Asymptotic Bounds The Differences Between (Big-O, Omega and Theta)

1
Asymptotic BoundsThe Differences Between
(Big-O, Omega and Theta)

• Properties

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Measuring Efficiency

• Scalability
• When algorithm is applied to a large data set,
  will finish relatively quickly.
• Speed and memory usage
• Measuring speed-we measure algorithm speed in
  terms of Operations relative to input size.

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Big O
y
g(x)
f(x)
Cg(x)
x
xo
4
Big O

• Definition Let f(x) and g(x) be two functions
  We say that
• f(x) ? O(g(x))
• if there exists a constant c, Xo gt 0 such that
• f(x)cg(x) for all X Xo.
• f (x) is asymptotically less than or equal to
  g(x)
• Big-O gives an asymptotic upper bound.

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Big ?
y
g(x)
f(x)
Cg(x)
x
xo
6
Big-Omega

• Definition Let f (x)and g(x) be two functions
  We say that
• f(x) ? ?(g(x))
• if there exists a constant c, Xo 0 such that
• f(x) cg(x) for all X Xo
• f(x) is asymptotically greater than or equal to
  g(x)
• Big-Omega gives an asymptotic lower bound

7
Big T
y
g(x)
C1g(x)
f(x)
C2g(x)
x
xo
8
Big T

• Definition Let f(x) and g(x) be two functions
  We say that
• f(x) ? ?(g(x))
• if there exists a constant c1, c2, Xo gt 0 such
  that for every integer x ? x0 we have
• c1g(x) f(x) c2g(x)
• F(x) is asymptotically equal to g(x)
• F(x) is bounded above and below by g(x)
• Big-Theta gives an asymptotic equivalence