Lesson 8: Functions and Growth

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Lesson 8 Functions and Growth

• Objectives
• Use big-O, big-Omega, and big-Theta notation
• Find witnesses for function relationships
• State basic properties of divisibility

• Outline
• Big O
• Function Combinations
• Big Omega
• Big Theta
• HW due Feb 15
• 2.2 1, 5, 7, 14, 22, 45, 46, 47, 48
• 2.3 4, 5, 7, 8, 9, 10, 13, 22
• Reading Section 2.2

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Section 2.2 Growth of Functions

• Tools used in analyzing operations of hardware
  and execution of algorithms
• Mainly concerned with asymptotic behavior
• Three Notations
• Big-O
• Big-Omega
• Big-Theta

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Considerations

• Three Costs
• f(n) (1)n 10,000
• g(n) (80)ln(n) 100,000
• h(n) (0.01)n2 1

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Big-O Notation

• f, g are functions (Z or ?) ? ?
• We say f(x) is O(g(x)) if there exists
  constants C and k
• f(x) Cg(x) whenever x gt k
• Reads f of x is big Oh of g of x
• C, k are called witnesses

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Big O

• Show that ( f(x) 3x2 2x 8 ) is O(x2)

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Big O

• Show that (9x2) is O(x4)

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Big-O estimates

• Find a Big-O estimate of the sum of the first n
  positive integers.

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Big O estimates

• Show that f(x) n! is O(nn)

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Common Growth functions

• Page 138, Figure 3

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Combinations

• If f1(x) is O(g1(x)) and f2(x) is O(g2(x))
• (f1 f2)(x)
• If f1(x) is O(g(x)) and f2(x) is O(g(x))
• (f1 f2)(x)
• If f1(x) is O(g1(x)) and f2(x) is O(g2(x))
• (f1 f2)(x)

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Using Combo Rules

• f(x) x3 x2 x 1
• f(x) (2x 1)log(x)

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Combinations In Class

• Give a Big-O estimate for
• n log(n) 2nnlog(n)(n! 2n)

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Big-O In Class

• We want to prove that 3x3 - 2x 6 is O(x4)
• If we choose k 3, what is the smallest witness
  C that can be called to show this relationship?

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Big Omega

• f, g are functions (Z or ?) ? ?
• We say f(x) is W(g(x)) if there exists
  constants C and k
• f(x) Cg(x) whenever x gt k
• Reads f of x is big Omega of g of x
• C, k are called witnesses

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Big Omega

• Find witnesses C, k to prove that f(x) 12x3
  2x 1 is big-Omega(x3)

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Big Theta

• f, g are functions (Z or ?) ? ?
• We say f(x) is Q(g(x)) if f(x) is O(g(x)) and
  f(x) is W(g(x))
• f of x is big-theta of g of x
• f of x is of order g of x

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Big Theta

• Show that 12x2 x 2 is Q(x2)

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Algorithm Complexity

• for (int i0 i lt x i)
• for (int j0 j lt i j)
• tmpVar1 x
• tmpVar2 x
• tmpVar3 tmpVar1 tmpVar3
• myArrayji tmpVar3

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Functions

• What does it mean for a function to be W(1)?
• If a function is Q(1), what can we say about the
  function?

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Algorithms

• A satisfactory algorithm must
• Be correct
• Efficient in time
• Efficient in space
• Time Complexity
• Space Complexity

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Binary Search Algorithm

• Check to see if n is in a list of 2k sorted
  elements
• a 2 18 19 20 29 30 102 3000

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Linear Search

• array an
• target object x
• j 0
• while(j lt n x ! aj) j
• if (j lt n) return j
• else return 1
• worst, best case

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Linear Search Average Case (assuming in list)

• if j1, 3 comparisons
• if j 2, 5 comparisons
• j n ____________
• Average assume that element is equally likely to
  be in any spot.

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Polynomial Calculation

• 2x3 4x2 x 10 (evaluated at x2)
• float a 10 1 4 2
• int power 1 int order 3
• float sum a0
• for (j1 jltorder j)
• power powerx
• sum sum ajpower
• return sum

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Horners Method

• 2x3 4x2 x 10 (evaluated at x2)
• float a 10 1 4 2
• int order 3
• float sum aorder
• for (j1 jltorder j)
• sum sumx aorder-j
• return sum

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Circuit-Sat

• Given boolean expression
• of (an, an-1, a1, a0)
• Is there a combination of ais which will make
  the circuit output a 1

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Hard Problems

• Polynomial Complexity
• Tractable
• Exponential Complexity
• Intractable
• Exponential algorithms may be polynomial in the
  average case
• Approximations to exponential algorithms may be
  polynomial
• Some problems are unsolvable
• there exists no algorithm to solve all cases

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NP-completeness

• Class NP problems
• Solutions can be checked in polynomial time
• Class NP-complete problems
• Can be converted to one another in polynomial
  time
• If a polynomial solution for one is found, all
  are solved in P