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Asymptotic Growth Rates

Asymptotic Growth Rates. Themes Analyzing the cost of programs Ignoring constants and Big-Oh Recurrence Relations & Sums Divide and Conquer Examples Sort Computing powers Euclidean algorithm (computing gcds) Integer Multiplication. Asymptotic Growth Rates – “Big-O” (upper bound).

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Asymptotic Growth Rates

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  1. Asymptotic Growth Rates • Themes • Analyzing the cost of programs • Ignoring constants and Big-Oh • Recurrence Relations & Sums • Divide and Conquer • Examples • Sort • Computing powers • Euclidean algorithm (computing gcds) • Integer Multiplication

  2. Asymptotic Growth Rates– “Big-O” (upper bound) • f(n) = O(g(n)) [f grows at the same rate or slower than g] iff: There exists positive constants c and n0 such that f(n)  c g(n) for all n  n0 • f is bound above by g • Note: Big-O does not imply a tight bound • Ignore constants and low order terms

  3. Big-O, Examples • E.G. 1: 5n2 = O(n3) c = 1, n0= 5: 5n2 nn2 = n3 • E.G. 2: 100n2 = O(n2) c = 100, n0= 1 • E.G. 3: n3 = O(2n) c = 1, n0= 12 n3  (2n/3)3, n  2n/3 for n  12 [use induction]

  4. Little-oLoose upper bound • f(n) = o(g(n)) [f grows strictly slower than g] f(n) = O(g(n)) and g(n)  O(f(n)) • limn f(n)/g(n) = 0 • “f is bound above by g, but not tightly”

  5. Little-o, restatement • limn f(n)/g(n) = 0  f(n) = o(g(n)) •   >0,  n0 s.t.  n  n0, f(n)/g(n) < 

  6. Equivalence - Theta • f(n) = (g(n)) [grows at the same rate] • f(n) = O(g(n)) and g(n) = O(f(n)) • g(n) = (f(n)) • limnf(n)/g(n) = c, c ≠ 0 f(n) = (g(n)) • “f is bound above by g, and below by g”

  7. Common Results • [j < k] limn nj/nk = limn 1/n(k-j) = 0 • nj = o(nk), if j<k • [c < d] limn cn/dn = limn (c/d)n = 0 • cn = o(dn), if c<d • limn ln(n)/n = / • limn ln(n)/n = limn (1/n)/1 = 0 [L’Hopital’s Rule] • ln(n) = o(n) • [ > 0] ln(n) = o(n) [similar calculation]

  8. Common Results • [c > 1, k an integer] limn nk/cn = / • limn knk-1/ cnln(c) • limn k(k-1)nk-2/ cnln(c)2 • … • limn k(k-1)…(k-1)/cnln(c)k = 0 • nk = o(cn)

  9. Asymptotic Growth Rates • (log(n)) – logarithmic [log(2n)/log(n) = 1 + log(2)/log(n)] • (n) – linear [double input  double output] • (n2) – quadratic [double input  quadruple output] • (n3) – cubit [double input  output increases by factor of 8] • (nk) – polynomial of degree k • (cn) – exponential [double input  square output]

  10. Asymptotic Manipulation • (cf(n)) = (f(n)) • (f(n) + g(n)) = (f(n)) if g(n) = O(f(n))

  11. Computing Time Functions • Computing time function is the time to execute a program as a function of its inputs • Typically the inputs are parameterized by their size [e.g. number of elements in an array, length of list, size of string,…] • Worst case = max runtime over all possible inputs of a given size • Best case = min runtime over all possible inputs of a given size • Average = avg. runtime over specified distribution of inputs

  12. Analysis of Running Time • We can only know the cost up to constants through analysis of code [number of instructions depends on compiler, flags, architecture, etc.] • Assume basic statements are O(1) • Sum over loops • Cost of function call depends on arguments • Recursive functions lead to recurrence relations

  13. Loops and Sums • for (i=0;i<n;i++) • for (j=i;j<n;j++) • S; // assume cost of S is O(1)

  14. Merge Sort and Insertion Sort • Insertion Sort • TI(n) = TI(n-1) + O(n) =(n2) [worst case] • TI(n) = TI(n-1) + O(1) =(n) [best case] • Merge Sort • TM(n) = 2TM(n/2) + O(n) =(nlogn) [worst case] • TM(n) = 2TM(n/2) + O(n) =(nlogn) [best case]

  15. Karatsuba’s Algorithm • Using the classical pen and paper algorithm two n digit integers can be multiplied in O(n2) operations. Karatsuba came up with a faster algorithm. • Let A and B be two integers with • A = A110k + A0, A0 < 10k • B = B110k + B0, B0 < 10k • C = A*B = (A110k + A0)(B110k + B0) = A1B1102k + (A1B0 + A0 B1)10k + A0B0 Instead this can be computed with 3 multiplications • T0 = A0B0 • T1 = (A1 + A0)(B1 + B0) • T2 = A1B1 • C = T2102k + (T1 - T0 - T2)10k + T0

  16. Complexity of Karatsuba’s Algorithm • Let T(n) be the time to compute the product of two n-digit numbers using Karatsuba’s algorithm. Assume n = 2k. T(n) = (nlg(3)), lg(3)  1.58 • T(n)  3T(n/2) + cn  3(3T(n/4) + c(n/2)) + cn = 32T(n/22) + cn(3/2 + 1)  32(3T(n/23) + c(n/4)) + cn(3/2 + 1) = 33T(n/23) + cn(32/22 + 3/2 + 1) …  3iT(n/2i) + cn(3i-1/2i-1 + … + 3/2 + 1) ...  cn[((3/2)k - 1)/(3/2 -1)] --- Assuming T(1)  c  2c(3k - 2k)  2c3lg(n) = 2cnlg(3)

  17. Divide & Conquer Recurrence Assume T(n) = aT(n/b) + (n) • T(n) = (n) [a < b] • T(n) = (nlog(n)) [a = b] • T(n) = (nlogb(a)) [a > b]

  18. (blank for notes)

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