Data Structures and Algorithms Lecture 2: Analysis of Algorithms, Asymptotic ...TechVision8
This document discusses analyzing the running time of algorithms. It introduces pseudocode as a way to describe algorithms, primitive operations that are used to count the number of basic steps an algorithm takes, and asymptotic analysis to determine an algorithm's growth rate as the input size increases. The key points covered are using big-O notation to focus on the dominant term and ignore lower-order terms and constants, and analyzing two algorithms for computing prefix averages to demonstrate asymptotic analysis.
This document provides an overview of algorithm analysis and asymptotic complexity. It discusses learning outcomes related to analyzing algorithm efficiency using Big O, Omega, and Theta notation. Key points covered include:
- Defining the problem size n and relating algorithm running time to n
- Distinguishing between best-case, worst-case, and average-case complexity
- Using asymptotic notation like Big O to give upper bounds on complexity rather than precise calculations
- Common asymptotic categories like O(n), O(n^2), O(n log n) that classify algorithm growth rates
The document discusses algorithm analysis and determining the efficiency of algorithms. It introduces key concepts such as:
- Algorithms must be correct and efficient to solve problems.
- The time and space complexity of algorithms is analyzed to compare efficiencies. Common growth rates include constant, logarithmic, linear, quadratic, and exponential time.
- The asymptotic worst-case time complexity of an algorithm (its "order") is expressed using Big O notation, such as O(n) for linear time. Higher order terms indicate slower growth.
Fundamentals of the Analysis of Algorithm EfficiencySaranya Natarajan
This document discusses analyzing the efficiency of algorithms. It introduces the framework for analyzing algorithms in terms of time and space complexity. Time complexity indicates how fast an algorithm runs, while space complexity measures the memory required. The document outlines steps for analyzing algorithms, including measuring input size, determining the basic operations, calculating frequency counts of operations, and expressing efficiency in Big O notation order of growth. Worst-case, best-case, and average-case time complexities are also discussed.
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
Order notation is a mathematical method used to analyze algorithms as the problem size increases. It allows comparison of performance independent of machine-specific factors. Common notations include Big-O (upper bound), Big-Omega (lower bound), and Theta (tight bound). These describe the limiting behavior of execution time as the problem size approaches infinity and are used to classify algorithms by their running time growth rates like constant, logarithmic, linear, quadratic, and exponential.
The document discusses algorithms and their analysis. It defines an algorithm as a step-by-step procedure to solve a problem and get a desired output. Key aspects of algorithms discussed include their time and space complexity, asymptotic analysis to determine best, average, and worst case running times, and common asymptotic notations like Big O that are used to analyze algorithms. Examples are provided to demonstrate how to determine the time and space complexity of different algorithms like those using loops, recursion, and nested loops.
The document discusses algorithms and their analysis. It defines an algorithm as a set of instructions to solve a problem and notes they must be correct, efficient, and independent of implementation or data used. The analysis of algorithms focuses on estimating time requirements using mathematical techniques like counting operations and expressing efficiency through growth functions. Common growth rates are discussed, with faster growing functions like O(n^2) and O(n^3) being less efficient than slower ones like O(n) and O(log n).
Data Structures and Algorithms Lecture 2: Analysis of Algorithms, Asymptotic ...TechVision8
This document discusses analyzing the running time of algorithms. It introduces pseudocode as a way to describe algorithms, primitive operations that are used to count the number of basic steps an algorithm takes, and asymptotic analysis to determine an algorithm's growth rate as the input size increases. The key points covered are using big-O notation to focus on the dominant term and ignore lower-order terms and constants, and analyzing two algorithms for computing prefix averages to demonstrate asymptotic analysis.
This document provides an overview of algorithm analysis and asymptotic complexity. It discusses learning outcomes related to analyzing algorithm efficiency using Big O, Omega, and Theta notation. Key points covered include:
- Defining the problem size n and relating algorithm running time to n
- Distinguishing between best-case, worst-case, and average-case complexity
- Using asymptotic notation like Big O to give upper bounds on complexity rather than precise calculations
- Common asymptotic categories like O(n), O(n^2), O(n log n) that classify algorithm growth rates
The document discusses algorithm analysis and determining the efficiency of algorithms. It introduces key concepts such as:
- Algorithms must be correct and efficient to solve problems.
