Algorithm big o

Algorithm big o

• 1.
  Unit 2.1 Big ONotation
• 2.
  Intro • When solvinga computer science problem there will usually be more than just one solution. • These solutions will often be in the form of different algorithms, and you will generally want to compare the algorithms to see which one is more efficient. • This is where Big O analysis helps – it gives us some basis for measuring the efficiency of an algorithm. • Big O measures the efficiency of an algorithm based on the time it takes for the algorithm to run as a function of the input size. Ashim Lamichhane 2
• 3.
  Asymptotic notation • Forexample, suppose that an algorithm, running on an input of size n, takes 6n2 + 100n + 300 machine instructions. • The 6n2 term becomes larger than the remaining terms, 100 n + 300, once n becomes large enough, 20 in this case. • Here's a chart showing values of 6n2 and 100n + 300 for values of n from 0 to 100: • We could say “f(n) grows at the order Of n2”. And write f(n)=O(n2) Ashim Lamichhane 3
• 4.
  Asymptotic notation • Bydropping the less significant terms and the constant coefficients, we can focus on the important part of an algorithm's running time. • When we drop the constant coefficients and the less significant terms, we use asymptotic notation. • We'll see three forms of it: big-Θ notation, big-O notation, and big-Ω notation. Ashim Lamichhane 4
• 5.
  Big-O Notation Ashim Lamichhane5 • A function f(x)=O(g(x)) (read as f(x) is big oh of g(x) ) iff there exists two positive constants c and x0 such that for all x >= x0, f(x) <= c*g(x) • Consider an example: f(x)=5x3+3x2+4 find big O of f(x) solution: f(x)= 5x3+3x2+4 we can exclude the lower order polynomial so, f(x)=5x3 since, f(x)<c*g(x), where c=5 and x=3 Thus by definition of big oh O(f(x))=O(x3)
• 6.
  • We use"big-O" notation for occasions such as "the running time grows at most this much, but it could grow more slowly.” Ashim Lamichhane 6
• 7.
  • We havea Fibonacci algorithm. • We're just going to run through it operation by operation and ask how long it takes. Ashim Lamichhane 7
• 8.
  Ashim Lamichhane 8
• 9.
  Big Omega (Ω)notation • Sometimes, we want to say that an algorithm takes at least a certain amount of time, without providing an upper bound. • We use big-Ω notation; that's the Greek letter "omega.” • A function f(x) =Ω (g(x)) (read as f(x) is big omega of g(x) ) iff there exists two positive constants c and x0 such that for all x >= x0, 0 <= c*g(x) <= f(x). • Therefore, f(x)>=c*g(x) • The above relation says that g(x) is a lower bound of f(x). Ashim Lamichhane 9
• 10.
  Big Omega (Ω)notation Ashim Lamichhane 10
• 11.
  Big Theta (Θ)notation • When we need asymptotically tight bound then we use notation. • A function f(x) = (g(x)) (read as f(x) is big theta of g(x) ) iff there exists three positive constants c1, c2 and x0 such that for all x >= x0, c1*g(x) <= f(x) <= c2*g(x) Ashim Lamichhane 11
• 12.
  Assignment • https://github.com/ashim888/dataStructureAndAlgorithm/tree/master/Assignme nts/assignment_4 NOTE: beforesubmitting please do check • https://github.com/ashim888/dataStructureAndAlgorithm/tree/master/Assignme nts Ashim Lamichhane 12