Asymptotic Notation
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This document discusses asymptotic notations which are mathematical representations used to describe the time complexity of algorithms. It introduces the Big O, Big Omega, and Big Theta notations. Big O notation provides an upper bound on running time. Big Omega provides a lower bound. Big Theta gives a tight bound between the upper and lower bounds such that the running time is equal to both. Examples are provided to demonstrate each notation.
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This document discusses various mathematical notations and asymptotic analysis used for analyzing algorithms. It covers floor and ceiling functions, remainder function, summation symbol, factorial function, permutations, exponents, logarithms, Big-O, Big-Omega and Theta notations. It provides examples of calculating time complexity of insertion sort and bubble sort using asymptotic notations. It also discusses space complexity analysis and how to calculate the space required by an algorithm.
Daa unit 6_efficiency of algorithms
Order notation is a mathematical method used to analyze algorithms as the problem size increases. It allows comparison of performance independent of machine-specific factors. Common notations include Big-O (upper bound), Big-Omega (lower bound), and Theta (tight bound). These describe the limiting behavior of execution time as the problem size approaches infinity and are used to classify algorithms by their running time growth rates like constant, logarithmic, linear, quadratic, and exponential.
Theta notation
The Theta (Θ) notation is a method of expressing the asymptotic tight bound on the growth rate of an
algorithm’s running time both from above and below ends i.e. upper bound and lower bound.
Basic Computer Engineering Unit II as per RGPV Syllabus
The document provides an overview of algorithms and computational complexity. It defines an algorithm as a set of unambiguous steps to solve a problem, and discusses how algorithms can be expressed using different languages. It then covers algorithmic complexity and how to analyze the time complexity of algorithms using asymptotic notation like Big-O notation. Specific time complexities like constant, linear, logarithmic, and quadratic time are defined. The document also discusses flowcharts as a way to represent algorithms graphically and introduces some basic programming concepts.
Asymptotic Notation and Complexity
The ppt will provide complete detail of different types of complexity and asymptotic notation with suitable example.
Asymptotic notation
This document discusses asymptotic notations that are used to characterize an algorithm's efficiency. It introduces Big-Oh, Big-Omega, and Theta notations. Big-Oh gives an upper bound on running time. Big-Omega gives a lower bound. Theta gives both upper and lower bounds, representing an algorithm's exact running time. Examples are provided for each notation. Time complexity is also introduced, which describes how the time to solve a problem changes with problem size. Worst-case time analysis provides an upper bound on runtime.
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Flowchart
A flowchart uses standard symbols to represent a set of instructions and the step-by-step solution to a problem. It contains symbols like a terminator that marks the start and end, action boxes that represent steps or subprocesses, input/output symbols that show materials or information entering or leaving the system, decision diamonds for branching points, and connector symbols to indicate where the flow continues. Flowcharts are diagrams that show the flow of a process through these different symbols.
Upload our ppt on internet
This document provides instructions for uploading a PowerPoint presentation to Google Slideshare in 3 steps: 1) Prepare the PowerPoint slides, 2) Go to the Slideshare website and register if needed, 3) Select the PowerPoint file, fill in the title, description, privacy settings, and category before pressing the upload button. The process allows users to share PowerPoint files publicly or privately on Slideshare.
Trees
Trees are connected acyclic graphs. Important properties of trees include having one fewer edges than vertices and the ability to designate a root node, creating a rooted tree. Basic tree terminology includes root, child, parent, siblings, leaves, height, and depth. Ordered trees are rooted trees where nodes are ordered left and right, such as in binary trees where each node has at most two children designated as left or right.
Graph therory
introduction and representation of graph.
graph is collection of points and vertices.
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Decrease and conquer technic
This technique involves breaking down a large problem into smaller subproblems, solving those subproblems, and combining the solutions to solve the original problem. It can be applied recursively or iteratively by decreasing the problem size by a constant amount each iteration. Examples where it is used include insertion sort, depth-first search, breadth-first search, and topological sorting.
Minimum spanning tree.
A minimum spanning tree is a subgraph of a weighted connected graph that is a tree containing all the vertices with no circuits, and has the minimum or smallest total weight among all the possible spanning trees of that graph.
Divide and conquer technic
Divide and conquer is an algorithmic strategy that breaks down a problem into smaller subproblems, solves those subproblems independently, and then combines their solutions into a solution for the original problem. The document describes the divide and conquer algorithm as dividing a problem p into subproblems p1, p2, ..., pn, recursively applying the divide and conquer strategy to solve each subproblem, and then combining the solutions to solve problem p.
Mathematical Analysis of Non-Recursive Algorithm.
Mathematically analysis of non recursive algorithm with using some basic guide lines forward and back word substitution method
Greedy technique.
The greedy technique constructs solutions to problems in a step-by-step manner, choosing the best local option at each step to arrive at a complete solution. It aims to find a feasible solution that satisfies all constraints and also provides an optimal solution with minimum cost. The algorithm selects the best option from the available choices at each step until a full solution is reached. For example, in traveling from city A to B, the greedy approach may select taking a train then plane to arrive within a 10 hour time constraint, providing a feasible and optimal solution.
Greedy technic.
The greedy technique is an approach that constructs a solution through a sequence of steps, each expanding the partially constructed solution until a complete solution is reached. At each step, it chooses the next part of the solution that locally seems best at the moment, in the hope of finding a global optimum. An optimization problem aims to find either the minimum or maximum result. For example, finding the fastest way for a person to travel between two cities using different transportation methods within a 10 hour time constraint. The greedy algorithm selects the next solution at each step from the set of available choices that is feasible and leads to an optimal solution with minimum cost.
Mathematical Analysis of Recursive Algorithm.
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using forward and backsword substitution method.
Brute force technic
The document discusses brute force algorithms. It provides examples of using brute force to compute exponentials, factorials, and matrix multiplication. Brute force involves directly implementing the problem statement without optimization. While inefficient, it can solve small problems by straightforwardly implementing the core logic.
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Asymptotic Notation
- 2. The mathametical representation of time c omplexity is called as Asymtotic Notation s. Big oh ( O)notation Big omegap(Ω)notation Big theta (θ) notation
- 3. Big oh (O) notation: It is a method repr esenting upper bound of algorithm's running time. Definition: Let f(n)& g(n) ---->two non negative function s. n0 & c -----> constants n0 is some input n>n0 , c>0
- 5. 5 YOUR TITLE Add your words here according to your need to draw the text box size YOUR TITLE Add your words here according to your need to draw the text box size YOUR TITLE Add your words here according to your need to draw the text box size YOUR TITLE Add your words here according to your need to draw the text box size Add your title here
- 6. Example: f(n)=2n+2 & g(n)=n2 If n=3 f(n)=2(3)+2=8 g(n)=(3)2=9 f(n)<g(n)
- 7. 90% Big omega(Ω)notation This represents lower boun d of algorithm running time. f(n)>=c*g(n)
- 9. f(n)=2n2+5 & g(n)=7n n=0 f(n)=5 & g(n)=0 f(n)>g(n) f(n)∈Ωg(n)
- 10. Theta notation :(θ ) By this method tbe running time is betw een upper bound and lower bound. C1*g(n)<=f(n)<=c2*g(n) Then say that f(n)∈θg(n)
- 12. Example: If f(n)=2n+8 & g(n)=7n I e 5n<2n+8<7n n>=2
- 14. Hint: 1< log n < n< n2<n3........< 2n Lower bound
- 15. Hint: 1< log n< n<n2<n3.........<2 n Exact