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Time Analysis Since the time it takes to execute an algorithm usually depends on the size of the input, we express the algorithm's time complexity as a function of the size of the input Question: For a given algorithm, will the same inputs size result in the same number of steps to run the program? Yes or no?

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No Consider a linear search on an array that does not contain the target as opposed to an array where the target is the first element

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Time Analysis (cont’d) Best case analysis – Given the algorithm and input of size n that makes it run fastest (compared to all other possible inputs of size n), what is the running time? Worst case analysis – Given the algorithm and input of size n that makes it run slowest (compared to all other possible inputs of size n), what is the running time? A bad worst-case complexity doesn't necessarily mean that the algorithm should be rejected Average case analysis – Given the algorithm and a typical, average input of size n, what is the running time?

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Worst-Case Analysis Worst case running time: Obtain bound on largest possible running time of algorithm on input of a given size n – Generally captures efficiency in practice Average case running time: Obtain bound on running time of algorithm on random input as a function of input size n – Cavg = sum I (P(input_i)*step(input_i)) – Hard (or impossible) to accurately model real instances by random distributions – Algorithm tuned for a certain distribution may perform poorly on other inputs

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Time Analysis (cont’d) Example: Consider searching sequentially an unordered array to find a number – Best Case? O(1) – Worst Case? O(n) – Average Case? O(n) why?

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Example Binary search Best case complexity Worst case complexity

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Chap 2 Exercise 1.Explain O(1) 2. Assume f1(n) is O(g1(n)) and f2(n) is o(g2(n)), prove that f1(n) + f2(n) is O(max(g1(n), g2(n))) 3. Find functions f1 and f2 such that both f1(n) and f2(n) are O(g(n)), but f1(n) is not O(f2(n)) 4. Page 74 10. 5.Page 74 11.

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