Lecture 2 role of algorithms in computing

Lecture 2 : Role of Algorithms in Computing Jayavignesh T Asst Professor SENSE

What is an Algorithm? • An algorithm is a set of rules for carrying out calculation either by hand or on a machine. • An algorithm is a finite step-by-step procedure to achieve a required result. • An algorithm is a sequence of computational steps that transform the input into the output. • An algorithm is a sequence of operations performed on data that have to be organized in data structures. • An algorithm is an abstraction of a program to be executed on a physical machine (model of Computation).

What is an Algorithm contd..? • An algorithm is a sequence of unambiguous instructions for solving a problem, – i.e., for obtaining a required output for any legitimate input in a finite amount of time

Requirements • Recipe, process, method, technique, procedure, routine,… with following requirements: 1. Finiteness – terminates after a finite number of steps 2. Definiteness – rigorously and unambiguously specified 3. Input – valid inputs are clearly specified 4. Output – can be proved to produce the correct output given a valid input 5. Effectiveness – steps are sufficiently simple and basic

Design and Analysis • Algorithmic is a branch of computer science that consists of designing and analyzing computer algorithms • The “design” pertain to – The description of algorithm at an abstract level by means of a pseudo language, and – Proof of correctness that is, the algorithm solves the given problem in all cases. • The “analysis” deals with performance evaluation(complexity analysis).

Design and Analysis • It is not depended on programming language, machine. • Are mathematical entities, which can be thought of as running on some sort of idealized computer with an infinite random access memory • Algorithm design is all about the mathematical theory behind the design of good programs.

Why Analyze? • Why analyze algorithms? – Evaluate Algorithm performance – Compare Different Algorithms • Analyze what about them? – Running time, Memory usage, Solution quality – Worst-case and “typical” case • Computational complexity – Understanding the intrinsic difficulty of computational problems – Classifying problems according to difficulty – Algorithms provide upper bound to show problem is hard, must show that any algorithm to solve it requires at least a given amount of resources

Algorithm Problem solving Understand the problem Decision Making Design an Algorithm Prove Correctness Analyze Algorithm Code the Algorithm

Understand the problem • Necessary Inputs to solve the problem? • Input – Instances • Decide range of inputs • To Fix Boundary values • Work correctly for all Valid inputs

Decision Making • Analyze the required inputs and decide on – Computational Means – Exact vs Approximate Solving – Data Structures – Algorithm Design Technique  Computational Means  Computational capability of running devices  Sequential Algorithm ( Random Access Machine RAM)  Parallel Algorithm (Piece executed on multiple processing devices and put back together to get correct result)  Proper choice of Computational device (Space/Time efficient)

Decision Making contd.. • Exact vs Approximate Solving • Exact Algorithm – Problem needs to be solved exactly or correctly • Approximation Algorithm – Solve a complex problem, we will not get exact solution – Ex : Travelling salesman problem • Data Structures – Algorithm + Data Structures – Programming Language – Proper Choice of Data Structures required before designing actual algorithm

Algorithm Design Technique • Strategy or Paradigm – General approach to solve program algorithmically – Different problems from Different areas of computing • Brute Force : – Straight forward technique with naïve approach • Divide and Conquer : – Problem is divided into smaller instances

Algorithm Design Technique contd.. • Decrease and Conquer : – Instance size is decreased to solve the problem • Transform and Conquer : – Instance is modified and then solved • Dynamic Programming : – Results of smaller, reoccurring instances are obtained to solve problem • Greedy Technique : – Solve the problem by making locally optimal decisions

Specification of Algorithm • Natural Language – Drawback : Specification by this means not clear • Pseudo Code – Mixture of Natural Language + Programming constructs ALGORITHM Sum(a,b) // Problem Description : This algorithm performs addition of two numbers // Input : Two integers a and b // Output : Addition of two integers c a + b return c • Flow Chart – Using Geometric Shapes containing descriptions of algorithm’s steps.

