The document discusses algorithm analysis and complexity. It defines a priori and a posteriori analysis, and explains that algorithm analysis deals with running time. There are two main complexity measures: time complexity, which describes how time scales with input size, and space complexity, which describes how memory usage scales with input size. Time complexity can be best-case, average-case, or worst-case. Asymptotic notation like Big-O, Big-Omega, and Big-Theta are used to describe these complexities. Common loop types like linear, logarithmic, quadratic, and dependent quadratic loops are covered along with their time complexities.
Analysis and design of algorithms part2Deepak John
Analysis of searching and sorting. Insertion sort, Quick sort, Merge sort and Heap sort. Binomial Heaps and Fibonacci Heaps, Lower bounds for sorting by comparison of keys. Comparison of sorting algorithms. Amortized Time Analysis. Red-Black Trees – Insertion & Deletion.
DSA Complexity.pptx What is Complexity Analysis? What is the need for Compl...2022cspaawan12556
What is Complexity Analysis?
What is the need for Complexity Analysis?
Asymptotic Notations
How to measure complexity?
1. Time Complexity
2. Space Complexity
3. Auxiliary Space
How does Complexity affect any algorithm?
How to optimize the time and space complexity of an Algorithm?
Different types of Complexity exist in the program:
1. Constant Complexity
2. Logarithmic Complexity
3. Linear Complexity
4. Quadratic Complexity
5. Factorial Complexity
6. Exponential Complexity
Worst Case time complexity of different data structures for different operations
Complexity Analysis Of Popular Algorithms
Practice some questions on Complexity Analysis
practice with giving Quiz
Conclusion
Analysis and design of algorithms part2Deepak John
Analysis of searching and sorting. Insertion sort, Quick sort, Merge sort and Heap sort. Binomial Heaps and Fibonacci Heaps, Lower bounds for sorting by comparison of keys. Comparison of sorting algorithms. Amortized Time Analysis. Red-Black Trees – Insertion & Deletion.
DSA Complexity.pptx What is Complexity Analysis? What is the need for Compl...2022cspaawan12556
What is Complexity Analysis?
What is the need for Complexity Analysis?
Asymptotic Notations
How to measure complexity?
1. Time Complexity
2. Space Complexity
3. Auxiliary Space
How does Complexity affect any algorithm?
How to optimize the time and space complexity of an Algorithm?
Different types of Complexity exist in the program:
1. Constant Complexity
2. Logarithmic Complexity
3. Linear Complexity
4. Quadratic Complexity
5. Factorial Complexity
6. Exponential Complexity
Worst Case time complexity of different data structures for different operations
Complexity Analysis Of Popular Algorithms
Practice some questions on Complexity Analysis
practice with giving Quiz
Conclusion
TIME EXECUTION OF DIFFERENT SORTED ALGORITHMSTanya Makkar
what is Algorithm and classification and its complexity
Time Complexity
Time Space trade-off
Asymptotic time complexity of algorithm and its notation
Why do we need to classify running time of algorithm into growth rates?
Big O-h notation and example
Big omega notation and example
Big theta notation and its example
best among the 3 notation
finding complexity f(n) for certain cases
1. Average case
2.Best case
3.Worst case
Searching
Sorting
complexity of Sorting
Conclusion
Measuring the performance of algorithm, in terms of Big(O), theta, and Omega notation with respect to their Scientific notation like Worst, Average, Best, cases
In this video you will learn about
--Introduction to Algorithms
--Characteristics of an Algorithm
--Algorithms Analysis
--Priori Analysis
-- Posterior Analysis
-- Algorithm Efficiency
--Time Complexity
--Space Complexity
--Algorithm Design Tools
-- Pseudocode
--Flowchart
--Asymptotic Analysis/ Notations
--Big-Oh Notation
--Omega Notation
--Theta Notation
Unit 1: Fundamentals of the Analysis of Algorithmic Efficiency, Units for Measuring Running Time, PROPERTIES OF AN ALGORITHM, Growth of Functions, Algorithm - Analysis, Asymptotic Notations, Recurrence Relation and problems
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
ACEP Magazine edition 4th launched on 05.06.2024Rahul
This document provides information about the third edition of the magazine "Sthapatya" published by the Association of Civil Engineers (Practicing) Aurangabad. It includes messages from current and past presidents of ACEP, memories and photos from past ACEP events, information on life time achievement awards given by ACEP, and a technical article on concrete maintenance, repairs and strengthening. The document highlights activities of ACEP and provides a technical educational article for members.
6th International Conference on Machine Learning & Applications (CMLA 2024)ClaraZara1
6th International Conference on Machine Learning & Applications (CMLA 2024) will provide an excellent international forum for sharing knowledge and results in theory, methodology and applications of on Machine Learning & Applications.
NUMERICAL SIMULATIONS OF HEAT AND MASS TRANSFER IN CONDENSING HEAT EXCHANGERS...ssuser7dcef0
Power plants release a large amount of water vapor into the
atmosphere through the stack. The flue gas can be a potential
source for obtaining much needed cooling water for a power
plant. If a power plant could recover and reuse a portion of this
moisture, it could reduce its total cooling water intake
requirement. One of the most practical way to recover water
from flue gas is to use a condensing heat exchanger. The power
plant could also recover latent heat due to condensation as well
as sensible heat due to lowering the flue gas exit temperature.
