Analysis of algorithms
- 1. CS8451- DESIGN AND ANALYSIS OF ALGORITHMS Prepared By Ms.P.Jothi Thilaga, AP/CSE Ramco Institute of Technology Rajapalayam
- 2. Why we need to study “ALGORITHMS”? As programs get complicated, thinking algorithmically allows us to: • reason about their correctness and efficiency before implementing them • focus on techniques for solving problems • understand relationship between different computational problems
- 3. Algorithms • The term algorithm originates from the name of the Persian mathematician “Al Khowarismi”. • He had presented generic methods for solving certain categories of problems for the first time in 825 A.D. • In 300 B.C., the Greek mathematician Euclid design an efficient algorithm for finding GCD.
- 4. • In Computer science, an algorithm is a set of unambiguous instructions to solve a problem. i.e., for obtaining a required output for any legitimate input in a finite amount of time. • Algorithms are used for calculation, data processing and automated reasoning. • An algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function.
- 5. Algorithms
- 6. NOTION OF ALGORITHM Properties: • Non-ambiguity • Multiplicity • Range of Inputs • Definiteness • Finiteness
- 7. Fundamentals of the Analysis of Algorithm Efficiency • Issues: Time efficiency Space efficiency correctness optimality • Approaches: theoretical analysis empirical analysis
- 8. Measuring an Input’s size • Algorithm efficiency is measured in terms of ‘n’ where n is the input size of an algorithm. • In some cases, more than one parameter is required to indicate the size of their inputs. • In this case the size is represented by the ‘n’ bits. • To measure size by the number b of bits in the n’s binary representation: b = floor(log2n)+1
- 9. Examples
- 10. Time Complexity • The execution time depends on, System load Number of other programs running Instruction set used Speed of underlying hardware
- 11. Orders of Growth
- 12. Units of measuring running time T(n)=cop C(n) • T(n) - running time of basic operation. • cop - time taken by the basic operation to execute. • C(n) – Number of times the operation needs to be executed.
- 13. Example: Sequential Search • Best Case: 1 comparisons • Worst Case: ‘n’ key comparisons • Average Case: (n+1)/2, assume k is in A
- 14. Amortized Efficiency • Amortized complexity is defined as the finding the average running time or operation over a worst case sequence of operations.
- 15. Empirical Analysis 1. Understand the experiment's purpose. 2. Decide on the efficiency metric M to be measured and the measurement unit 3. Decide on characteristics of the input sample 4. Prepare a program implementing the algorithm for the experimentation. 5. Generate a sample of inputs. 6. Run the algorithm on the sample's inputs and record the data observed. 7. Analyze the data obtained.
- 16. THEORETICAL ANALYSIS • Analyze before program is written • Independent of machine & programming environment • Independent of input instances • Need mathematical knowledge • For lengthy algorithms, analysis is difficult EMPIRICALANALYSIS • Analyze after the program is written • Dependent of programming Environment • Depends on input instances • Mathematical knowledge not required • Length of the program is immaterial
- 17. Thank You