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Design Analysis of Algorithm_Introduction-1.pptx
- 1. AD3351-DESIGN AND ANALYSIS OFALGORITHM UNIT-1 INTRODUCTION
- 2. NOTION OF ALGORITHM ALGORITHM Analgorithm 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.
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- 4. Some Important Pointson Algorithm
- 5. CHARACTERISTICS OF AN ALGORITHM Input - Valid inputs must be clearly specified Output – Produces at least one output- the result or solution of the problem Definiteness - Each step must be clear and unambiguously specified Finiteness - Terminates after a finite number of steps Effectiveness - Steps must be simple enough to be performed using basic operations.
- 6. GCD(m,n) - Definition gcd(m,n) – the largest integer that divides both m and n evenly, ie., with a remainder of zero. General method – gcd(m,n) Eg: Compute GCD(60,24) • Find common factors of 2 given numbers: • Product of those common factors – gcd => gcd(24,18)=?
- 7. To illustrate thenotion of the algorithm we consider 3 methods to solve the same problem – Computing the greatest common divisor of 2 non-negative, not-both- zero integers m,n => gcd(m,n) Euclid’s Algorithm Consecutive Integer Checking Algorithm Middle-School Algorithm
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- 10. Compute gcd(60,24) usingEuclid’s algorithm Given: m=60 and n=24 Iteration 1: n=24≠0 -> r=60 mod 24 -> r=12 Now, m=n=24 and n=r=12 Iteration 2: n=12≠0 -> r=24 mod 12 -> r=0 Now, m=n=12 and n=r=0 Iteration 3: n=0, condition fails and returns m value Therefore, gcd(60,24)=12.
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- 12. GCD(24,12) Given: m=24 andn=12 Step 1: prime factors of m => 2,2,2,3 Step 2: prime factors of n => 2,2,3 Step 3: common factors => 2,2,3 Step 4: product of common factors => 2x2x3=12
- 13. Consecutive Integer CheckingAlgorithm
- 14. Contd.. Eg: (i) Computegcd(9,6) using Consecutive Integer Checking Algorithm (i) Compute gcd(24,12) using Consecutive Integer Checking Algorithm (iii) Compute gcd(60,24) using Consecutive Integer Checking Algorithm Drawback: Doesn’t work properly when one of its inputs is ZERO
- 15. Sieve of Eratosthenes Simple algorithm for generating consecutive primes not exceeding any given integer n>1 Starts by initializing a list of primes from 2 to n First Iteration – eliminate all multiples of 2 except 2 Next Iteration - eliminate all multiples of 3 except 3 Continuous until no more elements to be eliminated Remaining integers – primes
- 16. Algorithm : Sieveof Eratosthenes Step1: Create a list of consecutive integers from 2 through n – (1,2,3, …..,n). Step 2: Initially let p=2 the smallest prime no. Step 3: Enumerate the multiples of p by counting in increments of p from 2p to n and mark them in the list (these will be 2p,3p,4p,…, the p itself should not be marked). Step 4: Find the smallest no in the list greater than p that is not marked. If there is no such no stop. Otherwise let p now equal to this new no and repeat from Step 3. Step 5: When the algorithm terminates the numbers remaining not marked in the list are all the primes below n.
- 19. FUNDAMENTALS OF ALGORITHMICPROBLEM SOLVING
- 20. Important Problem Types The different problem types are 1. Sorting 2. Searching 3. String processing 4. Graph problems 5. Combinatorial problems 6. Geometric problems 7. Numerical problems
- 21. Sorting Sorting problemis one which rearranges the items of a given list in ascending order. For example in student’s information, we can choose to sort student records in alphabetical order of names or by student register number. Such a specially chosen piece of information is called a key. Sorting solves many problems, the most important of them is searching: dictionaries, telephone books, class lists, and so on are sorted.
- 22. Contd.. There aretwo important properties of sorting algorithms. A sorting algorithm is called stable, if it preserves the relative order of any two equal elements in its input. For example, we have a list of students sorted alphabetically and if we sort the student list based on their GPA: a stable algorithm will yield a list in which students with same GPA will be sorted alphabetically. The second feature of an algorithm is ‘in place’, if it does not require extra memory. There are some sorting algorithms that are in place and those that are not.
- 23. Searching The searchingproblem deals with finding a given value, called a search key, in a given set. The searching can be either a straightforward sequential search algorithm or binary search algorithm based on different forms of representing the set. Some algorithms work faster but require more memory, some are very fast but applicable only to sorted arrays. Searching, mainly deals with addition and deletion of records.
- 24. String processing AString is a sequence of characters. It is mainly used in string handling algorithms. Most common ones are text strings, which consists of letters, numbers and special characters. Bit strings consist of zeroes and ones and gene sequences, which can be modeled by strings of four-character alphabet{A,C,G,T}
- 25. Graph Problems Agraph is a collection of points called vertices,some of which are connected by line segments called edges. Graphs can be used for modeling a wide variety of real-life applications such as transportation, communication networks, project scheduling, and games. It includes graph traversal, shortest-path and topological sorting algorithms. The most well-known examples of computationally hard problems are the traveling salesman problem and graph-coloring problem
- 26. Contd… The TSPis the problem of finding the shortest tour through n cities that visits every city exactly once. It arises in modern applications as circuit board and VLSI chip fabrication, X- ray crystallography and genetic engineering. The graph-coloring problem is to assign the smallest number of colors to vertices of a graph so that no two adjacent vertices are of the same color.
- 27. Combinatorial Problems Thetraveling salesman problem and the graph-coloring problem are examples of combinatorial problems. These are problems that ask us to find a combinatorial object such as permutation, combination or a subset that satisfies certain constraints and has some desired (e.g. maximizes a value or minimizes a cost).
- 28. Geometric Problems Geometricalgorithms deal with geometric objects such as points, lines and polygons. It also includes various geometric shapes such as triangles, circles etc. The applications for these algorithms are in computer graphic, robotics etc. The two problems most widely used are the closest-pair problems, given ‘n’ points in the plane, find the closest pair among them. The convex-hull problem is to find the smallest convex polygon that would include all the points of a given set.
- 29. Numerical Problems Thisis another large special area of applications, where the problems involve mathematical objects of continuous nature: solving equations, computing definite integrals and evaluating functions and so on. These problems can be solved only approximately. These require real numbers, which can be represented in a computer only approximately. It can also lead to an accumulation of round-off errors.
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