L01 intro-daa - ppt1
- 1. The Design and Analysis of Algorithms
- 2. 2 CHAPTER 1: INTRODUCTIONCHAPTER 1: INTRODUCTION What is an Algorithm Steps in Designing and Implementing an Algorithm Important Problem Types Fundamental Data Structures
- 3. 3 ALGORITHM A sequence of unambiguous instructions for solving a problem, i.e. for obtaining the required output for any legitimate input in a finite amount of time
- 4. 4 Important points Non-ambiguityNon-ambiguity Range of inputsRange of inputs The same algorithm can beThe same algorithm can be represented in different waysrepresented in different ways Several algorithms for solving theSeveral algorithms for solving the same problemsame problem Finiteness
- 5. 5 GCD 3 methods 1.euclid alg 2.integer check alg 3.find repititive factors ex : 60 and 24
- 7. 7 Understand the problem Decide on computational means Exact vs approximate solution Data structures Algorithm design technique Design an algorithm Prove correctness Analyze the algorithm Code the algorithm
- 8. 8 Understanding the problem: understand the problem completely by Read the problem description carefully and clear the doubts. Specify exactly the range of inputs the algorithm need to handle. Once the problem is clearly understandable, then determine the overall goals but it should be in a precise manner. Then divide the problem into smaller problems until they become manageable size
- 9. 9 Ascertaining the capabilities of a computational device () Sequential Algorithm: Instructions are executed one after another, one operation at a time. This is implemented in RAM model. Parallel Algorithm: Instructions are executed in parallel or concurrently
- 10. 10 Choosing between exact and appropriate problem solving : The algorithm used to solve the problem exactly called exact algorithm. The algorithm used to solve the problem approximately is called approximation algorithm Optimal and feasible solution ( explain example)
- 11. 11 Deciding on appropriate data structures Data structure is important for both design and analysis of algorithms. Algorithm + Data Structures = Programs. In Object Oriented Programming, the data structure is important for both design and analysis of algorithms. The variability in algorithm is due to the data structure in which the data of the program are stored such as 1.How the data are arranged in relation to each other. 2.Which data are kept in memory 3.Which data are kept in files and how the files are arranged. Which data are calculated when needed?
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- 13. 13 Design an algorithm • Build a computational model of the solving process Prove correctness • Correct output for every legitimate input in finite time • Based on correct math formula • By induction
- 14. 14 Analyze the algorithm Efficiency: time and space Simplicity Generality: range of inputs, special cases Optimality: no other algorithm can do better Coding How the objects and operations in the algorithm are represented in the chosen programming language?
- 15. 15 Important problem types Sorting Searching String processing Graph problems Combinatorial problems Geometric problems Numerical problems
- 16. Sorting Arranging the elements in increasing order or in decreasing order. The sorting can be done on numbers , characters (alphabets), string or employees record. EX :ANNA UNIV RANK,SPORTS,BANK 1. Bubble sort, 2. Selection sort 3. Shell sort , 4. Insertion sort 5. Quick sort , 6. Heap sort 16
- 17. Searching One of the important applications of array is searching, Searching is an activity by which we can find out the desired element from the list. The element which is to be searched is called search key. There are many searching algorithm such as sequential search , Fibonacci search and more. 17
- 18. String processing A string is a collection of characters from an alphabet. Different type of strings are Text String It is a collection of letters, numbers and special characters. Bit String It is collection of zeros and ones. 18
- 19. Graph problems Graph is a collection of vertices and edges. a graph G={ V, E }. The graph problems involve graph traversal algorithms, shortest path algorithm and topological sorting and so on. Some graph problems are very hard to solve. For example travelling salesman problem, graph colouring problems 19
- 20. Combinatorial problems The travelling salesman problem and the graph colouring problems are examples of combinatorial problems. A combinatorial object such as a permutation a combination or a subset that satisfies certain constraints and has some desired property such as maximizes a value or minimizes a cost should be find. 20
- 21. Combinatorial problems 2 Combinatorial problems are the most difficult problems. 1. As problem size grows the combinatorial objects grow rapidly and reach to huge value. size. 2. There is no algorithms available which can solve these problems in finite amount of time 3. Many of these problems fall in the category of unsolvable problem. Some combinatorial problems can be solved by efficient algorithms 21
- 22. Geometric problems Geometric algorithms deal with geometric objects such as points ,lines and polygons. The procedure for solving a variety of geometric problems includes the problems of constructing simple geometric shapes such as triangles, circles and so on. The two classic problems of computational geometry are the 1. Closest pair problem 2. Convex hull problem The closest pair problem is self explanatory. Given n points in the plane, find the closest pair among them.The convex hull problem is used to find the smallest convex polygon that would include all the points of a given set. The geometric problems are solved mainly in applications to computer graphics or in robotics 22
- 23. Numerical problems Numerical problems are problems that involve mathematical objects of continuous nature such as solving equations and systems of equations computing definite integrals evaluating functions and so on. Most of the mathematical problems can be solved approximate algorithms. These algorithms require manipulating of the real numbers; hence we may get wrong output many times. 23
- 24. 24 Fundamental data structures Linear data structures Array Linked list Stack Queue Operations: search, delete, insert Implementation: static, dynamic
- 25. 25 Non-linear data structures Graphs Trees : connected graph without cycles Rooted trees Ordered trees Binary treesGraph representation: adjacency lists, adjacency matrix Tree representation: as graphs; binary nodes Fundamental data structures
- 26. 26 Fundamental data structures Sets, Bags, Dictionaries Set: unordered collection of distinct elements Operations: membership, union, intersection Representation: bit string; linear structure Bag: unordered collection, elements may be repeated Dictionary: a bag with operations search, add, delete
- 27. 27 Conclusion Algorithm classification By types of problems By design technique Design techniques a general approach to solving problems