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Analysis of Algorithms II

Analysis of Algorithms II. Basics. Before we attempt to analyze an algorithm, we need to define two things: How we measure the size of the input How we measure the time (or space) requirements

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Analysis of Algorithms II

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  1. Analysis of Algorithms II

  2. Basics • Before we attempt to analyze an algorithm, we need to define two things: • How we measure the size of the input • How we measure the time (or space) requirements • Once we have done this, we find an equation that describes the time (or space) requirements in terms of the size of the input • We simplify the equation by discarding constants and discarding all but the fastest-growing term

  3. Size of the input • Usually it’s quite easy to define the size of the input • If we are sorting an array, it’s the size of the array • If we are computing n!, the number n is the “size” of the problem • Sometimes more than one number is required • If we are trying to pack objects into boxes, the results might depend on both the number of objects and the number of boxes • Sometimes it’s very hard to define “size of the input” • Consider:f(n) = if n is 1, then 1; else if n is even, then f(n/2); else f(3*n + 1) • The obvious measure of size, n, is not actually a very good measure • To see this, compute f(7) and f(8)

  4. Measuring requirements • If we want to know how much time or space an algorithm takes, we can do empirical tests—run the algorithm over different sizes of input, and measure the results • This is not analysis • However, empirical testing is useful as a check on analysis • Analysis means figuring out the time or space requirements • Measuring space is usually straightforward • Look at the sizes of the data structures • Measuring time is usually done by counting characteristic operations • Characteristic operation is a difficult term to define • In any algorithm, there is some code that is executed the most times • This is in an innermost loop, or a deepest recursion • This code requires “constant time” (time bounded by a constant) • Example: Counting the comparisons needed in an array search

  5. Big-O and friends • Informal definitions: • Given a complexity function f(n), • (f(n))is the set of complexity functions that are lower bounds on f(n) • O(f(n))is the set of complexity functions that are upper bounds on f(n) • (f(n))is the set of complexity functions that, given the correct constants, correctly describes f(n) • Example: If f(n) = 17x3 + 4x – 12, then • (f(n)) contains 1, x, x2, log x, x log x, etc. • O(f(n))contains x4, x5, 2x, etc. • (f(n))contains x3

  6. Formal definition of Big-O* • A function f(n) is O(g(n)) if there exist positive constants c and N such that, for all n > N, 0 < f(n) < cg(n) • That is, if n is big enough (larger than N—we don’t care about small problems), then cg(n) will be bigger than f(n) • Example: 5x2 + 6 is O(n3) because0 < 5n2 + 6 < 2n3 whenever n > 3 (c = 2, N = 3) • We could just as well use c = 1, N = 6, or c = 50, N = 50 • Of course, 5x2 + 6 is also O(n4), O(2n), and even O(n2)

  7. Formal definition of Big-* • A function f(n) is (g(n)) if there exist positive constants c and N such that, for all n > N, 0 < cg(n) < f(n) • That is, if n is big enough (larger than N—we don’t care about small problems), then cg(n) will be smaller than f(n) • Example: 5x2 + 6 is (n) because0 < 20n < 5n2 + 6 whenever n > 4 (c=20, N=4) • We could just as well use c = 50, N = 50 • Of course, 5x2 + 6 is also O(log n), O(n), and even O(n2)

  8. Formal definition of Big-* • A function f(n) is (g(n)) if there exist positive constants c1 andc2and N such that, for all n > N, 0 < c1g(n) < f(n) < c2g(n) • That is, if n is big enough (larger than N), then c1g(n) will be smaller than f(n) and c2g(n) will be larger than f(n) • In a sense,  is the “best” complexity of f(n) • Example: 5x2 + 6 is (n2) becausen2< 5n2 + 6 < 6n2whenever n > 5 (c1 = 1, c2 = 6)

  9. cg(n) f(n) is O(g(n)) f(n) is (g(n)) f(n) f(n) cg(n) N N c1g(n) f(n) is (g(n)) f(n) c2g(n) N Graphs • Points to notice: • What happens near the beginning(n < N) is not important • cg(n) always passes through 0, but f(n) might not (why?) • In the third diagram, c1g(n) and c2g(n) have the same “shape” (why?)

  10. Informal review • For any function f(n), and large enough values of n, • f(n) = O(g(n)) if cg(n) is greater than f(n), • f(n) = theta(g(n)) if c1g(n) is greater than f(n) and c2g(n) is less than f(n), • f(n) = omega(g(n)) if cg(n) is less than f(n), • ...for suitably chosen values of c, c1, and c2

  11. The End • The formal definitions were taken, with some slight modifications, from Introduction to Algorithms, by Thomas H. Cormen, Charles E. Leiserson, Donald L. Rivest, and Clifford Stein

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