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Advanced Algorithms Analysis By Dr. Attiqa Rehman Dr. Attiqa Rehman Advanced Algorithms Analysis 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 Dr. Attiqa Rehman Advanced Algorithms Analysis 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. Dr. Attiqa Rehman Advanced Algorithms Analysis Lecture No 1 Introduction (What, Why and Where Algorithms . . .) Dr. Attiqa Rehman Advanced Algorithms Analysis 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 Dr. Attiqa Rehman Advanced Algorithms Analysis What is Algorithm? • 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. Input Dr. Attiqa Rehman Algorithm output Advanced Algorithms Analysis 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. Dr. Attiqa Rehman Advanced Algorithms Analysis 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. Dr. Attiqa Rehman Advanced Algorithms Analysis 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 Dr. Attiqa Rehman Advanced Algorithms Analysis 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 Dr. Attiqa Rehman Advanced Algorithms Analysis 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 Dr. Attiqa Rehman Advanced Algorithms Analysis Problem Solving Process • Problem • Strategy • Algorithm – Input – Output – Steps • Analysis – Correctness – Time & Space – Optimality • Implementation • Verification Dr. Attiqa Rehman Advanced Algorithms Analysis 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. Dr. Attiqa Rehman Advanced Algorithms Analysis 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. Dr. Attiqa Rehman Advanced Algorithms Analysis Model of Computation (Assumptions) • In theoretical analysis, computational complexity – Estimated in asymptotic sense, 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. Dr. Attiqa Rehman Advanced Algorithms Analysis Drawbacks in Model of Computation First poor 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 Dr. Attiqa Rehman Advanced Algorithms Analysis 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 Dr. Attiqa Rehman Advanced Algorithms Analysis 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 Dr. Attiqa Rehman Advanced Algorithms Analysis 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 Dr. Attiqa Rehman Advanced Algorithms Analysis