- The time and space complexity of algorithms is analyzed to compare efficiencies. Common growth rates include constant, logarithmic, linear, quadratic, and exponential time.
- The asymptotic worst-case time complexity of an algorithm (its "order") is expressed using Big O notation, such as O(n) for linear time. Higher order terms indicate slower growth.
Fundamentals of the Analysis of Algorithm EfficiencySaranya Natarajan
This document discusses analyzing the efficiency of algorithms. It introduces the framework for analyzing algorithms in terms of time and space complexity. Time complexity indicates how fast an algorithm runs, while space complexity measures the memory required. The document outlines steps for analyzing algorithms, including measuring input size, determining the basic operations, calculating frequency counts of operations, and expressing efficiency in Big O notation order of growth. Worst-case, best-case, and average-case time complexities are also discussed.
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
Order notation is a mathematical method used to analyze algorithms as the problem size increases. It allows comparison of performance independent of machine-specific factors. Common notations include Big-O (upper bound), Big-Omega (lower bound), and Theta (tight bound). These describe the limiting behavior of execution time as the problem size approaches infinity and are used to classify algorithms by their running time growth rates like constant, logarithmic, linear, quadratic, and exponential.
The document discusses algorithms and their analysis. It defines an algorithm as a step-by-step procedure to solve a problem and get a desired output. Key aspects of algorithms discussed include their time and space complexity, asymptotic analysis to determine best, average, and worst case running times, and common asymptotic notations like Big O that are used to analyze algorithms. Examples are provided to demonstrate how to determine the time and space complexity of different algorithms like those using loops, recursion, and nested loops.
The document discusses algorithms and their analysis. It defines an algorithm as a set of instructions to solve a problem and notes they must be correct, efficient, and independent of implementation or data used. The analysis of algorithms focuses on estimating time requirements using mathematical techniques like counting operations and expressing efficiency through growth functions. Common growth rates are discussed, with faster growing functions like O(n^2) and O(n^3) being less efficient than slower ones like O(n) and O(log n).
This document provides an overview of algorithms including definitions, characteristics, design, and analysis. It defines an algorithm as a finite step-by-step procedure to solve a problem and discusses their key characteristics like input, definiteness, effectiveness, finiteness, and output. The document outlines the design of algorithms using pseudo-code and their analysis in terms of time and space complexity using asymptotic notations like Big O, Big Omega, and Big Theta. Examples are provided to illustrate linear search time complexity and the use of different notations to determine algorithm efficiency.
This document provides an overview of algorithm analysis. It discusses how to analyze the time efficiency of algorithms by counting the number of operations and expressing efficiency using growth functions. Different common growth rates like constant, linear, quadratic, and exponential are introduced. Examples are provided to demonstrate how to determine the growth rate of different algorithms, including recursive algorithms, by deriving their time complexity functions. The key aspects covered are estimating algorithm runtime, comparing growth rates of algorithms, and using Big O notation to classify algorithms by their asymptotic behavior.
This document provides an overview of algorithm analysis and determining the time complexity of algorithms. It discusses that the time an algorithm takes to run can be estimated by counting the number of basic operations and expressing the runtime using asymptotic notation. Examples are provided to demonstrate how to analyze the runtime of simple algorithms with loops and nested loops. The key growth rates like constant, linear, quadratic, and exponential are defined. Determining the highest order term provides the overall time complexity of an algorithm in Big O notation.
This document discusses algorithm analysis and asymptotic notation. It introduces algorithms for computing prefix averages in arrays. One algorithm runs in quadratic time O(n^2) by applying the definition directly. A more efficient linear time O(n) algorithm is also presented that maintains a running sum. Asymptotic analysis determines the worst-case running time of an algorithm as a function of the input size using big-O notation. This provides an analysis of algorithms that is independent of implementation details and hardware.
The document discusses algorithm analysis and asymptotic analysis. It introduces key concepts such as algorithms, running time analysis, experimental studies vs theoretical analysis, pseudocode, primitive operations, counting operations, big-O notation, and analyzing algorithms to determine asymptotic running time. As an example, it analyzes two algorithms for computing prefix averages - one with quadratic running time O(n^2) and one with linear running time O(n).
