Big O notation

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https://en.wikipedia.org/wiki/Big_O_notation 

Example of Big O notation: ${\displaystyle f(x)\in O(g(x))}$ as there exists ${\displaystyle M>0}$ (e.g., ${\displaystyle M=1}$) and ${\displaystyle x_0}$ (e.g.,${\displaystyle x_0=5}$) such that ${\displaystyle f(x)\le Mg(x)}$ whenever ${\displaystyle x\ge x_0}$.

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. Big O is a member of a family of notations invented by German mathematicians Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for Ordnung, meaning the order of approximation.

In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates.

Big O notation characterizes functions according to their growth rates: different functions with the same asymptotic growth rate may be represented using the same O notation. The letter O is used because the growth rate of a function is also referred to as the order of the function. A description of a function in terms of big O notation usually only provides an upper bound on the growth rate of the function.

Associated with big O notation are several related notations, using the symbols o, Ω, ω, and Θ, to describe other kinds of bounds on asymptotic growth rates.

Formal definition

Let ${\displaystyle f}$, the function to be estimated, be a real or complex valued function and let ${\displaystyle g}$, the comparison function, be a real valued function. Let both functions be defined on some unbounded subset of the positive real numbers, and ${\displaystyle g(x)}$ be strictly positive for all large enough values of ${\displaystyle x}$. One writes

$${\displaystyle f(x)=O(g(x))\text{ as }x\to \mathrm{\infty }}$$

and it is read "${\displaystyle f(x)}$ is big O of ${\displaystyle g(x)}$" if the absolute value of ${\displaystyle f(x)}$ is at most a positive constant multiple of ${\displaystyle g(x)}$ for all sufficiently large values of ${\displaystyle x}$. That is, ${\displaystyle f(x)=O(g(x))}$ if there exists a positive real number ${\displaystyle M}$ and a real number ${\displaystyle x_0}$ such that

$${\displaystyle |f(x)|\le Mg(x)\text{ for all }x\ge x_0.}$$

In many contexts, the assumption that we are interested in the growth rate as the variable ${\displaystyle x}$ goes to infinity is left unstated, and one writes more simply that

$${\displaystyle f(x)=O(g(x)).}$$

The notation can also be used to describe the behavior of ${\displaystyle f}$ near some real number ${\displaystyle a}$ (often, ${\displaystyle a=0}$): we say

$${\displaystyle f(x)=O(g(x))\text{ as }x\to a}$$

if there exist positive numbers ${\displaystyle \delta }$ and ${\displaystyle M}$ such that for all defined ${\displaystyle x}$ with ${\displaystyle 0<|x-a|<\delta }$,

$${\displaystyle |f(x)|\le Mg(x).}$$

As ${\displaystyle g(x)}$ is chosen to be strictly positive for such values of ${\displaystyle x}$, both of these definitions can be unified using the limit superior:

$${\displaystyle f(x)=O(g(x))\text{ as }x\to a}$$

if

$${\displaystyle \underset{x\to a}{lim sup}\frac{\left|f(x)\right|}{g(x)}<\mathrm{\infty }.}$$

And in both of these definitions the limit point ${\displaystyle a}$ (whether ${\displaystyle \mathrm{\infty }}$ or not) is a cluster point of the domains of ${\displaystyle f}$ and ${\displaystyle g}$, i. e., in every neighbourhood of ${\displaystyle a}$ there have to be infinitely many points in common. Moreover, as pointed out in the article about the limit inferior and limit superior, the ${\displaystyle \underset{x\to a}{lim sup}}$ (at least on the extended real number line) always exists.

In computer science, a slightly more restrictive definition is common: ${\displaystyle f}$ and ${\displaystyle g}$ are both required to be functions from some unbounded subset of the positive integers to the nonnegative real numbers; then ${\displaystyle f(x)=O(g(x))}$ if there exist positive integer numbers ${\displaystyle M}$ and ${\displaystyle n_0}$ such that ${\displaystyle f(n)\le Mg(n)}$ for all ${\displaystyle n\ge n_0}$.

