1. O-notation (upper bound)
Asymptotic running times of algorithms are usually defined by functions whose domain are N={0, 1, 2, …} (natural numbers)
Formal Definition of O-notation
f(n) = O(g(n)) if  positive constants c, n0 such that
0 ≤ f(n) ≤ cg(n), n ≥ n0
Ex. 2n2 = O(n3)
2n2 ≤ cn3  cn ≥ 2  c = 1 & n0 = 2 or c = 2 & n0 = 1
2n3 = O(n3)
2n3 ≤ cn3  c ≥ 2  c = 2 & n0 = 1
BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)

2. O-notation (upper bound)
“=” is funny; “one-way” equality
O-notation is sloppy, but convenient.
We must understand what O(n) really means
O(g(n)) is actually a set of functions.
O(g(n)) = { f(n) :  positive constants c, n0 such that
0 ≤ f(n) ≤ cg(n), n ≥ n0 }
2n2 = O(n3) means that 2n2  O(n3)
BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)

3. O-notation (upper bound)
O-notation is an upper-bound notation
It makes no sense to say “running time of an algorithm is at least O(n2)”.
let running time be T(n)
T(n) ≥ O(n2) means
T(n) ≥ h(n) for some h(n)  O(n2)
however, this is true for any T(n) since
h(n) = 0  O(n2), & running time > 0,
so stmt tells nothing about running time
BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)

4. O-notation (upper bound)
BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)

5. -notation (lower bound)
Formal Definition of -notation
f(n) = (g(n)) if  positive constants c, n0 such that
0 ≤ cg(n) ≤ f(n) , n ≥ n0
(g(n)) { f(n) :  positive constants c, n0 such that
0 ≤ cg(n) ≤ f(n) , n ≥ n0 }
BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)

6. -notation (lower bound)
BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)

7. Θ-notation (tight bound)
Formal Definition of Θ-notation
f(n) = Θ(g(n)) if  positive constants c1, c2, n0 such that
0 ≤ c1g(n) ≤ f(n) ≤ c2g(n) , n ≥ n0
Θ(g(n)) { f(n) :  positive constants c1, c2, n0 such that
0 ≤ c1g(n) ≤ f(n) ≤ c2g(n) , n ≥ n0 }
BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)

8. Θ-notation (tight bound) - Example
BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)

9. Θ-notation (tight bound) - 0 < c1 ≤ h(n) ≤ c2
BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)

10. Θ-notation (tight bound) - 0 < c1 ≤ h(n) ≤ c2
BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)

11. Θ-notation (tight bound)
Θ(g(n)) { f(n) :  positive constants c1, c2, n0 such that
0 ≤ c1g(n) ≤ f(n) ≤ c2g(n) , n ≥ n0 }
BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)

12. Θ-notation
BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)

13. Θ-notation
BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)

14. Using O-notation for Describing Running Times
O-notation is used to bound worst-case running times
bounds running time on arbitrary inputs as well
O(n2) bound on worst-case running time of insertion sort also applies to its running time on every input
What we really mean by “running time of insertion sort is O(n2)”
worst-case running time of insertion sort is O(n2)
or equivalently
no matter what particular input of size n is chosen (for each value of) running time on that set of inputs is O(n2)
BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)

15. Using Ω-notation for Describing Running Times
Ω(n) is used to bound the best-case running times
bounds the running time on arbitrary inputs as well
e.g., Ω(n) bound on best-case running time of insertion sort
running time of insertion sort is Ω(n)
“running time of an algorithm is Ω(g(n))” means
no matter what particular input of size n is chosen (for any n), running time on that set of inputs is at least a constant times g(n), for sufficiently large n
however, it is not contradictory to say “worst-case running time of insertion sort is Ω(n2)” since there exists an input that causes algorithm to take Ω(n2) time
BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)

16. Using Θ-notation for Describing Running Times
Case 1.
used to bound worst-case & best-case running times of an algorithm if they are not asymptotically equal
Case 2.
used to bound running time of an algorithm if its worst & best case running times are asymptotically equal
BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)

17. Using Θ-notation for Describing Running Times
Case (1)
a Θ-bound on worst-/best-case running time does not apply to its running time on arbitrary inputs
e.g., Θ(n2) bound on worst-case running time of insertion sort does not imply a Θ(n2) bound on running time of insertion sort on every input since T(n) = O(n2) & T(n) = Ω(n) for insertion sort
BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)

18. Using Θ-notation for Describing Running Times
Case (2)
implies a Θ-bound on every input
e.g., merge sort
T(n) = O(nlgn)
T(n) = Ω(nlgn)
 T(n) = Θ(nlgn)
BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)

19. Asymptotic Notation In Equations
BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)

20. Asymptotic notation appears on LHS of an equation
BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)

21. Other asymptotic notations – o-notation (small o)
BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)

22. o-notation
BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)

23. ω-notation (small omega notation)
BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)

24. Asymptotic comparison of functions
BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)

25. Analogy to the comparison of two real numbers
BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)

26. Analogy to the comparison of two real numbers
BIL741: Advanced Analysis of Algorithms I (İleri Algoritma Çözümleme I)