Algorithm_Analysis

# Algorithm_Analysis -   Algorithm Analysis ...

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Unformatted text preview: 1/27/10  Algorithm Analysis  COMP 2210  Algorithm Analysis  •  Characterize the cost of an algorithm as a  func;on of its problem size.  •  This func;on is the ;me complexity func;on  of the algorithm.  •  Time complexity func;ons are typically  derived by coun;ng the number of ;mes that  one or more basic opera;ons are performed.  1  1/27/10  Time Complexity  Time Complexity  2  1/27/10  Big O  •  The O stands for “order of”  •  Concerned with what happens for very large  values of n (as the problem size grows larger  and larger)  •  All coeﬃcients, low‐order terms, and  constants are ignored.  •  So, 15n2 + 50n + 35 is O(n2)  Big O  •  Given two func;ons f(n) and g(n), f(n) =  O(g(n)) if and only if there exists a constant c  such that c*g(n) grows asympto;cally faster  than f(n).  3  1/27/10  Big O  • Algorithm A is O(n)  • Algorithm B is O(n2)  • Algorithm C is O(1)  Big O  •  Linear search is O(n)  •  Binary search is O(logn)  4  1/27/10  Prac8cal Implica8ons  Assume your machine can execute one billion instructions per second, and sustain this rate indefinitely. Input Size 1,000 O(log n) O(n) Time Complexity O(nlogn) O(n2) O(n3) O(n4) O(2n) O(n!) < nano sec 1 ns 10 ns 1 ms 1 sec 17 min 3.4x10284 years ?? 10,000 10 ns 10 ns 0.1 ms 0.1 sec 17 min 4 months ?? ?? 100,000 132 ns 100 ns 2 ms 10 sec 12 days 3,200 years ?? ?? 1,000,000 166 ns 1 ms 20 ms 17 min 32 years 3.2 million yrs ?? ?? 10,000,000 199 ns 10 ms 0.2 sec 28 hours 32,000 years 3.17x1015 years ?? ?? Are data sets really that big?  •  Photos from a 10 megapixel camera have  10,000,000 pixels.   •  A 3‐minute song stored as a 16‐bit sound ﬁle  sampled 44,100 ;mes per second for each of  two channels would consist of about  8,000,000 data points.  •  Google indexes billions and billions of web  pages, documents, and images.  5  ...
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