Example

In typical usage the O notation is asymptotical, that is, it refers to very large x. In this setting, the contribution of the terms that grow "most quickly" will eventually make the other ones irrelevant. As a result, the following simplification rules can be applied:

• If f(x) is a sum of several terms, if there is one with largest growth rate, it can be kept, and all others omitted.
• If f(x) is a product of several factors, any constants (factors in the product that do not depend on x) can be omitted.

For example, let f(x) = 6x⁴ − 2x³ + 5, and suppose we wish to simplify this function, using O notation, to describe its growth rate as x approaches infinity. This function is the sum of three terms: 6x⁴, −2x³, and 5. Of these three terms, the one with the highest growth rate is the one with the largest exponent as a function of x, namely 6x⁴. Now one may apply the second rule: 6x⁴ is a product of 6 and x⁴ in which the first factor does not depend on x. Omitting this factor results in the simplified form x⁴. Thus, we say that f(x) is a "big O" of x⁴. Mathematically, we can write f(x) = O(x⁴). One may confirm this calculation using the formal definition: let f(x) = 6x⁴ − 2x³ + 5 and g(x) = x⁴. Applying the formal definition from above, the statement that f(x) = O(x⁴) is equivalent to its expansion,

$${\displaystyle |f(x)|\le M{x}^4}$$

for some suitable choice of a real number x₀ and a positive real number M and for all x > x₀. To prove this, let x₀ = 1 and M = 13. Then, for all x > x₀:

$${\displaystyle \begin{array}{rl}|6x^4-2x^3+5|& \le 6x^4+|2x^3|+5\\ & \le 6x^4+2x^4+5x^4\\ & =13x^4\end{array}}$$

so

$${\displaystyle |6x^4-2x^3+5|\le 13x^4.}$$

Usage

Big O notation has two main areas of application:

• In mathematics, it is commonly used to describe how closely a finite series approximates a given function, especially in the case of a truncated Taylor series or asymptotic expansion
• In computer science, it is useful in the analysis of algorithms

In both applications, the function g(x) appearing within the O(·) is typically chosen to be as simple as possible, omitting constant factors and lower order terms.

There are two formally close, but noticeably different, usages of this notation:

• infinite asymptotics
• infinitesimal asymptotics.

This distinction is only in application and not in principle, however—the formal definition for the "big O" is the same for both cases, only with different limits for the function argument.

Infinite asymptotics

Graphs of functions commonly used in the analysis of algorithms, showing the number of operations N versus input size n for each function

Big O notation is useful when analyzing algorithms for efficiency. For example, the time (or the number of steps) it takes to complete a problem of size n might be found to be T(n) = 4n² − 2n + 2. As n grows large, the n² term will come to dominate, so that all other terms can be neglected—for instance when n = 500, the term 4n² is 1000 times as large as the 2n term. Ignoring the latter would have negligible effect on the expression's value for most purposes. Further, the coefficients become irrelevant if we compare to any other order of expression, such as an expression containing a term n³ or n⁴. Even if T(n) = 1,000,000n², if U(n) = n³, the latter will always exceed the former once n grows larger than 1,000,000 (T(1,000,000) = 1,000,000³ = U(1,000,000)). Additionally, the number of steps depends on the details of the machine model on which the algorithm runs, but different types of machines typically vary by only a constant factor in the number of steps needed to execute an algorithm. So the big O notation captures what remains: we write either

${\displaystyle T(n)=O(n^2)}$

or

${\displaystyle T(n)\in O(n^2)}$

and say that the algorithm has order of n² time complexity. The sign "=" is not meant to express "is equal to" in its normal mathematical sense, but rather a more colloquial "is", so the second expression is sometimes considered more accurate (see the "Equals sign" discussion below) while the first is considered by some as an abuse of notation.