The document discusses algorithm analysis and asymptotic analysis. It introduces key concepts such as algorithms, running time analysis, experimental studies vs theoretical analysis, pseudocode, primitive operations, counting operations, big-O notation, and analyzing algorithms to determine asymptotic running time. As an example, it analyzes two algorithms for computing prefix averages - one with quadratic running time O(n^2) and one with linear running time O(n).
The document discusses algorithm analysis and asymptotic analysis. It introduces key concepts such as algorithms, running time analysis, experimental vs theoretical analysis, pseudocode, primitive operations, counting operations, and asymptotic analysis using big-O notation. As an example, it analyzes an algorithm for finding the maximum element in an array, showing that it runs in O(n) time. It also analyzes two algorithms for computing prefix averages, showing one runs in O(n^2) time while the other improves it to O(n) time.
The document discusses algorithm analysis and asymptotic notation. It introduces algorithms for computing prefix averages of an array in quadratic and linear time. Specifically:
- An algorithm that computes prefix averages by directly applying the definition runs in O(n^2) time as its inner loop iterates over i elements n times.
- A more efficient algorithm that maintains a running sum runs in O(n) time, as each of its n iterations performs a constant number of operations.
- Asymptotic analysis allows algorithms to be classified based on growth rate, ignoring constant factors. This provides an algorithm-independent analysis of computational complexity.
The document discusses algorithm analysis and asymptotic analysis. It introduces key concepts like best case, worst case, and average case running times. It explains that worst case analysis is most important and easiest to analyze. The document covers analyzing algorithms using pseudocode, counting primitive operations, and determining asymptotic running time using Big-O notation. Examples are provided to illustrate these concepts, including analyzing algorithms for finding the maximum element in an array and computing prefix averages.
Data Structure & Algorithms - Mathematicalbabuk110
This document discusses various mathematical notations and asymptotic analysis used for analyzing algorithms. It covers floor and ceiling functions, remainder function, summation symbol, factorial function, permutations, exponents, logarithms, Big-O, Big-Omega and Theta notations. It provides examples of calculating time complexity of insertion sort and bubble sort using asymptotic notations. It also discusses space complexity analysis and how to calculate the space required by an algorithm.
The document discusses analyzing the efficiency of algorithms. It notes that algorithms can be correct but the best depends on efficiency. Efficiency depends on time and space complexity, where time complexity is the amount of time needed for complete execution and space complexity is the amount of memory required. Different algorithms have different efficiencies that can be analyzed based on how processing time and memory usage increase with larger input sizes.
The document discusses algorithm analysis and computational complexity, specifically focusing on time complexity and big O notation. It defines key concepts like best case, average case, and worst case scenarios. Common time complexities like constant, logarithmic, linear, quadratic, and exponential functions are examined. Examples are provided to demonstrate how to calculate the time complexity of different algorithms using big O notation. The document emphasizes that worst case analysis is most useful for program design and comparing algorithms.
Data Structure & Algorithms - Introductionbabuk110
This document introduces key concepts in data structures and algorithms including algorithms, programs, data structures, objects, and relations between data structures and objects. It discusses analyzing algorithms through measuring running time experimentally and theoretically using asymptotic analysis. Examples are provided of insertion sort and prefix averages algorithms, analyzing their best, worst, and average cases, and calculating their asymptotic running times as O(n) and O(n^2) respectively. The document outlines criteria for good algorithms and techniques for asymptotic analysis including Big-O notation.
This document discusses complexity analysis of algorithms. It defines an algorithm and lists properties like being correct, unambiguous, terminating, and simple. It describes common algorithm design techniques like divide and conquer, dynamic programming, greedy method, and backtracking. It compares divide and conquer with dynamic programming. It discusses algorithm analysis in terms of time and space complexity to predict resource usage and compare algorithms. It introduces asymptotic notations like Big-O notation to describe upper bounds of algorithms as input size increases.
The document discusses data structures and algorithms. It defines key concepts like algorithms, programs, data structures, and asymptotic analysis. It explains how to analyze algorithms to determine their efficiency, including analyzing best, worst, and average cases. Common notations for describing asymptotic running time like Big-O, Big-Omega, and Big-Theta are introduced. The document provides examples of analyzing sorting algorithms like insertion sort and calculating running times. It also discusses techniques for proving an algorithm's correctness like assertions and loop invariants.
