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CS 234 (31)
Lecture

notes06a.pdf

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Department
Computer Science
Course Code
CS 234
Professor
Robert Sproule

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Lecture 6 - May 23, 2013 Chapter 4– Algorithm Analysis Algorithm: It is a sequence of well-define steps for solving a given problem. Inputs -> Algorithm -> Outputs For a problem P, we may have multiple algorithms. For every algorithm A, we may havemultiple implementations. How do we assess the efficiency of A? Experimentation • Implement and run • Influenced by processor, memory, OS, complier, PL… • You cannot test for all inputs Complexity Analysis (pencil and paper approach) • Examine the algorithm • Implementation independent • Considers all input instances Example: Write an algorithm to find the sum of the first n numbers Solution 1: Solution 2: Random Access Machine (RAM) Model • Memory accesses take constant time • Primitive operations take constant time o Comparisons o Arithmetic operations o Assignment Measuring Time Complexity • Given the size of input n • Sum the number of memory accesses • Sum the number of primitive operations in the algorithm • Derive a formula T(n) T(n) is approximately the number of steps in the algorithm in terms of the input size. Analysis Types • Worst case – most common approach, simpler, crucial • Best case – Almost always faster on certain inputs • Average case – difficult Big-O Notation (AsymptoticNotation) • Approximates time complexity of algorithms • Given a function T(n), T(n) is O(f(n)) if there exists a function f(n) such that T(n) ≤ c.f(n) for some c > 0 and some m > 0, n ≥ m where m is a sufficiently large value for n. T(n) = 1 + n + 1 = n + 2 T(n) is O(f(n)) – T(n) is bounded from above by f(n) Determining the time complexity Considerations when calculating T(n) • Most interested in the dominant term • Group together similar expressions • For a loop - Determine the # of iterations T(n) = 1
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