Binary search design and ana algorithm.pptx
- 1. 1 ASYMPTOTICS NOTATIONS O-notation (Big Oh) Asymptotic Upper Bound For a given function g(n), we denote O(g(n)) as the set of functions: O(g(n)) = { f(n)| there exists positive constants c and n0 such that 0 ≤ f(n) ≤ c g(n) for all n ≥ n0 } It is used to represent the worst case growth of an algorithm in time or a data structure in space when they are dependent on n, where n is big enough Example : 8/10/2021
- 2. 2 Big Oh - Example f(n) = n2 + 5n = O(n2) g(n) = n2 ……… c = 2 n n2 + 5n 2n2 1 5 2 2 14 8 5 50 50 f(n) <= c g(n) for all n> = n0 where c=2 & n0 =5 8/10/2021
- 3. 3 Big Oh - Example Let f(N) = 2N2 . Then ◦ f(N) = O(N4 ) (loose bound) ◦ f(N) = O(N3 ) (loose bound) ◦ f(N) = O(N2 ) (It is the best answer and the bound is asymptotically tight.) O(N2 ): reads “order N-squared” or “Big-Oh N-squared”. 8/10/2021
- 4. 4 Big Oh - Example Prove that if T(n) = 15n3 + n2 + 4, T(n) = O(n3 ). Proof. Let c = 20 and n0 = 1. Must show that 0 ≤ f (n) and f (n) ≤ cg(n). 0≤15n3 + n2 + 4 for all n ≥ n0 = 1. f (n) = 15n3 + n2 + 4 ≤ 20n3 (20g(n) = cg(n)) As per defination of Big O, hence T(n) = O(n3 ) 8/10/2021
- 5. 5 ASYMPTOTICS NOTATIONS Ω-notation Asymptotic lower bound Ω (g(n)) represents a set of functions such that: Ω(g(n)) = {f(n): there exist positive constants c and n0 such that 0 ≤ c g(n) ≤ f(n) for all n≥ n0} 8/10/2021
- 6. 6 Big Omega - Example Example 1 : f(n) = n2 + 5n g(n) = n2 ……… c = 1 n n2 + 5n c*n2 1 5 1 2 14 4 5 50 25 f(n) >= c g(n) for all n> = n0 where c=1 & n0 =1 Example 1 : Prove that if T(n) = 15n3 + n2 + 4, T(n) = Ω (n3 ). Proof. Let c = 15 and n0 = 1. Must show that 0 ≤ cg(n) and cg(n) ≤ f (n). 0 ≤ 15n3 for all n ≥ n0 = 1. cg(n) = 15n3 ≤ 15n3 + n2 + 4 = f (n) 8/10/2021
- 7. 7 ASYMPTOTICS NOTATIONS Θ-notation Asymptotic tight bound Θ (g(n)) represents a set of functions such that: Θ (g(n)) = {f(n): there exist positive constants c1, c2, and n0such that 0 ≤ c1g(n) ≤ f(n) ≤ c2g(n) for all n≥ n0 } 8/10/2021
- 8. 8 Theta Example f(N) = Θ(g(N)) iff f(N) = O(g(N)) and f(N) = Ω(g(N)) It can be read as “f(N) has order exactly g(N)”. The growth rate of f(N) equals the growth rate of g(N). The growth rate of f(N) is the same as the growth rate of g(N) for large N. Theta means the bound is the tightest possible. If T(N) is a polynomial of degree k, T(N) = Θ(Nk ). For logarithmic functions, T(logm N) = Θ(log N). 8/10/2021
- 9. 9 o notation (Little oh) The function f(n) = o(g(n)) iff o(g(n)) = {f(n): c > 0, n0 > 0 such that n n0, we have 0 f(n) < cg(n)}. Example: The function 3n + 2 = o(n2 ) as 0 ) ( ) ( lim n g n f n 0 2 3 lim 2 n n n 8/10/2021
- 10. 10 ω notation (Little omega) The function f(n) = ω(g(n)) iff w(g(n)) = {f(n): c > 0, n0 > 0 such that n n0, we have 0 cg(n) < f(n)}. 0 ) ( ) ( lim n f n g n 8/10/2021
- 11. 11 Asymptotic Notations It is a way to compare “sizes” of functions O-notation ------------------Less than equal to (“≤”) Θ-notation ------------------Equal to (“=“) Ω-notation ------------------Greater than equal to (“≥”) O ≈ ≤ Ω ≈ ≥ Θ ≈ = o ≈ < ω ≈ > 8/10/2021
- 12. 12 Examples On Asymptotic Notations Explain How is f(x) = 4n^2 – 5n + 3 is O(n^2) Show that 1) 30n+8 is O(n) 2) 100n + 5 ≠ (n2 ) 3) 5n2 = (n) 4) 100n + 5 = O(n2 ) 5) n2 /2 –n/2 = (n2 ) 8/10/2021
- 13. 13 Algorithm Design Techniques/Strategies Brute force Divide and conquer Space and time tradeoffs Greedy approach Dynamic programming Backtracking Branch and bound 8/10/2021
- 14. 14 Methods for Solving recurrences Substitution Method We guess a bound and then use mathematical induction to prove our guess correct. Recursion Tree Convert recurrence into tree Master Method 8/10/2021
- 15. 15 Math You need to Review 8/10/2021
- 19. 19 Master Theorem Master method provides a “cookbook” method for solving recurrences of the following form T(n) = a T(n/b) + f(n) where a 1, b > 1. If f(n) is asymptotically positive function. T(n) has following asymptotic bounds: There are 3 cases: 8/10/2021