Lecture

Complexity of Algorithms: Advanced Big-O Facts

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In course

CS-101: Advanced information, computation, communication I

Discrete mathematics is a discipline with applications to almost all areas of study. It provides a set of indispensable tools to computer science in particular. This course reviews (familiar) topics as diverse as mathematical reasoning, combinatorics, disc ...

Description

This lecture covers advanced big-O facts, including big-O estimates for the factorial function, combinations of functions, and the big-O for powers, logarithms, and factorials. It also discusses the big-O for the sum and product of functions.

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Instructor

Karl Aberer

Karl Aberer received his PhD in mathematics in 1991 from the ETH Zürich. From 1991 to 1992 he was postdoctoral fellow at the International Computer Science Institute (ICSI) at the University of California, Berkeley. In 1992, he joined the Integrated Publication and Information Systems institute (IPSI) of GMD in Germany, where he was leading the research division Open Adaptive Information Management Systems. In 2000 he joined EPFL as full professor. Since 2005 he is the director of the Swiss National Research Center for Mobile Information and Communication Systems ( NCCR-MICS, www.mics.ch ). He is member of the editorial boards of VLDB Journal, ACM Transaction on Autonomous and Adaptive Systems and World Wide Web Journal. He has been consulting for the Swiss government in research and science policy as a member of the Swiss Research and Technology Council ( SWTR ) from 2003 - 2011.

Official source

Ontological neighbourhood

Mathematics

Analysis: Complex analysis

Related concepts (32)

Logarithm

In mathematics, the logarithm is the inverse function to exponentiation. That means that the logarithm of a number x to the base b is the exponent to which b must be raised to produce x. For example, since 1000 = 103, the logarithm base 10 of 1000 is 3, or log10 (1000) = 3. The logarithm of x to base b is denoted as logb (x), or without parentheses, logb x, or even without the explicit base, log x, when no confusion is possible, or when the base does not matter such as in big O notation.

Natural logarithm

The natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, loge x, or sometimes, if the base e is implicit, simply log x. Parentheses are sometimes added for clarity, giving ln(x), loge(x), or log(x). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.

Common logarithm

In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered its use, as well as standard logarithm. Historically, it was known as logarithmus decimalis or logarithmus decadis. It is indicated by log(x), log10 (x), or sometimes Log(x) with a capital L (however, this notation is ambiguous, since it can also mean the complex natural logarithmic multi-valued function).

Gamma function

In mathematics, the gamma function (represented by Γ, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer n, Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles.

Double factorial

In mathematics, the double factorial of a number n, denoted by n!!, is the product of all the positive integers up to n that have the same parity (odd or even) as n. That is, Restated, this says that for even n, the double factorial is while for odd n it is For example, 9!! = 9 × 7 × 5 × 3 × 1 = 945. The zero double factorial 0!! = 1 as an empty product. The sequence of double factorials for even n = 0, 2, 4, 6, 8,... starts as The sequence of double factorials for odd n = 1, 3, 5, 7, 9,...

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