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Introduction to Analysis of Algorithms

Introduction to Analysis of Algorithms. Design Fundamentals. Key points Machines that hold data Language for describing data manipulation. Functions that describe what kind of refined data can be produced from raw data. Structures for representing data. Analyzing Algorithms. Why

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Introduction to Analysis of Algorithms

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  1. Introduction to Analysis of Algorithms

  2. Design Fundamentals • Key points • Machines that hold data • Language for describing data manipulation. • Functions that describe what kind of refined data can be produced from raw data. • Structures for representing data.

  3. Analyzing Algorithms • Why • Measuring efficiency of an Algorithm. • Efficiency checked by • Correctness • Implementation • Simplicity • Execution time and Memory space req. • New ways of doing same task better.

  4. Time and Space • Time Complexity • Amount of computer time an algorithm needs to execute the program and get the intended result. • Space Complexity • Amount of memory required for running an algorithm.

  5. Types of Analysis • Priori Analysis • Machine independent • Done before implementation • Posteriori Analysis • Target Machine dependent • Done after implementation

  6. Analysis of an Algorithm • Lets take an example • Finding maximum of n numbers and its location in the vector. • Requirements • Vector say A of size n. • Variables to store largest value and its index location (say large and j) • Design the Algorithm • Calculate the time complexity/Order of Magnitude

  7. Order of Magnitude • Order of Magnitude of an Algorithm is the sum of number of occurrences of statements contained in it. • Other terms • Worst case Running Time • Best case Running Time • Average case Running Time

  8. Insertion Sort (A, n) Another algorithm

  9. Asymptotic Notations • Concerned with how the running time of an algorithm increases with the size of input in the limit, as the size of input increases without bound. • Notations • O notation • Ω notation • Θ notation

  10. O notation • Formally defined as • For non-negative functions T(n) and f(n), the function T(n) = O(f(n)) if there are positive constants c and n0such that T(n) ≤ c*f(n) for all n, n ≥ n0 • Known as the Big-Oh • If graphed, f(n) serves as an upper bound to the curve you are analysing. • Describes the worst that can happen.

  11. c*f(n) T(n) n0 n  Big Oh Notation O notation

  12. Ω Notation • Formally defined as • For non-negative functions, T(n) and f(n), the function T(n) = Ω(f(n)) if there are positive constants c and n0 such that T(n) ≥ c*f(n) for all n, n ≥ n0. • f(n) is the lower bound for T(n). • Describes best that can happen for a given data input size.

  13. Ω Notation Omega Notation T(n) c*f(n) n0 n 

  14. Θ Notation • Formally defined as • For non-negative functions, T(n) and g(n), the function T(n) = Θ(g(n)) if there exist positive constants c1, c2 and n0 such that c1*g(n) ≤ T(n) ≤ c2*g(n) for all n, n ≥ n0. • Describes the average case for the input data size n.

  15. Θ Notation c2*g(n) T(n) c1*g(n) n0 n  Θ Notation

  16. Assumptions while finding Time Complexity • The leading constant of highest power of n and all lower powers of n are ignored in f(n) • Example for Insertion sort • T(n) = O(f(n)) • Best case f(n) = (c1+c2+c3+c4+c7)n – (c2+c3+c7) • Therefore T(n) = O(n)

  17. Some Problems….

  18. Randomized Algorithms • A randomized algorithm is just one that depends on random numbers for its operation. • Normally use a randomizer. • Decisions made in the algorithm depend on the output of the randomizer.

  19. Pseudorandom Numbers • The computer is not capable of generating truly random numbers • The computer can only generate pseudorandom numbers--numbers that are generated by a formula • Pseudorandom numbers look random, but are perfectly predictable if you know the formula • Devices for generating truly random numbers do exist

  20. Getting (pseudo)random numbers in Java • import java.util.Random; • new Random(long seed) // constructor • new Random() // constructor, uses System.timeInMillis() as seed • void setSeed(long seed) • nextBoolean() • nextFloat(), nextDouble() // 0.0 < return value < 1.0 • nextInt(), nextLong() // all 232 or 264 possibilities • nextInt(int n) // 0 < return value < n • nextGaussian()

  21. Randomized Algorithms …contd • Result may not always be 100% correct. • Correctness characterized by some randomness or probability. • Need to run the algorithms for a longer time • Categories: Numerical, Monte-Carlo, Las Vegas, Sherwood.

  22. Randomized Algorithms… contd • Examples • Primality Testing • Identifying the Repeated Element • Randomized Quicksort • Advantages • Simple • Efficient • Disadvantages • Does not always result in better performance

  23. Points to be noted • We generate random numbers • For their own sake • For simulation • Experimental in nature • Longer you run them, the better your results • It is what you do when you do not • know what else to do • Should be in every programmers toolbox

  24. Recursive Algorithms • Algorithms which use the concept of recursion to solve problems • Examples • Fibonacci series • Factorial • Algorithms with divide and conquer approach

  25. Recursive Algorithms… contd • Advantages • Easier to implement • Efficient • Disadvantages • Load on compiler.

  26. Next … • Divide and Conquer

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