Algorithms.pdf
- 1. CSC 102 : Introduction to Algorithm Techniques Department of Computer Science College of Physical Sciences Federal University of Agriculture, Abeokuta. WEEK 2 2019/2020 session DR. O.E OJO
- 2. Steps involved in algorithm development ■ Step1. Identification of input: For an algorithm, there are quantities to be supplied called input and these are fed externally. The input is to be identified first for any specified problem. ■ Step2: Identification of output: From an algorithm, at least one quantity is produced, called for any specified problem. ■ Step3 : Identification the processing operations : All the calculations to be performed in order to lead to output from the input are to be identified in an orderly manner.
- 3. Steps involved in algorithm development ■ Step4 : Processing Definiteness : The instructions composing the algorithm must be clear and there should not be any ambiguity in them. ■ Step5 : Processing Finiteness : If we go through the algorithm, then for all cases, the algorithm should terminate after a finite number of steps. ■ Step6 : Possessing Effectiveness : The instructions in the algorithm must be sufficiently basic and in practice they can be carries out easily.
- 4. Method for developing an algorithm ■ 1. Define the problem: State the problem that you are trying to solve in clear and concise terms. ■ 2. List the inputs (information needed to solve the problem) and the outputs (what the algorithm will produce as a result) ■ 3. Describe the steps needed to convert or manipulate the inputs or produce the outputs. Start at a high level first and keep refining the steps until they are effectively computable operations. ■ 4. Test the algorithm: Choose data sets and verify that your algorithm works.
- 5. Example of an Algorithm Write an algorithm to find the area of a Circle of radius r. Inputs to the algorithm: Radius r of the Circle. Expected output: Area of the Circle Algorithm: Step1: Start Step 2: Input the Radius r and PI of the Circle Step3: PI 3.142 Step4: Area PI * r * r Step5: Output Area Step6: Stop
- 6. Example of an Algorithm Write an algorithm to find the Sum of two numbers Initialization Inputs to the algorithm: First number = A. Second number = B Expected output: Sum of A and B Algorithm: Step 1: Start Step 2: Sum = 0 Step 3: Input the value of A Step 4: Input the value of B Step 5: Sum = A + B Step 6: Output Sum Step 7: End
- 7. Example ■ Design an algorithm to add these test scores: 26, 49, 98, 87, 62, 75 and obtain the Average score 1.Start 2. Sum 0 3. Input 26, 49, 98, 87, 62, 75 4. Sum 26+49+98+87+62+75 5. Average Sum/6 6. Output Average 7. Stop
- 8. Properties of Algorithm ■ Correctness Correctness is the property reflecting the extent to which the algorithm is able to reach a solution without errors. ■ Efficiency The efficiency of an algorithm is the property that regards the rapidity by which a solution is reached. ■ Generality This means that it must solve every instance of the problem. For example, a program that computes the area of a rectangle should work on all possible dimensions of the rectangle.
- 9. ■ Effectiveness This means that an algorithm must provide the correct answer to the problem. ■ Comprehensible Finally the algorithm must be comprehensible by the one who performs it. It must describe in all details, the actions and the entities necessary for the resolution of a problem.
- 10. Categories of Algorithm ■ From data structure point of view, following are some important categories of algorithms − ■ Search − Algorithm to search an item in a data structure. ■ Sort − Algorithm to sort items in certain order ■ Insert − Algorithm to insert item in a data structure ■ Update − Algorithm to update an existing item in a data structure ■ Delete − Algorithm to delete an existing item from a data structure
- 11. One problem(multiple solutions) ■ We design an algorithm to get solution of a given problem. A problem can be solved in more than one ways.
- 12. Algorithm Analysis ■ In computer science, the analysis of algorithms is the determination of the amount of computing resources (such as time and space) necessary to execute them. ■ We want to be able to consider two algorithms and say that one is better than the other because it is more efficient in its use of those resources or perhaps because it simply uses fewer.
- 13. Algorithm Analysis ■ Algorithm analysis deals with the execution or running time of various operations involved. ■ Running time of an operation can be defined as no. of computer instructions executed per operation.
- 14. Algorithm Analysis ■ Efficiency of an algorithm can be analyzed at two different stages, ■ before implementation(priori analysis) ■ and after implementation(posterior analysis).
- 15. Algorithm Analysis ■ It is important to consider the amount of space or memory an algorithm requires to solve the problem. ■ The amount of space required by a problem solution is typically dictated by the problem instance itself.
- 16. Priori Analysis ■ This is theoretical analysis of an algorithm. ■ Efficiency of algorithm is measured by assuming that all other factors (e.g. Processor speed), are constant and have no effect on implementation.
- 17. Posterior Analysis ■ This is empirical analysis of an algorithm. ■ The selected algorithm is implemented using programming language. ■ This is then executed on target computer machine. ■ In this analysis, actual statistics like running time and space required, are collected. ■ Our focus in this course is priori algorithm analysis
- 18. ALGORITHM COMPLEXITY ■ Suppose X is an algorithm and ■ n is the size of input data, ■ the time and space used by the Algorithm X are the two main factors which decide the efficiency of X. ■ Time Factor − The time is measured by counting the number of key operations such as comparisons in sorting algorithm
- 19. ALGORITHM COMPLEXITY ■ Space Factor − The space is measured by counting the maximum memory space required by the algorithm. ■ The complexity of an algorithm f(n) gives the running time and / or storage space required by the algorithm in terms of n as the size of input data.
- 20. SPACE COMPLEXITY ■ Space complexity of an algorithm represents the amount of memory space required by the algorithm in its life cycle. ■ Space required by an algorithm = Fixed Part + Variable Part.
- 21. SPACE COMPLEXITY ■ A fixed part is a space required to store certain data and variables, that are independent of the size of the problem. Eg: Simple variables & constant used, program size etc. ■ A variable part is a space required by variables, whose size depends on the size of the problem. Eg. Dynamic memory allocation, recursion stack space etc.
- 22. SPACE COMPLEXITY ■ Space complexity S(P) of any algorithm P is, S(P) = C + SP(I) ■ Where C is the fixed part and ■ S(I) is the variable part of the algorithm which depends on instance characteristic I.
- 23. SPACE COMPLEXITY ■ Example Algorithm: SUM (A, B) Step 1 – START Step 2 - C ← A + B + 10 Step 3 – Stop. ■ Here we have three variables A, B and C and one constant. ■ Hence S(P) = 1+3. ■ Therefore, space depends on data types of given variables and constant types.
- 24. CLASS WORK 2 ■ Question: Find the space complexity in the algorithm given below: Algorithm: Addition of numbers Step 1 – START Step 2 – a = 40 Step 3- M← a + b +c+ d Step 4 – Stop.