General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

(f ≤a g), iff there is an n0 ∈ N, such that f(n) ≤ g(n) for all n > n0.
Definition 259 The three Landau sets O(g), Ω(g), Θ(g) are defined as
O(g) = {f | ∃k > 0.f ≤a k · g}
Ω(g) = {f | ∃k > 0.f ≥a k · g}
Θ(g) = O(g) ∩ Ω(g)
Intuition: The Landau sets express the “shape of growth” of the graph of a function.
If f ∈ O(g), then f grows at most as fast as g. (“f is in the order of g”)
If f ∈ Ω(g), then f grows at least as fast as g. (“f is at least in the order of g”)
If f ∈ Θ(g), then f grows as fast as g. (“f is strictly in the order of g”)
Commonly used Landau Sets
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Landau set class name rank Landau set class name rank
O(1) constant 1 O(n 2 ) quadratic 4
O(log 2(n)) logarithmic 2 O(n k ) polynomial 5
O(n) linear 3 O(k n ) exponential 6
Theorem 260 These Ω-classes establish a ranking (increasing rank increasing growth)
O(1)⊂O(log2(n))⊂O(n)⊂O(n 2 )⊂O(n k′
)⊂O(k n )
where k ′ > 2 and k > 1. The reverse holds for the Ω-classes
Ω(1)⊃Ω(log2(n))⊃Ω(n)⊃Ω(n 2 )⊃Ω(n k′
)⊃Ω(k n )
Idea: Use O-classes for worst-case complexity analysis and Ω-classes for best-case.
Examples
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Idea: the fastest growth function in sum determines the O-class
Example 261 (λn.263748) ∈ O(1)
Example 262 (λn.26n + 372) ∈ O(n)
Example 263 (λn.7n 2 − 372n + 92) ∈ O(n 2 )
Example 264 (λn.857n 10 + 7342n 7 + 26n 2 + 902) ∈ O(n 10 )
Example 265 (λn.3 · 2 n + 72) ∈ O(2 n )
Example 266 (λn.3 · 2 n + 7342n 7 + 26n 2 + 722) ∈ O(2 n )
c○: Michael Kohlhase 147
With the basics of complexity theory well-understood, we can now analyze the cost-complexity of
Boolean expressions that realize Boolean functions. We will first derive two upper bounds for the
cost of Boolean functions with n variables, and then a lower bound for the cost.
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