Major objective of this course is: Design and analysis of modern algorithms Different variants

1. Objective of This Course Major objective of this course is: • Design and analysis of modern algorithms • Different variants • Accuracy • Efficiency • Comparing efficiencies • Motivation thinking new algorithms • Advanced designing techniques • Real world problems will be taken as examples • To create feelings about usefulness of this course

2. Expected Results On successful completion, students will be able to • Argue and prove correctness of algorithms • Derive and solve mathematical models of problems • Reasoning when an algorithm calls certain approach • Analyze average and worst-case running times • Integrating approaches in dynamic and greedy algos. • Use of graph theory in problems solving • Advanced topics such as • Computational geometry, number theory etc. • Several other algorithms such as • String matching, NP completeness, approximate algorithms etc.

3. Lecture No 1Introduction(What, Why and Where Algorithms . . .)

4. Today Covered In this lecture we will cover the following • What is Algorithm? • Designing Techniques • Model of Computation • Algorithms as a technology • Algorithms and other technologies • Importance of algorithms • Difference in Users and Developers • Kinds of problems solved by algorithms • Conclusion

5. What is Algorithm? Algorithm Input output • A computer algorithm is a detailed step-by-step method for solving a problem by using a computer. • An algorithm is a sequence of unambiguous instructions for solving a problem in a finite amount of time. • An Algorithm is well defined computational procedure that takes some value, or set of values, as input and produces some value, or set of values as output. • More generally, an Algorithm is any well defined computational procedure that takes collection of elements as input and produces a collection of elements as output.

6. Popular Algorithms, Factors of Dependence • Most basic and popular algorithms are • Sorting algorithms • Searching algorithms Which algorithm is best? • Mainly, it depends upon various factors, for example in case of sorting • The number of items to be sorted • The extent to which the items are already sorted • Possible restrictions on the item values • The kind of storage device to be used etc.

7. One Problem, Many Algorithms Problem • The statement of the problem specifies, in general terms, the desired input/output relationship. Algorithm • The algorithm describes a specific computational procedure for achieving input/output relationship. Example • One might need to sort a sequence of numbers into non-decreasing order. Algorithms • Various algorithms e.g. merge sort, quick sort, heap sorts etc.

8. Important Designing Techniques • Brute Force • Straightforward, naive approach • Mostly expensive • Divide-and-Conquer • Divide into smaller sub-problems • Iterative Improvement • Improve one change at a time • Decrease-and-Conquer • Decrease instance size • Transform-and-Conquer • Modify problem first and then solve it • Space and Time Tradeoffs • Use more space now to save time later

9. Some of the Important Designing Techniques • Greedy Approach • Locally optimal decisions, can not change once made. • Efficient • Easy to implement • The solution is expected to be optimal • Every problem may not have greedy solution • Dynamic programming • Decompose into sub-problems like divide and conquer • Sub-problems are dependant • Record results of smaller sub-problems • Re-use it for further occurrence • Mostly reduces complexity exponential to polynomial

10. Problem Solving Phases • Analysis • How does system work? • Breaking a system down to known components • How components (processes) relate to each other • Breaking a process down to known functions • Synthesis • Building tools • Building functions with supporting tools • Composing functions to form a process • How components should be put together? • Final solution

11. Problem Solving Process • Problem • Strategy • Algorithm • Input • Output • Steps • Analysis • Correctness • Time & Space • Optimality • Implementation • Verification

12. Model of Computation (Assumptions) • Design assumption • Level of abstraction which meets our requirements • Neither more nor less e.g. [0, 1] infinite continuous interval • Analysis independent of the variations in • Machine • Operating system • Programming languages • Compiler etc. • Low-level details will not be considered • Our model will be an abstraction of a standard generic single-processor machine, called a random access machine or RAM.

13. Model of Computation (Assumptions) • A RAM is assumed to be an idealized machine • Infinitely large random-access memory • Instructions execute sequentially • Every instruction is in fact a basic operation on two values in the machines memory which takes unit time. • These might be characters or integers. • Example of basic operations include • Assigning a value to a variable • Arithmetic operation (+, - , × , /) on integers • Performing any comparison e.g. a < b • Boolean operations • Accessing an element of an array.

14. Model of Computation (Assumptions) • In theoretical analysis, computational complexity • Estimated in asymptoticsense, i.e. • Estimating for large inputs • Big O, Omega, Theta etc. notations are used to compute the complexity • Asymptotic notations are used because different implementations of algorithm may differ in efficiency • Efficiencies of two given algorithm are related • By a constant multiplicative factor • Called hidden constant.

15. Drawbacks in Model of Computation Firstpoor assumption • We assumed that each basic operation takes constant time, i.e. model allows • Adding • Multiplying • Comparing etc. two numbers of any length in constant time • Addition of two numbers takes a unit time! • Not good because numbers may be arbitrarily • Addition and multiplication both take unit time! • Again very bad assumption

16. Model of Computation not so Bad Finally what about Our Model? • But with all these weaknesses, our model is not so bad because we have to give the • Comparison not the absolute analysis of any algorithm. • We have to deal with large inputs not with the small size • Model seems to work well describing computational power of modern nonparallel machines Can we do Exact Measure of Efficiency ? • Exact, not asymptotic, measure of efficiency can be sometimes computed but it usually requires certain assumptions concerning implementation

17. Summary : Computational Model • Analysis will be performed with respect to this computational model for comparison of algorithms • We will give asymptotic analysis not detailed comparison i.e. for large inputs • We will use generic uniprocessor random-access machine (RAM) in analysis • All memory equally expensive to access • No concurrent operations • All reasonable instructions take unit time, except, of course, function calls

18. Conclusion • What, Why and Where Algorithms? • Designing Techniques • Problem solving Phases and Procedure • Model of computations • Major assumptions at design and analysis level • Merits and demerits, justification of assumptions taken • We proved that algorithm is a technology • Compared algorithmic technology with others • Discussed importance of algorithms • In almost all areas of computer science and engineering • Algorithms make difference in users and developers