Additionally, harmful acids released from the stack can be
reduced in a condensing heat exchanger by acid condensation. reduced in a condensing heat exchanger by acid condensation.
Condensation of vapors in flue gas is a complicated
phenomenon since heat and mass transfer of water vapor and
various acids simultaneously occur in the presence of noncondensable
gases such as nitrogen and oxygen. Design of a
condenser depends on the knowledge and understanding of the
heat and mass transfer processes. A computer program for
numerical simulations of water (H2O) and sulfuric acid (H2SO4)
condensation in a flue gas condensing heat exchanger was
developed using MATLAB. Governing equations based on
mass and energy balances for the system were derived to
predict variables such as flue gas exit temperature, cooling
water outlet temperature, mole fraction and condensation rates
of water and sulfuric acid vapors. The equations were solved
using an iterative solution technique with calculations of heat
and mass transfer coefficients and physical properties.
HEAP SORT ILLUSTRATED WITH HEAPIFY, BUILD HEAP FOR DYNAMIC ARRAYS.
Heap sort is a comparison-based sorting technique based on Binary Heap data structure. It is similar to the selection sort where we first find the minimum element and place the minimum element at the beginning. Repeat the same process for the remaining elements.
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
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We have compiled the most important slides from each speaker's presentation. This year’s compilation, available for free, captures the key insights and contributions shared during the DfMAy 2024 conference.
3. Algorithm Analysis
Analysis of efficiency of an algorithm can be performed at two different stages, before
implementation and after implementation, as
■ A priori analysis − This is defined as theoretical analysis of an algorithm. Efficiency of
algorithm is measured by assuming that all other factors e.g. speed of processor, are
constant and have no effect on implementation.
■ A posterior analysis − This is defined as empirical analysis of an algorithm. The chosen
algorithm is implemented using programming language. Next the chosen algorithm is
executed on target computer machine. In this analysis, actual statistics like running time
and space needed are collected.
Algorithm analysis is dealt with the execution or running time of various operations
involved. Running time of an operation can be defined as number of computer instructions
executed per operation.
4. The complexity of an algorithm is a function describing the efficiency of the algorithm in
terms of the amount of data the algorithm must process. There are two main complexity
measures of the efficiency of an algorithm:
■ Time complexity is a function describing the amount of time an algorithm takes in
terms of the amount of input to the algorithm.
■ Space complexity is a function describing the amount of memory (space) an algorithm
takes in terms of the amount of input to the algorithm.
5. Time Complexity
There are three types of time complexities, which can be found in the analysis of an
algorithm:
Best case time complexity
Average case time complexity
Worst case time complexity
6. Best-case time complexity
The best-case time complexity of an algorithm is a measure of the minimum time that the algorithm
will require. For example, the best case for a simple linear search on a list occurs when the desired
element is the first element of the list.
Worst-case time complexity
The worst-case time complexity of an algorithm is a measure of the maximum time that the
algorithm will require. A worst-case estimate is normally computed because it provides an upper
bound for all inputs including the extreme slowest case also. For example, the worst case for a simple
linear search on a list occurs when the desired element is found at the last position of the list or not
on the list.
Average-case time complexity
The average-case time complexity of an algorithm is a measure of average time of all instances taken
by an algorithm. Average case analysis does not provide the upper bound and sometimes it is difficult
to compute.
Average-case time complexity and worst-case time complexity are the most used in algorithm
analysis. Best-case time complexity is rarely found but is does not have any uses.
7. What is Asymptotic Notation?
■ Whenever we want to perform analysis of an algorithm, we need to calculate the
complexity of that algorithm. But when we calculate the complexity of an algorithm it
does not provide the exact amount of resource required. So instead of taking the exact
amount of resource, we represent that complexity in a general form (Notation) which
produces the basic nature of that algorithm. We use that general form (Notation) for
analysis process.
■ Asymptotic notation of an algorithm is a mathematical representation of its complexity.
Majorly, we use THREE types of Asymptotic Notations and those are as follows...
■ Big - Oh (O)
■ Big - Omega (Ω)
■ Big - Theta (Θ)
8. Big - Oh Notation (O)
■ Big - Oh notation is used to define the upper bound of
an algorithm in terms of Time Complexity.
■ That means Big - Oh notation always indicates the
maximum time required by an algorithm for all input
values. That means Big - Oh notation describes the
worst case of an algorithm time complexity.
Big - Oh Notation can be defined as follows...
■ Consider function f(n) as time complexity of an
algorithm and g(n) is the most significant term. If f(n)
<= C g(n) for all n >= n0, C > 0 and n0 >= 1. Then we
can represent f(n) as O(g(n)).
f(n) = O(g(n))
9. Big - Omega Notation (Ω)
■ Big - Omega notation is used to define the lower bound
of an algorithm in terms of Time Complexity.
■ That means Big-Omega notation always indicates the
minimum time required by an algorithm for all input
values. That means Big-Omega notation describes the best
case of an algorithm time complexity.