Infinitesimal asymptotics

Big O can also be used to describe the error term in an approximation to a mathematical function. The most significant terms are written explicitly, and then the least-significant terms are summarized in a single big O term. Consider, for example, the exponential series and two expressions of it that are valid when x is small:

${\displaystyle \begin{array}{rlr}{e}^x& =1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots & \text{for all }x\\ & =1+x+\frac{x^2}{2}+O(x^3)& \text{as }x\to 0\\ & =1+x+O(x^2)& \text{as }x\to 0\end{array}}$

The second expression (the one with O(x³)) means the absolute-value of the error e^x − (1 + x + x²/2) is at most some constant times |x³| when x is close enough to 0.

Properties

If the function f can be written as a finite sum of other functions, then the fastest growing one determines the order of f(n). For example,

${\displaystyle f(n)=9\log n+5(\log n{)}^4+3n^2+2n^3=O(n^3)\text{as }n\to \mathrm{\infty }.}$

In particular, 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. The sets O(n^c) and O(cⁿ) are very different. If c is greater than one, then the latter grows much faster. A function that grows faster than n^c for any c is called superpolynomial. One that grows more slowly than any exponential function of the form cⁿ is called subexponential. An algorithm can require time that is both superpolynomial and subexponential; examples of this include the fastest known algorithms for integer factorization and the function n^{log n}.

We may ignore any powers of n inside of the logarithms. The set O(log n) is exactly the same as O(log(n^c)). The logarithms differ only by a constant factor (since log(n^c) = c log n) and thus the big O notation ignores that. Similarly, logs with different constant bases are equivalent. On the other hand, exponentials with different bases are not of the same order. For example, 2ⁿ and 3ⁿ are not of the same order.

Changing units may or may not affect the order of the resulting algorithm. Changing units is equivalent to multiplying the appropriate variable by a constant wherever it appears. For example, if an algorithm runs in the order of n², replacing n by cn means the algorithm runs in the order of c²n², and the big O notation ignores the constant c². This can be written as c²n² = O(n²). If, however, an algorithm runs in the order of 2ⁿ, replacing n with cn gives 2^cn = (2^c)ⁿ. This is not equivalent to 2ⁿ in general. Changing variables may also affect the order of the resulting algorithm. For example, if an algorithm's run time is O(n) when measured in terms of the number n of digits of an input number x, then its run time is O(log x) when measured as a function of the input number x itself, because n = O(log x).

Product

${\displaystyle f_1=O(g_1)\text{ and }{f}_2=O(g_2)\Rightarrow f_1{f}_2=O(g_1{g}_2)}$

${\displaystyle f\cdot O(g)=O(fg)}$

Sum

If ${\displaystyle f_1=O(g_1)}$ and ${\displaystyle f_2=O(g_2)}$ then ${\displaystyle f_1+f_2=O(max(g_1,g_2))}$. It follows that if ${\displaystyle f_1=O(g)}$ and ${\displaystyle f_2=O(g)}$ then ${\displaystyle f_1+f_2\in O(g)}$. In other words, this second statement says that ${\displaystyle O(g)}$ is a convex cone.

Multiplication by a constant

Let k be a nonzero constant. Then ${\displaystyle O(|k|\cdot g)=O(g)}$. In other words, if ${\displaystyle f=O(g)}$, then ${\displaystyle k\cdot f=O(g).}$

Multiple variables

Big O (and little o, Ω, etc.) can also be used with multiple variables. To define big O formally for multiple variables, suppose ${\displaystyle f}$ and ${\displaystyle g}$ are two functions defined on some subset of ${\displaystyle {\mathbb{R}}^n}$. We say

