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Analysis And Design Of Algorithms


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Asymptotic Notations

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  Asymptotic Analysis of algorithms ( asymptotic notations ) Resources for an algorithm are usually expressed as a function regarding input. Often this function is messy and complicated to work. To study Function growth efficiently, we reduce the function down to the important part. Let f (n) = an 2 +bn+c In this function, the n 2  term dominates the function that is when n gets sufficiently large. Dominate terms are what we are interested in reducing a function, in this; we ignore all constants and coefficient and look at the highest order term concerning n. Asymptotic notation: The word  Asymptotic  means approaching a value or curve arbitrarily closely (i.e., as some sort of limit is taken). Asymptotic analysis It is a technique of representing limiting behavior. The methodology has the applications across science. It can be used to analyze the performance of an algorithm for some large data set. 1. In computer science in the analysis of algorithms, considering ...
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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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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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