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: 1 Chapter 3. Growth of function Chapter 4. Recurrences

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Contents of Table 1. Technicalities for abbreviation 2. Recurrence solution methods The substitution method 2 2. Notation of asymptotically larger/smaller o-notation ω-notation 3. Relationship between this notations A. Growth of function B. Recurrence

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f(n) is asymptotically smaller than g(n) o -notation Asymptotically smaller - Definition for all n, n n 0, any positive constant c, c>0, 0 f(n)< cg(n), the function f(n) is smaller to g(n) to within constant factor Asymptotically larger/smaller notation f(n) o(g(n)) n0n0

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4 2. Asymptotically larger/smaller notation o -notation - Definition o(g(n)) = { f(n) : for any positive constant c>0, there exist a constant n 0 >0 such that 0 f(n) < cg(n) for all n n 0 } this limit shows the function f(n) becomes insignificant relative to g(n) as n approaches infinity

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o -notation Example 1. Asymptotic notation Use the definition of o-notation to prove the following property. f(n) = o(n 2 ) Use the definition of o-notation to prove the following property. f(n) = o(n 2 )

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6 2. Notation of asymptotically larger/smaller o -notation Solution to example 2n = o(n 2 ) If f(n) = o((g(n)) then

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f(n) is asymptotically larger than g(n) ω -notation Asymptotically larger - Definition for all n, n n 0, any positive constant c, c>0, 0 cg(n) < f(n), the function f(n) is larger to g(n) to within constant factor Notation of asymptotically larger/smaller f(n) ω(g(n))

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8 2. Notation of asymptotically larger/smaller ω -notation - Definition ω(g(n)) = { f(n) : for any positive constant c>0, there exist a constant n 0 >0 such that 0 cg(n) < f(n) for all n n 0 } if the limit exists. That is, f(n) becomes arbitrarily large relative to g(n) as n approaches infinity.

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ω -notation - We use ω-notation to denote a lower bound that is not asymptotically tight. - We use o-notation to denote an upper bound that is not asymptotically tight. - f(n) ω(g(n)) if and only if g(n) o(f(n)) Notation of asymptotically larger/smaller

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10 2. Notation of asymptotically larger/smaller ω -notation Example Use the definition of o-notation to prove the following property. f(n) = ω(n) Use the definition of o-notation to prove the following property. f(n) = ω(n)

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11 2. Notation of asymptotically larger/smaller ω -notation Solution to example =ω(n) If f(n) = ω((g(n)) then

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12 3. Relationship between this notations f(n) O(g(n)) o(g(n)) ω(g(n)) Ω(g(n)) θ(g(n)) O(g(n))Ω(g(n)) ω(g(n)) o(g(n))

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13 3. Relationship between this notations f(n) O(g(n)) o(g(n)) ω(g(n)) Ω(g(n)) Transitivity Transpose symmetry f(n) = Θ(g(n)) and g(n) = Θ(h(n)) imply f(n) = Θ(h(n)), f(n) = O(g(n)) and g(n) = O(h(n)) imply f(n) = O(h(n)), f(n) = (g(n)) and g(n) = (h(n)) imply f(n) = (h(n)), f(n) = o(g(n)) and g(n) = o(h(n)) imply f(n) = o(h(n)), f(n) = ω(g(n)) and g(n) = ω(h(n)) imply f(n) = ω(h(n)). f(n) = O(g(n)) if and only if g(n) = (f(n)), f(n) = o(g(n)) if and only if g(n) = ω(f(n)).

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14 3. Relationship between this notations Reflexivity f(n) = Θ(f(n)) f(n) = O(f(n)) f(n) = (f(n)) Symmetry f(n) = Θ(g(n)) if and only if g(n) = Θ(f(n)). θ(g(n)) O(g(n))Ω(g(n))

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Recurrences 15 Recurrence When should we use the recurrence ? When algorithm contains a recursive call to itself, Its running time can often be described by a recurrence.

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Recurrences 16 Recurrence - Definition Equation or inequality that describes a function in terms of its value on smaller inputs.

