Master the Art of Asymptotic Notation

Understanding Big O Notation

• Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity.

• It is part of a family of notations called asymptotic notation and was invented by Paul Bachmann, Edmund Landau, and others.

• Big O is used in computer science to classify algorithms according to how their run time or space requirements grow as the input size grows.

• It characterizes functions according to their growth rates and different functions with the same asymptotic growth rate may be represented using the same O notation.

• Big O notation usually only provides an upper bound on the growth rate of a function and is associated with several related notations, using the symbols o, Ω, ω, and Θ, to describe other kinds of bounds on asymptotic growth rates.

• The notation can also be used to describe the behavior of a function near some real number.

• In computer science, a slightly more restrictive definition is common where both the function to be estimated and the comparison function are defined on some unbounded subset of the positive integers.

• Big O notation is useful when analyzing algorithms for efficiency and can be used to describe the error term in an approximation to a mathematical function.

• The sets O(nc) and O(cn) are very different and if a function may be bounded by a polynomial in n, then as n tends to infinity, one may disregard lower-order terms of the polynomial.

• Changing units may or may not affect the order of the resulting algorithm and changing variables may also affect the order of the resulting algorithm.

• Big O (and little o, Ω, etc.) can also be used with multiple variables and the subset on which a function is defined is significant when generalizing statements from the univariate setting to the multivariate setting.

• Big O notation is an important tool for algorithm analysis and is widely used in computer science, mathematics, and other fields to provide estimates of growth rates and describe the limiting behavior of functions.Understanding Big O Notation

• Big O notation is a mathematical notation used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity.

• It is widely used in computer science to describe the time complexity of an algorithm, and to compare the efficiency of different algorithms.

• Big O notation is written as O(f(n)), where f(n) is a function that describes the growth rate of the algorithm's running time as the input size n increases.

• The notation is used to describe the upper bound of a function's growth rate, or the maximum amount of time an algorithm will take to run.

• O(f(n)) notation can be extended to multivariate functions, but there is some inconsistency in the choice of definition.

• The equals sign used in the notation is considered by some to be an abuse of notation, and set notation is a more precise way to express it.

• Big O notation can be used in conjunction with other arithmetic operators, such as addition and subtraction.

• It can be used to express the time complexity of an algorithm in terms of the number of elements in the input set.

• Big O notation can appear in different places in an equation, even several times on each side, and the meaning of such statements is that the class of functions represented by the left side is a subset of the class of functions represented by the right side.

• Little-o notation is a related notation that describes the growth rate of a function as being much smaller than another function.

• Big Omega notation is another asymptotic notation that describes the lower bound of a function's growth rate.

• There are several related asymptotic notations in the family of Bachmann-Landau notations, including big Theta, little omega, and soft-O notation.Notation for Asymptotic Analysis

• O notation was introduced by Paul Bachmann in 1894 and later adopted by Edmund Landau. It is used in asymptotic analysis to describe the upper bound of a function.

• Similarly, o notation was introduced by Landau in 1909 to describe the lower bound of a function.

• The symbol Ω was introduced by Hardy and Littlewood in 1914 to mean "is not an o of." They also introduced symbols ΩR and ΩL in 1916, which later became Ω+ and Ω-, respectively.

• The big O notation is often used to ignore logarithmic factors because the growth-rate effects of some other super-logarithmic function indicate a growth-rate explosion for large-sized input parameters that is more important to predicting bad run-time performance than the finer-point effects contributed by the logarithmic-growth factor(s).

• The L notation is used for functions that are between polynomial and exponential in terms of ln n.

• The o notation can be used to define derivatives and differentiability in quite general spaces, and also (asymptotical) equivalence of functions.

• The Omega symbols (with their original meanings) are sometimes also referred to as "Landau symbols".

• Donald Knuth popularized the big O notation in computer science in the 1970s and introduced the related Theta notation.

• Landau never used the big Theta and small omega symbols, and Hardy's notation is not used anymore.

• Ivan Matveyevich Vinogradov introduced his notation ≪, which has been increasingly used in number theory instead of the O notation.

• The big-O originally stands for "order of" and is thus a Latin letter. Neither Bachmann nor Landau ever call it "Omicron".