Proving Algorithm’s Correctness • Prove algorithm yields required results • For Every Legitimate Input in finite amount of time • If one instance of valid input, the algorithm performs incorrectly !! – Ex : Mathematical Induction

Analysis of Algorithm • Time Efficiency – Indicates how fast algorithm runs • Space Efficiency – How much Extra memory the algorithm needs to complete its execution • Simplicity – Generating Sequence of instructions which are easy to understand • Generality – Range of inputs it can accept – Generality of problem which algorithm can solve

Coding an Algorithm • Implementation of algorithm done by suitable programming language – C – C++ – Java, – Verilog etc etc…

Problem Types to solve • Sorting Algorithms –Arranging elements in increasing (ascending) or decreasing (descending) order –Numbers, Characters, Strings –Sorting makes easier to search elements –Ex : Dictionary, Telephone books, Class lists, Employee database etc..

Problem Types to solve • Sorting Algorithms –Properties • Stable Property –Preserves the relative order of any two equal elements. Position i , j where i < j , sorted list i’ and j’ such that i’ < j’ • In Place Property –Does not require extra memory, except for possibly a few memory unit

Searching • To find the desired element from the list • Element to be Searched – Search Key • Sequential Search • Binary Search etc. • Some Search algorithms work faster than others but – Requires more memory – Works only in sorted arrays etc..

Linear Search Algorithm • Input: An array of n numbers and an element which is required to be searched in the given list • Output: Number exists in the list or not. Algorithm: 1. Input the size of list i.e. n 2. Read the n elements of array A 3. Input the item/element to be searched in the given list. 4. for each element in the array i=1 to n 5. if A[i]==item 6. Search successful, return 7. if i==n+1 8. Search unsuccessful. 9. Stop

Graph Problems • Collection of points called vertices • Some are connected by line segments – edges • Graph traversal algorithms, shortest path algorithms etc… • String Processing – String – sequence of characters – Text Strings = Letters, numbers, special characters, bit strings (0 & 1)

Combinatorial Problems • Computing Permutations and combinations • Complex computing due to – As problem size grows the combinatorial objects grow rapidly and reach to huge value – No algorithm available which solves these problems in finite amount of time. – Many fall under unsolvable problems

Geometric Problems • Deal with geometric objects such as points, lines and polygons • Applications in Computer Graphics, Robotics and topography

Time Complexity • Amount of computer time required by an algorithm to run on completion • Difficult to compute time complexity in terms of physically clocked time. • Drawbacks of measuring running time in-terms of seconds, millisecond etc are – Dependence of speed of a underlying hardware – Number of other programs running (System load) – Dependence of compiler used in generating machine code

How to calculate running time then? • Time complexity given in terms of FREQUENCY COUNT • Count denoting number of times of execution of statement. For (i=0; i <n;i++) { // St1 : 1, St 2 : n+1 , St 3 : n times sum = sum + a[i]; // n times } 3n + 2 ; O(n) neglecting constants and lower order terms

How to calculate running time then? for (i=0; i < n ; i ++) // 1 ; n+1 ; n times { for (j=0; j < n ; j ++) // n ; n(n+1) ; n(n) { c[i][j] = a[i][j] + b[i][j]; } } 3n2+4n+ 2 = O(n2)

How to calculate running time then? • All Algorithms run longer on larger inputs • Algorithm’s efficiency - f(n) • Identify the most important operations of the algorithm – BASIC Operation • Basic operation – contributing to most of total running time • Compute the number of times basic operation is executed (mostly in inner loop) Ex : Sorting Algorithms – Comparison (< >) Matrix Multiplication, Polynomial evaluation – Arithmetic Operations ( *, +) = (assignment), ==(equality) etc..

Order of Growth of Algorithm • Measuring the performance of an algorithm in relation with input size n

Rate of Growth of Algorithm as fn of i/p size

Lecture 2 role of algorithms in computing

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Lecture 2 role of algorithms in computing