log base 2 of n so

the big o notation here

on the note 9 okay guys so

just comment down below and subscribe to

like this video and of course subscribe

for more tutorial videos

bye