HOW TO COMPUTE TIME COMPLEXITY AND SPACE COMPLEXITY OF AN ALGORITHM
Summary
TLDRThis 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
📚 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.
🔍 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.
🔄 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.
🌐 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.
Mindmap
Keywords
💡Algorithm Analysis
💡Computational Complexity Theory
💡Time Complexity
💡Space Complexity
💡Big O Notation
💡Initialization
💡Condition
💡Increment
💡Constant
💡Highest Order Term
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
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