• The digit zero should not be used with the O notation.

Understanding Big O Notation

• Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity.

• It is part of a family of notations called asymptotic notation and was invented by Paul Bachmann, Edmund Landau, and others.

• Big O is used in computer science to classify algorithms according to how their run time or space requirements grow as the input size grows.

• It characterizes functions according to their growth rates and different functions with the same asymptotic growth rate may be represented using the same O notation.

• Big O notation usually only provides an upper bound on the growth rate of a function and is associated with several related notations, using the symbols o, Ω, ω, and Θ, to describe other kinds of bounds on asymptotic growth rates.

• The notation can also be used to describe the behavior of a function near some real number.

• In computer science, a slightly more restrictive definition is common where both the function to be estimated and the comparison function are defined on some unbounded subset of the positive integers.

• Big O notation is useful when analyzing algorithms for efficiency and can be used to describe the error term in an approximation to a mathematical function.

• The sets O(nc) and O(cn) are very different and if a function may be bounded by a polynomial in n, then as n tends to infinity, one may disregard lower-order terms of the polynomial.

• Changing units may or may not affect the order of the resulting algorithm and changing variables may also affect the order of the resulting algorithm.

• Big O (and little o, Ω, etc.) can also be used with multiple variables and the subset on which a function is defined is significant when generalizing statements from the univariate setting to the multivariate setting.

• Big O notation is an important tool for algorithm analysis and is widely used in computer science, mathematics, and other fields to provide estimates of growth rates and describe the limiting behavior of functions.Understanding Big O Notation

• Big O notation is a mathematical notation used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity.

• It is widely used in computer science to describe the time complexity of an algorithm, and to compare the efficiency of different algorithms.

• Big O notation is written as O(f(n)), where f(n) is a function that describes the growth rate of the algorithm's running time as the input size n increases.

• The notation is used to describe the upper bound of a function's growth rate, or the maximum amount of time an algorithm will take to run.

• O(f(n)) notation can be extended to multivariate functions, but there is some inconsistency in the choice of definition.

• The equals sign used in the notation is considered by some to be an abuse of notation, and set notation is a more precise way to express it.

• Big O notation can be used in conjunction with other arithmetic operators, such as addition and subtraction.

• It can be used to express the time complexity of an algorithm in terms of the number of elements in the input set.

• Big O notation can appear in different places in an equation, even several times on each side, and the meaning of such statements is that the class of functions represented by the left side is a subset of the class of functions represented by the right side.

• Little-o notation is a related notation that describes the growth rate of a function as being much smaller than another function.

• Big Omega notation is another asymptotic notation that describes the lower bound of a function's growth rate.

• There are several related asymptotic notations in the family of Bachmann-Landau notations, including big Theta, little omega, and soft-O notation.Notation for Asymptotic Analysis

• O notation was introduced by Paul Bachmann in 1894 and later adopted by Edmund Landau. It is used in asymptotic analysis to describe the upper bound of a function.

• Similarly, o notation was introduced by Landau in 1909 to describe the lower bound of a function.

• The symbol Ω was introduced by Hardy and Littlewood in 1914 to mean "is not an o of." They also introduced symbols ΩR and ΩL in 1916, which later became Ω+ and Ω-, respectively.

• The big O notation is often used to ignore logarithmic factors because the growth-rate effects of some other super-logarithmic function indicate a growth-rate explosion for large-sized input parameters that is more important to predicting bad run-time performance than the finer-point effects contributed by the logarithmic-growth factor(s).

• The L notation is used for functions that are between polynomial and exponential in terms of ln n.

• The o notation can be used to define derivatives and differentiability in quite general spaces, and also (asymptotical) equivalence of functions.

• The Omega symbols (with their original meanings) are sometimes also referred to as "Landau symbols".

• Donald Knuth popularized the big O notation in computer science in the 1970s and introduced the related Theta notation.

• Landau never used the big Theta and small omega symbols, and Hardy's notation is not used anymore.

• Ivan Matveyevich Vinogradov introduced his notation ≪, which has been increasingly used in number theory instead of the O notation.

• The big-O originally stands for "order of" and is thus a Latin letter. Neither Bachmann nor Landau ever call it "Omicron".

• The digit zero should not be used with the O notation.