Design Analysis of Alogorithm 1 ppt 2024.pptxrajesshs31r
This document discusses algorithms and their analysis. It begins by defining an algorithm as a sequence of unambiguous instructions to solve a problem in a finite amount of time. It then provides examples of Euclid's algorithm for computing the greatest common divisor. The document goes on to discuss the fundamentals of algorithmic problem solving, including understanding the problem, choosing exact or approximate solutions, and algorithm design techniques. It also covers analyzing algorithms by measuring time and space complexity using asymptotic notations.
This document provides an overview of algorithms including definitions, characteristics, design, and analysis. It defines an algorithm as a finite step-by-step procedure to solve a problem and discusses their key characteristics like input, definiteness, effectiveness, finiteness, and output. The document outlines the design of algorithms using pseudo-code and their analysis in terms of time and space complexity using asymptotic notations like Big O, Big Omega, and Big Theta. Examples are provided to illustrate linear search time complexity and the use of different notations to determine algorithm efficiency.
This document provides an overview of algorithm analysis. It discusses how to analyze the time efficiency of algorithms by counting the number of operations and expressing efficiency using growth functions. Different common growth rates like constant, linear, quadratic, and exponential are introduced. Examples are provided to demonstrate how to determine the growth rate of different algorithms, including recursive algorithms, by deriving their time complexity functions. The key aspects covered are estimating algorithm runtime, comparing growth rates of algorithms, and using Big O notation to classify algorithms by their asymptotic behavior.
This document provides an overview of algorithm analysis and determining the time complexity of algorithms. It discusses that the time an algorithm takes to run can be estimated by counting the number of basic operations and expressing the runtime using asymptotic notation. Examples are provided to demonstrate how to analyze the runtime of simple algorithms with loops and nested loops. The key growth rates like constant, linear, quadratic, and exponential are defined. Determining the highest order term provides the overall time complexity of an algorithm in Big O notation.
This document discusses algorithm analysis and asymptotic notation. It introduces algorithms for computing prefix averages in arrays. One algorithm runs in quadratic time O(n^2) by applying the definition directly. A more efficient linear time O(n) algorithm is also presented that maintains a running sum. Asymptotic analysis determines the worst-case running time of an algorithm as a function of the input size using big-O notation. This provides an analysis of algorithms that is independent of implementation details and hardware.
The document discusses algorithm analysis and asymptotic analysis. It introduces key concepts such as algorithms, running time analysis, experimental studies vs theoretical analysis, pseudocode, primitive operations, counting operations, big-O notation, and analyzing algorithms to determine asymptotic running time. As an example, it analyzes two algorithms for computing prefix averages - one with quadratic running time O(n^2) and one with linear running time O(n).
The document discusses algorithm analysis and asymptotic analysis. It introduces key concepts such as algorithms, running time analysis, experimental studies vs theoretical analysis, pseudocode, primitive operations, counting operations, big-O notation, and analyzing algorithms to determine asymptotic running time. As an example, it analyzes two algorithms for computing prefix averages - one with quadratic running time O(n^2) and one with linear running time O(n).
The document discusses algorithm analysis and asymptotic analysis. It introduces key concepts such as algorithms, running time analysis, experimental vs theoretical analysis, pseudocode, primitive operations, counting operations, and asymptotic analysis using big-O notation. As an example, it analyzes an algorithm for finding the maximum element in an array, showing that it runs in O(n) time. It also analyzes two algorithms for computing prefix averages, showing one runs in O(n^2) time while the other improves it to O(n) time.
The document discusses algorithm analysis and asymptotic notation. It introduces algorithms for computing prefix averages of an array in quadratic and linear time. Specifically:
- An algorithm that computes prefix averages by directly applying the definition runs in O(n^2) time as its inner loop iterates over i elements n times.
- A more efficient algorithm that maintains a running sum runs in O(n) time, as each of its n iterations performs a constant number of operations.
- Asymptotic analysis allows algorithms to be classified based on growth rate, ignoring constant factors. This provides an algorithm-independent analysis of computational complexity.
The document discusses algorithm analysis and asymptotic analysis. It introduces key concepts like best case, worst case, and average case running times. It explains that worst case analysis is most important and easiest to analyze. The document covers analyzing algorithms using pseudocode, counting primitive operations, and determining asymptotic running time using Big-O notation. Examples are provided to illustrate these concepts, including analyzing algorithms for finding the maximum element in an array and computing prefix averages.
