2. Introduction to Algorithm.pptx
- 2. Algorithms Basics • Algorithm is a step by step procedure, which defines a set of instructions to be executed in certain order to get the desired output. • Algorithms are generally created independent of underlying languages, • i.e. an algorithm can be implemented in more than one programming language.
- 3. • 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
- 4. Characteristics of an Algorithm Not all procedures can be called an algorithm. An algorithm should have the below mentioned characteristics • Unambiguous − Algorithm should be clear and unambiguous. Each of its steps (or phases), and their input/outputs should be clear and must lead to only one meaning. • Input − An algorithm should have 0ne or more well defined inputs. • Output − An algorithm should have 1 or more well defined outputs, and should match the desired output. • Finiteness − Algorithms must terminate after a finite number of steps. • Feasibility − Should be feasible with the available resources. • Independent − An algorithm should have step-by-step directions which should be independent of any programming code.
- 5. How to write an algorithm? • There are no well-defined standards for writing algorithms. • Rather, it is problem and resource dependent. • Algorithms are never written to support a particular programming code. • As we know that all programming languages share basic code constructs like loops (do, for, while), flow-control (if-else) etc. These common constructs can be used to write an algorithm. • We write algorithms in step by step manner, but it is not always the case. • Algorithm writing is a process and is executed after the problem domain is well-defined. • That is, we should know the problem domain, for which we are designing a solution.
- 6. Example • Let's try to learn algorithm-writing by using an example. • Problem − Design an algorithm to add two numbers and display result. • step 1 − START • step 2 − declare three integers a, b & c • step 3 − define values of a & b • step 4 − add values of a & b • step 5 − store output of step 4 to c • step 6 − print c • step 7 − STOP
- 7. • Algorithms tell the programmers how to code the program. Alternatively the algorithm can be written as − • step 1 − START ADD • step 2 − get values of a & b • step 3 − c ← a + b • step 4 − display c • step 5 − STOP
- 8. • In design and analysis of algorithms, usually the second method is used to describe an algorithm. • It makes it easy of the analyst to analyze the algorithm ignoring all unwanted definitions. • He can observe what operations are being used and how the process is flowing. • Writing step numbers, is optional. • We design an algorithm to get solution of a given problem. A problem can be solved in more than one ways.
- 9. • Hence, many solution algorithms can be derived for a given problem. • Next step is to analyze those proposed solution algorithms and implement the best suitable.
- 10. 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 • 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.
- 11. 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 is equal to the sum of the following two components − • A fixed part that is a space required to store certain data and variables, that are independent of the size of the problem. For example 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. For example dynamic memory allocation, recursion stack space etc.
- 12. Asymptotic Analysis • Asymptotic analysis of an algorithm, refers to defining the mathematical boundation/framing of its run-time performance. Using asymptotic analysis, we can very well conclude the best case, average case and worst case scenario of an algorithm. • Asymptotic analysis are input bound i.e., if there's no input to the algorithm it is concluded to work in a constant time. Other than the "input" all other factors are considered constant.
- 13. • Asymptotic analysis refers to computing the running time of any operation in mathematical units of computation. For example, running time of one operation is computed as f(n) and may be for another operation it is computed as g(n2). Which means first operation running time will increase linearly with the increase in n and running time of second operation will increase exponentially when n increases. Similarly the running time of both operations will be nearly same if n is significantly small.
- 14. • Usually, time required by an algorithm falls under three types − • Best Case − Minimum time required for program execution. • Average Case − Average time required for program execution. • Worst Case − Maximum time required for program execution.
- 15. Asymptotic Notations • Following are commonly used asymptotic notations used in calculating running time complexity of an algorithm. • Ο Notation • Ω Notation • θ Notation
- 16. Big Oh Notation, Ο • The Ο(n) is the formal way to express the upper bound of an algorithm's running time. • It measures the worst case time complexity or longest amount of time an algorithm can possibly take to complete.
- 17. • For example, for a function f(n) • Ο(f(n)) < { g(n) : there exists c > 0 and n0 such that f(n) ≤ c.g(n) for all n > n0. }
- 18. Omega Notation, Ω • The Ω(n) is the formal way to express the lower bound of an algorithm's running time. • It measures the best case time complexity or best amount of time an algorithm can possibly take to complete.
- 19. • For example, for a function f(n) • Ω(f(n)) ≥ { g(n) : there exists c > 0 and n0 such that g(n) ≤ c.f(n) for all n > n0. }
- 20. Theta Notation, θ • The θ(n) is the formal way to express both the lower bound and upper bound of an algorithm's running time. It is represented as following − • θ(f(n)) = { g(n) if and only if g(n) = Ο(f(n)) and g(n) = Ω(f(n)) for all n > n0. }
- 21. Common Asymptotic Notations constant − Ο(1) logarithmic − Ο(log n) linear − Ο(n) n log n − Ο(n log n) quadratic − Ο(n 2 ) cubic − Ο(n 3 ) polynomial − n Ο(1) exponential − 2 Ο(n)