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O log n in Data Structures Best Guide

Here is complete Guide on O log n in Data Structures, a very important topic to learn about Asymptotic Notations in Algorithms.

O log n in Data Structures

Big O notation

  • Big O is a mathematical notation describing constraints within any function when it reaches a certain value, or infinite. It’s about infinity.
  • Big O are members of a notation group formed by Paul Bachman. The process can be described in computer science by classification based upon ‘running time or space needs’ when input volume increases.
  • In analytic number theory big O notation may be used for expressing the boundary between arithmetic function and understanding.
  • The letter O is employed in order that growth is called the order of the function.

Tell me the meaning of O(log n)?

  • Logarithmic running time (O(log n)) means that the running time grows in proportion to the logarithm of the input size.

Properties

  • If f(n) = Θ(g(n)), then there exists positive constants c1, c2, n0 such that 0 ≤ c1.g(n) ≤ f(n) ≤ c2.g(n), for all n ≥ n0
  • If f(n) = O(g(n)), then there exists positive constants c, n0 such that 0 ≤ f(n) ≤ c.g(n), for all n ≥ n0
  • If f(n) = Ω(g(n)), then there exists positive constants c, n0 such that 0 ≤ c.g(n) ≤ f(n), for all n ≥ n0
  • If f(n) = o(g(n)), then there exists positive constants c, n0 such that 0 ≤ f(n) < c.g(n), for all n ≥ n0
  • If f(n) = ω(g(n)), then there exists positive constants c, n0 such that 0 ≤ c.g(n) < f(n), for all n ≥ n0

Orders of common functions

NotationMeaning
T(n) = O(f(n))Asymptotically, T does not grow faster than f.
T(n) = Omega(f(n))Asymptotically, T does not grow slower than f.
T(n) = Theta(f(n)),when T(n) = O(f(n)) and T(n) = Omega(f(n))Asymptotically, T grows equally fast as f.
T(n) = o(f(n)),means that lim_{n ->infinity}T(n)/f(n)=0Asymptotically, T grows slower than f.
T(n) = omega(f(n)),means that lim_{n ->infinity} T(n)/f(n)=infty.Asymptotically, T grows faster than f.
Table: Standard notations and their meanings

Related Asymptotic Notations

  1. O (Big Oh),
  2. Ω (Big omega),
  3. ϴ (Big theta).

Little-o notation

  • Little o notation is used to describe an upper bound that cannot be tight.

Asymptotic Notations:

  • Asymptotic notations are languages that allow us to analyze an algorithm’s running time by identifying its behavior as the input size for the algorithm increases.

Big Omega Notation:

  • Big Omega(Ω) function is used in computer science to describe the performance or complexity of an algorithm. If a running time is Ω(f(n)), then for large enough n, the running time is at least k⋅f(n) for some constant k.

Other arithmetic operators

TerminologyMeaning
f(n) = O(1)f is constant.
f(n) = O(log n)f is logarithmic.
f(n) = O(n)f is linear.
f(n) = O(n^2)f is quadratic.
f(n) = O(n^3)f is cubic.
f(n) = O(a^{g(n)}), for some a > 0 and
g(n) = omega(log n),
sometimes it is understood that
g(n) = Omega(n)
f is exponential
f(n) = O(n^a), for some constant a < infinityf is polynomial.
f(n) = o(n)f is sublinear.

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