Data Structure & Algorithms - Mathematicalbabuk110
This document discusses various mathematical notations and asymptotic analysis used for analyzing algorithms. It covers floor and ceiling functions, remainder function, summation symbol, factorial function, permutations, exponents, logarithms, Big-O, Big-Omega and Theta notations. It provides examples of calculating time complexity of insertion sort and bubble sort using asymptotic notations. It also discusses space complexity analysis and how to calculate the space required by an algorithm.
The document discusses analyzing the efficiency of algorithms. It notes that algorithms can be correct but the best depends on efficiency. Efficiency depends on time and space complexity, where time complexity is the amount of time needed for complete execution and space complexity is the amount of memory required. Different algorithms have different efficiencies that can be analyzed based on how processing time and memory usage increase with larger input sizes.
The document discusses algorithm analysis and computational complexity, specifically focusing on time complexity and big O notation. It defines key concepts like best case, average case, and worst case scenarios. Common time complexities like constant, logarithmic, linear, quadratic, and exponential functions are examined. Examples are provided to demonstrate how to calculate the time complexity of different algorithms using big O notation. The document emphasizes that worst case analysis is most useful for program design and comparing algorithms.
Data Structure & Algorithms - Introductionbabuk110
This document introduces key concepts in data structures and algorithms including algorithms, programs, data structures, objects, and relations between data structures and objects. It discusses analyzing algorithms through measuring running time experimentally and theoretically using asymptotic analysis. Examples are provided of insertion sort and prefix averages algorithms, analyzing their best, worst, and average cases, and calculating their asymptotic running times as O(n) and O(n^2) respectively. The document outlines criteria for good algorithms and techniques for asymptotic analysis including Big-O notation.
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The document discusses data structures and algorithms. It defines key concepts like algorithms, programs, data structures, and asymptotic analysis. It explains how to analyze algorithms to determine their efficiency, including analyzing best, worst, and average cases. Common notations for describing asymptotic running time like Big-O, Big-Omega, and Big-Theta are introduced. The document provides examples of analyzing sorting algorithms like insertion sort and calculating running times. It also discusses techniques for proving an algorithm's correctness like assertions and loop invariants.
Design Analysis of Alogorithm 1 ppt 2024.pptxrajesshs31r
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Data Structures and Agorithm: DS 22 Analysis of Algorithm.pptx
1. International Islamic University H-10, Islamabad, Pakistan
Data Structure
Lecture No. 22
Analysis of Algorithm
Engr. Rashid Farid Chishti
http://youtube.com/rfchishti
http://sites.google.com/site/chishti
2. An algorithm is a set of instructions to be followed to solve a problem.
There can be more than one solution (more than one algorithm) to solve a
given problem.
An algorithm can be implemented using different programming languages
on different platforms.
An algorithm must be correct. It should correctly solve the problem.
e.g. For sorting, this algorithm should work even if the input is already
sorted, or it contains repeated elements.
Once we have a correct algorithm for a problem, we have to determine the
efficiency of that algorithm.
Program: An Implementation of Algorithm in some programming language
Data Structure: Organization of data needed to solve the problem
Algorithm
3. There are two aspects of algorithmic performance:
Time
Instructions take time.
How fast does the algorithm perform ?
What affects its runtime ?
Space
Data structures take space (RAM)
What kind of data structures can be used ?
How does choice of data structure affect the runtime ?
We will focus on time:
How to estimate the time required for an algorithm ?
How to reduce the time required ?
Efficiency of an Algorithm
4. Analysis of Algorithms is the area of computer science that provides tools to
analyze the efficiency of different methods of solutions.
How do we compare the time efficiency of two algorithms that solve the same
problem?
Experimental Study: Implement these algorithms in a programming language
(C++), and run them to compare their time requirements. Comparing the
programs (instead of algorithms) has difficulties.
How are the algorithms coded?
Comparing running times means comparing the implementations.
We should not compare implementations, because they are sensitive to programming style that may
cloud the issue of which algorithm is inherently more efficient.
What computer should we use?
We should compare the efficiency of the algorithms independently of a particular computer.
What data should the program use?