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Recurrences 17 Recurrence Example Worst-case running time of T(n) could be described by the recurrence Merge-Sort procedure Show that the solution of T(n) = Θ(n log n) Merge-Sort procedure Show that the solution of T(n) = Θ(n log n) if n=1 if n>1

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Recurrences 18 Recurrence Solution method to example - The substitution method guess a bound and prove our guess correct for small input. - The recursion-tree method convert the recurrence into a tree whose nodes represent the costs incurred at various level of recursion - The master method determining asymptotic bounds for many simple recurrences of the form ( a 1, b 1, and f(n) is given function )

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1. Technicalities for abbreviation Technicalities - assumption of integer The running time T(n) of algorithm is only defined when n is an integer - floors / ceilings We often omit floors and ceilings - boundary condition The running time of an algorithm on a constant-sized input is a constant that we typically ignore.

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1. Technicalities for abbreviation Technicalities Example The recurrence describing the worst-case running time of Merge-Sort Omit assumption of integer Omit boundary condition if n=1 if n>1 if n=1 if n>1 if n=1 if n>1 Omit floors and ceilings

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2. Recurrence solution methods 21 Substitution Method - Definition When recurrence is easy to guess form of the answer with the inductive hypothesis is applied to smaller values, substitution of the guessed answer for the function. - It can be used to establish either upper or lower bounds on a recurrence.

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2. Recurrence solution methods 22 Substitution Method Steps 1. Guess the form of the solution 2. Use mathematical induction to find the constant and show that the solution works. 1. Guess the form of the solution 2. Use mathematical induction to find the constant and show that the solution works.

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2. Recurrence solution methods 23 Substitution Method - Making a good guess prove loose upper and lower bounds on the recurrence and then reduce the range of uncertainty. - Guessing a solution takes : experience, occasionally, creativity - we can use recursion tree

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2. Recurrence solution methods 24 Substitution Method - Subtleties we can prove something stronger for given value by assuming something stronger for a smaller values. We guess that the solution is O(n), and show that T(n) cn Overcome difficulty by subtracting a lower-order term from our previous guess ( b1, c must be chosen large enough )

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2. Recurrence solution methods 25 Substitution Method - Avoid pitfalls In recurrence we can falsely prove T(n)= O(n) by guessing T(n) cn

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2. Recurrence solution methods 26 Substitution Method Example Let us determine an upper bound on the recurrence Let us determine an upper bound on the recurrence

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2. Recurrence solution methods 27 Substitution Method Solution to example 1. guess that the solution is 2. prove that ( c > 0 ) 1. guess that the solution is 2. prove that ( c > 0 ) - Assume that this bound holds for n/2, that is, T ( n/2 ) c n/2 lg( n/2 ). T(n) 2(c n/2 lg( n/2 )) + n cn lg(n/2) + n ( because, n/2 < n/2 ) = cn lg n - cn lg 2 + n = cn lg n - cn + n cn lg n (as long as c 1 ) We need to find constant c and n 0

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2. Recurrence solution methods 28 Substitution Method Solution to example - Find the constant n 0 - we can take advantage of asymptotic notation, we can replace base case T(1) cn lg n T(1) = 1 but, c1 lg1 = 0 - So, we only have to prove T (n) = cn lg n for n n 0 for n (n 0 =2) Replace T(1) by T(2) and T(3) as the base cases by letting n 0 =2.

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2. Recurrence solution methods 29 Substitution Method Solution to example - Find the constant c - choosing c large enough T (2) = 4 and T (3) = 5. T (2) = 4 c ( 2 lg 2 ) T (3) = 5 c ( 3 lg 3 ). Any choice of c 2 suffices for base case of n=2 and n =3

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2. Recurrence solution methods 30 Substitution Method Example Let us determine an upper bound on the recurrence Let us determine an upper bound on the recurrence

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2. Recurrence solution methods 31 Substitution Method Solution to example - Simplify this recurrence by changing variable Renaming m = lg n S(m) = T(2 m ) This new recurrence has same solution : S(m) = O(m log m)

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2. Recurrence solution methods 32 Substitution Method Solution to example -Simplify this recurrence by changing variable This new recurrence has same solution : S(m) = O(m log m) Changing back from S(m) to T(n) T(n) = T(2 m ) = S(m) = O(m lg m) = O( lg n lg(lg n) )

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