Analysis of Algorithms IIBasicsSize of the inputMeasuring requirementsBig-O and friendsFormal definition of Big-OFormal definition of Big-*Formal definition of Big-*GraphsInformal reviewThe EndJan 14, 2019Analysis of Algorithms II2BasicsBefore we attempt to analyze an algorithm, we need to define two things:How we measure the size of the inputHow we measure the time (or space) requirementsOnce we have done this, we find an equation that describes the time (or space) requirements in terms of the size of the inputWe simplify the equation by discarding constants and discarding all but the fastest-growing term3Size of the inputUsually it’s quite easy to define the size of the inputIf we are sorting an array, it’s the size of the arrayIf we are computing n!, the number n is the “size” of the problemSometimes more than one number is requiredIf we are trying to pack objects into boxes, the results might depend on both the number of objects and the number of boxesSometimes 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 measureTo see this, compute f(7) and f(8)4Measuring requirementsIf 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 resultsThis is not analysisHowever, empirical testing is useful as a check on analysisAnalysis means figuring out the time or space requirementsMeasuring space is usually straightforwardLook at the sizes of the data structuresMeasuring time is usually done by counting characteristic operationsCharacteristic operation is a difficult term to defineIn any algorithm, there is some code that is executed the most timesThis is in an innermost loop, or a deepest recursionThis code requires “constant time” (time bounded by a constant)Example: Counting the comparisons needed in an array search5Big-O and friendsInformal 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 x36Formal definition of Big-OA 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 = 50Of course, 5x2 + 6 is also O(n4), O(2n), and even O(n2)7Formal 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 = 50Of course, 5x2 + 6 is also O(log n), O(n), and even O(n2)* “omega”8Formal 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 (n) because n2 < 5n2 + 6 < 6n2 whenever n > 5 (c1 = 1, c2 = 6)* “theta”9GraphsPoints to notice:What happens near the beginning(n < N) is not importantcg(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?)f(n)cg(n)f(n) is O(g(n))Nf(n)cg(n)f(n) is (g(n))Nf(n)c2g(n)c1g(n)f(n) is (g(n))N10Informal reviewFor 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) = (g(n)) if c1g(n) is greater than f(n) and c2g(n) is less than f(n),f(n) = (g(n)) if cg(n) is less than f(n),...for suitably chosen values of c, c1, and c2O11The 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
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