Chapter1.1 Introduction to design and analysis of algorithm.ppt
- 1. Course Title: Design and Analysis of Algorithms
- 2. CHAPTER 1: INTRODUCTION • Algorithms • Analyzing Algorithms • Designing Algorithms
- 3. Algorithms? Why Algorithms is needed? computational problems in general and of the algorithms needed to solve them What is Algorithms? An algorithm is any well-defined computational procedure With What? It takes some value, or set of values, as input and produces some value, or set of values, as output
- 4. Definition: Algorithms • An algorithm is thus a sequence of computational steps that transform the input into the output. And also view an algorithm as a tool for solving a well-specified computational problem.
- 5. Example: • Input: A sequence of n numbers a1, a2, . . . , an . • Output: A permutation (reordering) of the input sequence such that Ex: Given an input sequence such as 31, 41, 59, 26, 41, 58 , a sorting algorithm returns as output the sequence 26, 31, 41, 41, 58, 59
- 6. Law of Algorithms • An algorithm is said to be correct if, for every input instance, it halts with the correct output. We say that a correct algorithm solves the given computational problem. • An incorrect algorithm might not halt at all on some input instances, or it might halt with other than the desired answer.
- 7. Steps to design the Algorithm 1. Under stand the problem 2. Decide about the way to solve the problem 3. Design the algorithm 4. Prove the correctness 5. Analysis the algorithm 6. Code the algorithm
- 8. Analyzing the Algorithms Why is it Important? It has come to mean predicting the resources that the algorithm requires. Occasionally, resources such as memory, communication bandwidth, or logic gates are of primary concern, but most often it is computational time that we want to measure
- 9. Analysis of Algorithms • 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? Naïve Approach: 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.
- 10. CENG 213 Data Structures 10 Analysis of Algorithms • 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.
- 11. Before Analyzing • What to do? have a model of the implementation technology that will be used it means, resources of that technology and their costs and running time Example: random-access machine (RAM)
- 12. Requirement for Analayze • mathematical tools required like, discrete combinatorics, elementary probability theory, algebraic dexterity, most significant terms in a formula. (behavior of the algorithm may change different I/P) • find a means of expression that is simple to write and manipulate, shows the important characteristics of an algorithm's resource requirements, and suppresses tedious details.
- 13. 13 General Rules for Estimation • 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.
- 14. Importance of Analyze Algorithm • Need to recognize limitations of various algorithms for solving a problem • Need to understand relationship between problem size and running time – When is a running program not good enough? • Need to learn how to analyze an algorithm's running time without coding it • Need to learn techniques for writing more efficient code • Need to recognize bottlenecks in code as well as which parts of code are easiest to optimize
- 15. Measuring Complexity Again • The worst case running time of an algorithm is the function defined by the maximum number of steps taken on any instance of size n. • The best case running time of an algorithm is the function defined by the minimum number of steps taken on any instance of size n. • The average-case running time of an algorithm is the function defined by an average number of steps taken on any instance of size n. • Which of these is the best to use?
- 16. Order of Growth • It is the rate of growth, or order of growth, of the running time • we write that insertion sort, for example, has a worst-case running time of θ (n2)
- 17. Best, Worst, and Average Case
- 18. Basic Common Growth Rates 1 constant log n logarithmic n linear n log n n-log-n n2 quadratic n3 cubic 2n exponential n! factorial
- 19. Running times for small inputs
- 20. • many ways to design algorithms. • Learn general approaches to algorithm design – Divide and conquer – Greedy method – Dynamic Programming – Basic Search and Traversal Technique – Graph Theory – Linear Programming – Approximation Algorithm – NP Problem Designing algorithms
- 21. Designing algorithms • Two approaches: 1. incremental approach 2. Divide and conquer approach The divide-and-conquer approach recursive in structure →to solve a given problem, they call themselves recursively one or more times to deal with closely related sub problems.
- 22. Three steps at each level of the recursion: • Divide • Conquer (recurcively) • Combine
- 23. Analyzing divide-and-conquer algorithms • When an algorithm contains a recursive call to itself, its running time can often be described by a recurrence equation or recurrence, which describes the overall running time on a problem of size n in terms of the running time on smaller inputs.
- 24. A recurrence for the running time of a divide- and-conquer algorithm is based on the three steps of the basic paradigm
- 25. • Examine methods of analyzing algorithm correctness and efficiency – Recursion equations – Lower bound techniques – O,Omega and Theta notations for best/worst/average case analysis • Decide whether some problems have no solution in reasonable time – List all permutations of n objects (takes n! steps) – Travelling salesman problem • Investigate memory usage as a different measure of efficiency