Any analysis must be independent of specific data.
Analysis of Algorithm
5. Experimental Study
Write a program that implements
an algorithm
Run the program with data sets of
varying size Use the method like
GetLocalTime() to get an
accurate measure of the actual
running time.
Measuring the Running Time
#include <windows.h>
#include <stdio.h>
#include <iostream>
using namespace std;
int main(){
SYSTEMTIME st;
GetLocalTime (&st);
cout << "The system time is: "
<< st.wHour <<":"
<< st.wMinute <<":"
<< st.wMilliseconds << endl;
system("PAUSE");
return 0;
}
6. #include <windows.h>
#include <stdio.h>
#include <iostream>
using namespace std;
long double factorial(long double n){
if (n < 0) // if n is negative
exit(-1); // close the program
long double f = 1;
while (n > 1)
f *= n--;
return f;
}
Measuring the Running Time
int main(){
LARGE_INTEGER t1, t2, t3;
long double n, answer;
cout << "Enter a positive integer: ";// 170
cin >> n;
QueryPerformanceCounter(&t1);
answer = factorial(n);
QueryPerformanceCounter(&t2);
t3.QuadPart = t2.QuadPart - t1.QuadPart;
cout << n << "! = " << answer << endl;
cout << "Time taken = "
<< t3.QuadPart <<" nano seconds"
<< endl;
system("PAUSE");
return 0;
}
7. Limitations of Experimental Study
This experiment can be performed only on a particular dataset.
It is necessary to implement and test the algorithm in order to determine its
running time.
Experiments can be done only on a limited sets of inputs.
In order to compare algorithms, the same set of hardware and software
should be used.
Measuring the Running Time
8. Beyond Experimental Study
When we analyze algorithms, we should employ mathematical techniques that
analyze algorithms independently of specific implementations, computers, or
data.
To analyze algorithms:
First, we start to count the number of significant operations in a particular solution to
assess its efficiency.
Then, we will express the efficiency of algorithms using growth functions.
Measuring the Running Time
9. Each operation in an algorithm (or a program) has a cost.
Each operation takes a certain of time.
count = count + 1; // take a certain amount of time, but it is constant
A sequence of operations:
count = count + 1; Cost: c1
sum = sum + count; Cost: c2
Total Cost = c1 + c2
Example 1: Simple If-Statement
Total Cost
= c1 + c2+max(c3,c4)
The Execution Time of an Algorithm
Simple if Statement Cost Running Time
int abs_value; C1 1
if ( n < 0 ) C2 1
abs_value = -n; C3 1
else
abs_value = n; C4 1
10. Example 2: Simple Loop
Total Cost = c1 + c2 + (n+1)*c3 + n*c4 + n*c5
= c1 + c2 + c3 + c3*n + c4*n + c5*n
= (c3 + c4 + c5)n + (c1 + c2 + c3) = an + b
The time required for this algorithm is proportional to n
The Execution Time of an Algorithm
Simple Loop Cost Running Time
int i = 1; C1 1
int sum = 0; C2 1
while ( i<= n) C3 n + 1
{
i = i + 1; C4 n
sum = sum + i; C5 n
}
11. Example 3: Nested Loop
Total Cost
= c1 + c2 + (n+1)*c3 + n*c4 + n*(n+1)*c5
+ n*n*c6 + n*n*c7 + n*c8
= (c5+c6+c7)n2 + (c3+c4+c5+c8)n + (c1+c2+c3)
The time required for this algorithm is
proportional to n2
The Execution Time of an Algorithm
Nested Loop Cost Running
Time
int i = 1; C1 1
int sum = 0; C2 1
while ( i<= n) C3 n + 1
{
int j = 1; C4 n
while ( j<= n) C5 n*(n+1)
{
sum = sum + i; C6 n*n
j = j + 1; C7 n*n
}
i = i + 1; C8 n
}
12. Loops: The running time of a loop is at most the running time of the
statements inside of that loop times the number of iterations.
Nested Loops: Running time of a nested loop containing a statement in the
inner most loop is the running time of statement multiplied by the product of
the sized of all loops.
Consecutive Statements: Just add the running times of those consecutive
statements.
If Else: Never more than the running time of the test plus the larger of running
times of S1 and S2.
