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Analysis And Design Of Algorithms


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Recurrence Relation

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  Recurrence Relation A recurrence is an equation or inequality that describes a function in terms of its values on smaller inputs. To solve a Recurrence Relation means to obtain a function defined on the natural numbers that satisfy the recurrence. For Example , the Worst Case Running Time T(n) of the MERGE SORT Procedures is described by the recurrence. T (n) = θ (1) if n=1 2T + θ (n) if n>1 There are four methods for solving Recurrence: Substitution Method Iteration Method Recursion Tree Method Master Method 1. Substitution Method: The Substitution Method Consists of two main steps: Guess the Solution. Use the mathematical induction to find the boundary condition and shows that the guess is correct. For Example1  Solve the equation by Substitution Method. T (n) = T + n We have to show that it is asymptotically bound by O (log n). Solution: For T (n) = O (log n) We have to show that for some constant c T (n) ≤c logn.   Put this in given Recurr...
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Master Method

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  Master Method The Master Method is used for solving the following types of recurrence T (n) = a T + f (n) with a≥1 and b≥1 be constant & f(n) be a function and  can be interpreted as Let T (n) is defined on non-negative integers by the recurrence. T (n) = a T + f (n) In the function to the analysis of a recursive algorithm, the constants and function take on the following significance: n is the size of the problem. a is the number of subproblems in the recursion. n/b is the size of each subproblem. (Here it is assumed that all subproblems are essentially the same size.) f (n) is the sum of the work done outside the recursive calls, which includes the sum of dividing the problem and the sum of combining the solutions to the subproblems. It is not possible always bound the function according to the requirement, so we make three cases which will tell us what kind of bound we can apply on the function. Master Theorem: It is possible to complete an asymptotic tight bound in ...
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Algorithm Design Techniques

  Algorithm Design Techniques The following is a list of several popular design approaches: What is an algorithm?   An Algorithm is a procedure to solve a particular problem in a finite number of steps for a finite-sized input.  The algorithms can be classified in various ways. They are:    Implementation Method Design Method Design Approaches Other Classifications In this article, the different algorithms in each classification method are discussed.  The classification of algorithms is important for several reasons: Organization:  Algorithms can be very complex and by classifying them, it becomes easier to organize, understand, and compare different algorithms. Problem Solving:  Different problems require different algorithms, and by having a classification, it can help identify the best algorithm for a particular problem. Performance Comparison:  By classifying algorithms, it is possible to compare their performance in terms of time and spa...
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