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Asymptotes: Why? How to describe an algorithm’s running time? (or space, …) How does the running time depend on the input? T(x) = running time for instance x Problem: Impractical to use, e.g., “15 steps to sort [3 9 1 7], 13 steps to sort [1 2 0 3 9], …” Need to abstract away from the individual instances. 1 Asymptotes: Why? Standard solution: Abstract based on size of input. How does the running time depend on the input? T(n) = running time for instances of size n Problem: Time also depends on other factors. E.g., on sortedness of array. 2 Asymptotes: Why? Solution: Provide a bound over these instances. Most common. Default. Worst case Best case Average case T(n) = max{T(x) | x is an instance of size n} T(n) = min{T(x) | x is an instance of size n} T(n) = |x|=n Pr{x} T(x) Determining the input probability distribution can be difficult. 3 Asymptotes: Why? What’s confusing about this notation? Worst case Best case Average case T(n) = max{T(x) | x is an instance of size n} T(n) = min{T(x) | x is an instance of size n} T(n) = |x|=n Pr{x} T(x) Two different kinds of functions: T(instance) T(size of instance) Won’t use T(instance) notation again, so can ignore. 4 Asymptotes: Why? Problem: T(n) = 3n2 + 14n + 27 Too much detail: constants may reflect implementation details & lower terms are insignificant. 3n2 n Solution: Ignore the constants & low-order terms. (Omitted details still important pragmatically.) 14n+17 1 3 31 10 300 157 100 30,000 1,417 1000 3,000,000 14,017 10000 300,000,000 140,017 3n2 > 14n+17 “large enough” n 5 Upper Bounds Creating an algorithm proves we can solve the problem within a given bound. But another algorithm might be faster. E.g., sorting an array. Insertion sort O(n2) What are example algorithms for O(1), O(log n), O(n), O(n log n), O(n2), O(n3), O(2n)? 6 Lower Bounds Sometimes can prove that we cannot compute something without a sufficient amount of time. That doesn't necessarily mean we know how to compute it in this lower bound. E.g., sorting an array. # comparisons needed in worst case (n log n) Shown in COMP 482. 7 Definitions: O, T(n) O(g(n)) constants C,k > 0 T(n) (g’(n)) constants C’,k’ > 0 such that such that nk, T(n) Cg(n) nk’, T(n) C’g’(n) Cg(n) T(n) C’g’(n) k k’ 8 Examples: O, 2n+13 O( ? ) O(n) 2n+13 ( ? ) (n), also (log n), (1), … 2n O(n) ? (n) ? Given a C, 2n Cn, for all but small n. (n), not O(n). nlog n O(n5) ? No. Given a C, log n C5, for all large enough n. Thus, (n5). Also, O(n2), O(5n), … Can always weaken the bound. 9 Definitions: T(n) (g(n)) T(n) O(g(n)) and T(n) (g(n)) Ideally, find algorithms that are asymptotically as good as possible. 10 Notation O(), (), () are sets of functions. But common to abuse notation, writing T(n) = O(…) instead of T(n) O(…) as well as T(n) = f(n) + O(…) 11