General Rules for Estimation
13. Algorithm’s time requirement is the function of the problem size.
Problem size depends on the application: e.g. number of elements in a list for a sorting
algorithm, the number users for a social network search.
So, for instance, we say that (if the problem size is n)
Algorithm A requires 5n2 time units to solve a problem of size n.
Algorithm B requires 7n time units to solve a problem of size n.
The most important thing to learn is how quickly the algorithm’s time
requirement grows as a function of the problem size.
Algorithm A requires time proportional to n2.
Algorithm B requires time proportional to n.
An algorithm’s proportional time requirement is known as growth rate.
We can compare the efficiency of two algorithms by comparing their growth
rates.
Algorithm Growth Rates
14. Algorithm Growth Rates
Graph of time requirements as
a function of the problem size n
Common Growth Rates
Function Growth Rate
Name
c Constant
log2 N Logarithmic
log2 N Log-squared
N Linear
N log2 N Log-linear
N2 Quadratic
N3 Cubic
2N Exponential
2
15. Running Time for Small Inputs
Running
time
Input size (x = n)
x3 2x x2
log2(x)
16. Running Time for Large Inputs
Running
time
Input size (x = n)
x.log2(x)
x
x2
2x x3
17. Searching a number from array,
(Best Case) The number found is at index 0
(Worst Case) The Number was Not Fount or the number found is at index N-1, Where N
is the size of an array.
(Average Case) The number found is at index between 1 and N-1
Best, Average and Worst Case
0
1
2
3
4
5
6
A B C D E F G
Running
Time
(ms)
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Worst Case
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Best Case
}Average Case
dataset
18. Asymptotic notation of an algorithm is a mathematical representation of its
complexity.
There are three types of Asymptotic Notations...
Big - O (O)
Big - Omega (Ω)
Big - Theta (θ)
Big - Oh Notation (O): Big Oh notation is used to define the upper bound of an
algorithm in terms of Time Complexity. It provides us with an asymptotic
upper bound.
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.
Asymptotic Notations
19. f(n) is your algorithm runtime, and g(n) is an arbitrary time complexity. you are
trying to relate to your algorithm.
f(n) is O(g(n)), if for some real constants c (c > 0) and number n0 , f(n) ≤ c g(n)
for every input size n (n > n0).
O(g(n))
= { f(n): there exist positive constants c and n0
such that 0 ≤ f(n) ≤ cg(n) for all n ≥ n0 }
f(n) = O(g(n)) means f(n) grows slower than or
equal to c g(n)
If an Algo A requires time proportional to n2, it is O(n2).
If an Algo A requires time proportional to n, it is O(n).
Big - Oh Notation (O)
21. To Simply the analysis of running time by getting the rid of details which may
be affected by specific implementation hardware.
Capturing the Essence: how the running time of an algorithm increases with
the size of the input in the limit.
Two Basic Rules:
Drop all lower order terms
Drop all constants
Big - Oh Notation (O)
f(n) = 5n2 + 6n + 10
= 5n2 + 6n + 10
= 5n2 + 10
= 5n2 + 10
= n2
f(n) = 7n2 + 2n + 4
= 7n2 + 2n + 4
= 7n2 + 4
= 7n2 + 4
= n2
22. If an algorithm requires f(n) = n2-3n+10 seconds to solve a problem size n. If
constants k and n0 exist such that
k*n2 >= n2-3n+10 for all n >= n0 .
the algorithm is order n2 (In fact, k is 2 and n0 is 2)
2*n2 >= n2-3n+10 for all n >= 2 .
Thus, the algorithm requires no more than k*n2 time units for n >= n0 ,
So it is O(n2)
Big - Oh Notation (O)
24. Big - Oh Notation (O)
Show 2x + 17 is O(2x)
2x + 17 ≤ 2x + 2x = 2*2x for x > 5
Hence k = 2 and n0 = 5
2*2x
n0 = 5
25. 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.
It provides us with an asymptotic lower bound.
f(n) is Ω(g(n)), if for some real constants c (c > 0)
and number n0 (n0 > 0), f(n) ≥ c g(n) for every
input size n (n > n0)
f(n) = Ω(g(n)) means f(n) grows faster than or
equal to g(n)
Big - Omega Notation (Ω)
26. Big - Theta notation is used to define the average bound of an algorithm in
terms of Time Complexity and denote the asymptotically tight bound
(Sandwich between best case and worst case)
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.
