Algorithm.pptx

Algorithm.pptx

• 1.
  Algorithm An algorithm isa well defined list of steps for solving a particular problem. (or) An Algorithm is a step by step procedure for performing some task in a finite amount of time. Every problem can be solved using various different methods. Each method can be represented by an algorithm. The choice of the best algorithm depends on the following factors: An algorithm should be easy to understand code and debug (design). An algorithm should make use of the computer resources efficiently.
• 2.
  Algorithm Complexity: Complexity ofan algorithm is a measure of time and/space required by an algorithm for an input of a given size(n). In order to find the complexity of algorithm Space complexity: It indicates the amount of storage required for running the algorithm i.e the amount of memory needed by the algorithm to run to completion. Time Complexity: Time complexity of an algorithm is the amount of time it needs in order to run to completion. Time –Space tradeoff: Time space trade off is a way of solving a problem in less time by using more storage space or by solving a problem in very little space by spending a long time.
• 3.
  • All Algorithmsmust satisfy the following criteria: • Input: Each algorithm should have zero or more inputs. • Output: The algorithm should produce correct results at least one output has to be produced. • Definiteness: Each instruction should be clear and unambiguous • Finiteness: The algorithm must terminate after a finite number of steps. • Effectiveness: Every instruction in the algorithm must be a basic statement so that it can be executed by a person using pen and paper.
• 4.
  Mathematical Notations AndFunctions Various mathematical notations and functions used in the analysis of algorithm. 1. Floor and ceiling functions: 2. Remainder Function 3. Integer and absolute Value functions 4. Factorial Function 5. Summation Symbol 6. Permutations 7. Exponents and logarithms
• 5.
  • Algorithmic Notations: 1.Name of the algorithm 2. Comments 3. Data type 4. Variable Names 5. Steps 6. Assignment Statement 7. Expression 8. Input and Output 9. Go to Statement 10. End statement 11. Functions or procedures.
• 6.
  Name of thealgorithm: • Every algorithm is given an Identifying name Comments • Every algorithm can have brief description of the tasks. But it gives the name and types of the variables used in the algorithm. Comments begin with // and continues until the end of line. Comments are included only for clarity. Data type • Data types such as integer,real,char,boolean and other data structures such as array, pointers,structure are used. • Examples: int max,min,sum • int a[ n] Variable Names • A Variable can have alphabets , digits and underscore • It can’t start with a digit • A variable name can start with the alphabet and underscore only. • No white Spaces is allowed
• 7.
  • A variablename must not be reserved word or keyword. • Ex: go to, int Steps • Each Algorithm is made of a sequence of numbered steps and each step has an ordered sequence of statements which steps describe tasks to be performed. • The assignment statement is indicated by placing equal(=) between the variable and expression. Expression • There are three Expressions • Arithmetic expression used arithmetic operator such as (+,-,/,*) • Relational expression used relational operator such as (>,<,<=,>=,==) • Logical expression used logical operator such as (AND,OR,NOT) Input and Output • For input of data READ statement and for output of data WRITE statement are used. If more than one input or output data then we can use comma as a separator among the variable names. Example : READ x READ x,y WRITE x,y
• 8.
  Go to Statement •The go to statement causes unconditional transfer of control to the step referenced. Thus statement goto step N will cause transfer of control to step N. • Example: GOTO Step4; End statement • The end statement is used to terminate an algorithm. It is usually the last step and algorithm name is written after the end. Example: END LINEARSEARCH Functions or procedures • A function is used to return single value to the calling function transfer of control and returning of the value are accomplished by return statement. • Example: function function_name(parameters list).
• 9.
  Control Structures • Sequence •Selection • Iteration • Branching
• 10.
  Analysis of Algorithm Analysisof an algorithm is refers to the task of determining how much computing time and storage an algorithm requires. To Compute the analysis of algorithm two phases are required. 1. Priori Analysis 2. Posterior Analysis Priori Analysis: This is one of the creative analysis of algorithms. It determine the total time that statement will spend for the execution. It requires essentially two items of information. 1. The statements frequency count 2. The time taken for one execution. 3. The product of these two numbers is the total time. A priori Analysis limits itself to determine the frequency count of each statement. The notations used in priori Analysis are Big Oh notation,Omega notation,Theta notation.
• 11.
  • Posterior Analysis: •Here we collect actual statistics about the algorithm ,conjunction of the time and space while executing. Once the algorithm is written it has to be tested. • Testing a program consist of two major phases. • A) Debugging • B) Profiling Debugging : It is the process of executing programs on sample data sets that determine whether we get proper results . If faulty occurs it has to be corrected. Profiling: It is the process of executing programs on actual data sets and measuring the time and space it takes to compute the results during execution.
• 12.
  Asymptotic Notations Asymptotic analysisrefers to computing the running time of any operation in mathematical units of computation. Using asymptotic analysis we can conclude the best case ,average case, and worst case scenario of an algorithm. The efficiency of the algorithm normally expressed using asymptotic notations. Following are the commonly used asymptotic notations. 1. Big Oh Notation(O) 2. Omega Notation(Ὠ) 3. Theta Notation(θ)
• 13.
  • Big OhNotation(O) • Big –Oh- Notation gives an upper bound on function f(n).The upper bound of f(n) indicates that the function f(n) will be the worst –case that it does not consume more than this computing time. • We define an algorithm’s worst-case time complexity by using the Big-O notation, which determines the set of functions grows slower than or at the same rate as the expression. Furthermore, it explains the maximum amount of time an algorithm requires considering all input values.
• 15.
  • Omega Notation(Ὠ) Itis used to find lower bound behaviour of f(n). The lower bound implies that the algorithm will take at least amount of time that to run its completion. • It defines the best case of an algorithm’s time complexity, the Omega notation defines whether the set of functions will grow faster or at the same rate as the expression. Furthermore, it explains the minimum amount of time an algorithm requires considering all input values.
• 17.
  • Theta Notation(θ) •It is used when the function f(n) is bounded between big Oh and Omega. • For some functions the lower bound and upper bound may be same. • There exist a special Notation to denote for function having the same time complexity for lower bound and upper bound and this notation is called theta notation. • It defines the average case of an algorithm’s time complexity. • f(n)= θ(g(n)) if and only if there exists three positive constants c1, c2 and a natural number n0 such that c1|g(n)| ≤ f(n) ≤ c2 | g(n)| for all n ≥ n0
• 18.
  Theta Notation
• 19.
  Worst case, Bestcase, and Average case efficiencies Best case: • The best case occurs when the element to be searched is found at the first location. In that case The algorithm make only one comparisons. • The number of comparisons = 1 • Tbest (n)=1 Worst Case: • The worst case occurs when the element to be searched is found at the last position or element to be searched is not found at any location. In both the cases the number of comparisons required is n. • Tworst(n)=n
• 20.
  Average case : •To find the average case efficiency, the following assumptions are made. a) The probability of successful search is equal to P (0< P< 1) b) The probability of first match occurring in the i th position of the array is the same for every i. TAvg(n)=(n+1)/2