Big - Omega Notation can be defined as follows...
■ Consider function f(n) as time complexity of an algorithm
and g(n) is the most significant term. If f(n) >= C g(n) for
all n >= n0, C > 0 and n0 >= 1. Then we can represent
f(n) as Ω(g(n)).
f(n) = Ω(g(n))
10. Big - Theta Notation (Θ)
■ Big - Theta notation is used to define the average bound
of an algorithm in terms of Time Complexity.
■ That means Big - Theta notation always indicates the
average time required by an algorithm for all input values.
That means Big - Theta notation describes the average
case of an algorithm time complexity.
■ Consider function f(n) as time complexity of an
algorithm and g(n) is the most significant term. If
C1 g(n) <= f(n) <= C2 g(n) for all n >= n0, C1 > 0, C2 >
0 and n0 >= 1. Then we can represent f(n) as Θ(g(n)).
f(n) = Θ(g(n))
11. What effects run time of an algorithm?
Complexity of an algorithm is a measure of the amount of time and/or space
required by an algorithm for an input of a given size (n). computer used, the hardware
platform.
Representation of abstract data types (ADT’s)
Efficiency of compiler
Competence of implementer (programming skills)
Complexity of underlying algorithm
Size of the input
12. Linear Loops
To calculate the efficiency of an algorithm that has a single loop, we need to first determine the
number of times the statements in the loop will be executed. This is because the number of
iterations is directly proportional to the loop factor. Greater the loop factor, more is the number of
iterations. For example, consider the loop given below:
Here, 100 is the loop factor. We have already said that efficiency is directly proportional to the
number of iterations. Hence, the general formula in the case of linear loops may be given as
However calculating efficiency is not as simple as is shown in the above example. Consider the
loop given below:
Here, the number of iterations is half the number of the loop factor. So, here the efficiency can be
given as
13. Logarithmic Loops
We have seen that in linear loops, the loop updation statement either adds or subtracts the loop-
controlling variable. However, in logarithmic loops, the loop-controlling variable is either multiplied or
divided during each iteration of the loop. For example, look at the loops given below:
■ Consider the first for loop in which the loop-controlling variable i is multiplied by 2. The loop will
be executed only 10 times and not 1000 times because in each iteration the value of I doubles.
Now, consider the second loop in which the loop-controlling variable i is divided by 2.
■ In this case also, the loop will be executed 10 times. Thus, the number of iterations is a function of
the number by which the loop-controlling variable is divided or multiplied. In the examples
discussed, it is 2. That is, when n = 1000, the number of iterations can be given by log 1000 which
is approximately equal to 10.
■ Therefore, putting this analysis in general terms, we can conclude that the efficiency of loops in
which iterations divide or multiply the loop-controlling variables can be given as
f(n) = log n
14. Linear logarithmic loop
■ Consider the following code in which the loop-controlling variable of the inner loop is
multiplied after each iteration. The number of iterations in the inner loop is log 10. This
inner loop is controlled by an outer loop which iterates 10 times. Therefore, according to
the formula, the number of iterations for this code can be given as 10 log 10.
■ In more general terms, the efficiency of such loops can be given as f(n) = n log n.
15. Quadratic loop
In a quadratic loop, the number of iterations in the inner loop is equal to the number of
iterations in the outer loop. Consider the following code in which the outer loop executes 10
times and for each iteration of the outer loop, the inner loop also executes 10 times. Therefore,
the efficiency here is 100.
The generalized formula for quadratic loop can be given as f(n) = n2
16. Dependent quadratic loop
In a dependent quadratic loop, the number of iterations in the inner loop is dependent on
the outer loop. Consider the code given below:
In this code, the inner loop will execute just once in the first iteration, twice in the second
iteration, thrice in the third iteration, so on and so forth. In this way, the number of
iterations can be calculated as
If we calculate the average of this loop (55/10 = 5.5), we will observe that it is equal to the
number of iterations in the outer loop (10) plus 1 divided by 2. In general terms, the inner
loop iterates (n + 1)/2 times. Therefore, the efficiency of such a code can be given as
17.
18. Space complexity
Space complexity of an algorithm represents the amount of memory space needed the
algorithm in its life cycle. Space needed by an algorithm is equal to the sum of the
following two components
■ A fixed part that is a space required to store certain data and variables (i.e. simple
variables and constants, program size etc.), that are not dependent of the size of
the problem.
■ A variable part is a space required by variables, whose size is totally dependent on
the size of the problem. For example, recursion stack space, dynamic memory
allocation etc.
19. Example of Space Complexity
Space complexity S(p) of any algorithm p is S(p) = A + Sp(I) Where A is treated as the fixed
part and S(I) is treated as the variable part of the algorithm which depends on instance
characteristic I. Following is a simple example that tries to explain the concept
Algorithm
■ SUM(P, Q)
■ Step 1 - START
■ Step 2 - R ← P + Q + 10
■ Step 3 - Stop
Here we have three variables P, Q and R and one constant. Hence S(p) = 1+3. Now space is
dependent on data types of given constant types and variables and it will be multiplied
accordingly.