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)).
Big - Theta Notation (θ)
Average
Worst
Best
}
}
27. Big - Theta Notation (θ)
g(n) = 8n2
g(n) = 7n2
Show f(n) = 7n2 + 1 is 𝚹(n2)
You need to show f(n) is O(n2) and f(n) is Ω(n2)
f(n) is O(n2) because 7n2 + 1 ≤ 7n2 + n2 ∀ n ≥ 1
k1 = 8 n0 = 1
f(n) is Ω (n2) because 7n2 + 1 ≥ 7n2 ∀n ≥ 0
k2 = 7 n0 = 0
Pick the largest n0 to satisfy both conditions naturally
k1 = 8, k2 = 7, n0 = 1
k1 = 8
n0 = 1
n0 = 0
28. Common Asymptotic Notations
Complexity Terminology
O (n!) Factorial
O (2n), n>1 Exponential
O (nb) Polynomial
O (n2) Quadratic
O (n log2 n) Linearithmic
O (n) Linear
O (log2 n) Logarithmic
O (1) Constant
Complexity
Dataset Size (n)
Running
Time
(sec)
2
2
30. Complexity Function:
F(n) = C1*1 + C2*1 + Max(C3,C4)*1
= (C1+C2+C3)1 = 1 (a fixed value)
So algorithm complexity is O(1)
Running Time Analysis Example 1
Simple if Statement Cost Running Time
int abs_value; C1 1
if ( n < 0 ) C2 1
abs_value = -n; C3 1
else
abs_value = n; C4 1
31. Complexity Function:
F(n) = C1 + C2 + (n+1)*C3 + n*C4 + n*C5
= (C3+C4+C5)*n + (C1+C2+C3) = an+b = n
So algorithm complexity is O(n)
Running Time Analysis Example 2
Simple Loop Cost Running Time
int i = 1; C1 1
int sum = 0; C2 1
while ( i<= n) C3 n + 1
{
i = i +1; C4 n
sum = sum +1; C5 n
}
32. Complexity Function:
F(n) = c1 + c2 + (n+1)*c3 + n*c4 + n*(n+1)c5
+ n*n*c6 + n*n*c7 + n*c8
= (c5+c6+c7)*n2 + (c3+c4+c5+c8)*n + (c1+c2+c3)
= a*n2 + b*n + c
So algorithm complexity is O(n2)
Running Time Analysis Example 3
Nested Loop Cost Running
Time
int i = 1; C1 1
int sum = 0; C2 1
while ( i<= n) C3 n + 1
{
int j = 1; C4 n
while ( j<= n) C5 n*(n+1)
{
sum = sum +1; C6 n*n
j = j + 1; C7 n*n
}
i = i + 1; C8 n
}
33. Some mathematical equalities are:
Some Mathematical Facts
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34. Complexity Function:
T(n) = c1*(n+1) + c2*( ) + c3* ( ) + c4*( )
= a*n3 + b*n2 + c*n + d
So, the growth-rate function for
this algorithm is O(n3)
Running Time Analysis Example 4
Nested Loop Cost Running Time
for (int i=1; i<=n; i++) C1 n+1
for (int j=1; j<=i; j++) C2
for (int k=1; k<=j; k++) C3
x = x+1; C4
n
j
j
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j
k
k
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Editor's Notes
#include <windows.h>
#include <stdio.h>
#include <iostream>
using namespace std;
long double factorial(long double n){
if (n < 0) // if n is negative
exit(-1); // close the program
long double f = 1;
while (n > 1)
f *= n--; //1st f=f*n then n decrements
return f;
}
int main(){
LARGE_INTEGER StartingTime, EndingTime, ElapsedNanoseconds;
long double n, answer;
cout << "Enter a positive integer: "; // e.g. 170
cin >> n;
QueryPerformanceCounter(&StartingTime);
answer = factorial(n);
QueryPerformanceCounter(&EndingTime);
ElapsedNanoseconds.QuadPart = EndingTime.QuadPart - StartingTime.QuadPart;
cout << n << "! = " << answer << endl;
cout << "Time taken = " << ElapsedNanoseconds.QuadPart <<" nano seconds" << endl;
system("PAUSE");
return 0;
}