Analysis of Algorithms II 22-Feb-19

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

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)

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

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

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) because 0 < 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)

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) because 0 < 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)

Formal definition of Big-* A function f(n) is (g(n)) if there exist positive constants c1 and c2 and 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) because n2 < 5n2 + 6 < 6n2 whenever n > 5 (c1 = 1, c2 = 6)

Graphs Points to notice: f(n) cg(n) f(n) is O(g(n)) N f(n) cg(n) f(n) is (g(n)) N f(n) c2g(n) c1g(n) f(n) is (g(n)) N 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?)

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

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