${\displaystyle f(\mathbf{x})\text{ is }O(g(\mathbf{x}))\text{ as }\mathbf{x}\to \mathrm{\infty }}$

if and only if there exist constants ${\displaystyle M}$ and ${\displaystyle C>0}$ such that ${\displaystyle |f(\mathbf{x})|\le C|g(\mathbf{x})|}$ for all ${\displaystyle \mathbf{x}}$ with ${\displaystyle x_i\ge M}$ for some ${\displaystyle i.}$ Equivalently, the condition that ${\displaystyle x_i\ge M}$ for some ${\displaystyle i}$ can be written ${\displaystyle \Vert \mathbf{x}{\Vert }_{\mathrm{\infty }}\ge M}$, where ${\displaystyle \Vert \mathbf{x}{\Vert }_{\mathrm{\infty }}}$ denotes the Chebyshev norm. For example, the statement

${\displaystyle f(n,m)=n^2+m^3+O(n+m)\text{ as }n,m\to \mathrm{\infty }}$

asserts that there exist constants C and M such that

${\displaystyle |f(n,m)-(n^2+m^3)|\le C|n+m|}$

whenever either ${\displaystyle m\ge M}$ or ${\displaystyle n\ge M}$ holds. This definition allows all of the coordinates of ${\displaystyle \mathbf{x}}$ to increase to infinity. In particular, the statement

${\displaystyle f(n,m)=O(n^m)\text{ as }n,m\to \mathrm{\infty }}$

(i.e., ${\displaystyle \mathrm{\exists }C\mathrm{\exists }M\mathrm{\forall }n\mathrm{\forall }m\cdots }$) is quite different from

${\displaystyle \mathrm{\forall }m:f(n,m)=O(n^m)\text{ as }n\to \mathrm{\infty }}$

(i.e., ${\displaystyle \mathrm{\forall }m\mathrm{\exists }C\mathrm{\exists }M\mathrm{\forall }n\cdots }$).

Under this definition, the subset on which a function is defined is significant when generalizing statements from the univariate setting to the multivariate setting. For example, if ${\displaystyle f(n,m)=1}$ and ${\displaystyle g(n,m)=n}$, then ${\displaystyle f(n,m)=O(g(n,m))}$ if we restrict ${\displaystyle f}$ and ${\displaystyle g}$ to ${\displaystyle [1,\mathrm{\infty }{)}^2}$, but not if they are defined on ${\displaystyle [0,\mathrm{\infty }{)}^2}$.

This is not the only generalization of big O to multivariate functions, and in practice, there is some inconsistency in the choice of definition.

Matters of notation

Equals sign

The statement "f(x) is O(g(x))" as defined above is usually written as f(x) = O(g(x)). Some consider this to be an abuse of notation, since the use of the equals sign could be misleading as it suggests a symmetry that this statement does not have. As de Bruijn says, O(x) = O(x²) is true but O(x²) = O(x) is not. Knuth describes such statements as "one-way equalities", since if the sides could be reversed, "we could deduce ridiculous things like n = n² from the identities n = O(n²) and n² = O(n²)." In another letter, Knuth also pointed out that "the equality sign is not symmetric with respect to such notations", as, in this notation, "mathematicians customarily use the = sign as they use the word "is" in English: Aristotle is a man, but a man isn't necessarily Aristotle".

For these reasons, it would be more precise to use set notation and write f(x) ∈ O(g(x)) (read as: "f(x) is an element of O(g(x))", or "f(x) is in the set O(g(x))"), thinking of O(g(x)) as the class of all functions h(x) such that |h(x)| ≤ Cg(x) for some positive real number C. However, the use of the equals sign is customary.

Other arithmetic operators

Big O notation can also be used in conjunction with other arithmetic operators in more complicated equations. For example, h(x) + O(f(x)) denotes the collection of functions having the growth of h(x) plus a part whose growth is limited to that of f(x). Thus,

${\displaystyle g(x)=h(x)+O(f(x))}$

expresses the same as

${\displaystyle g(x)-h(x)=O(f(x)).}$

Example

Suppose an algorithm is being developed to operate on a set of n elements. Its developers are interested in finding a function T(n) that will express how long the algorithm will take to run (in some arbitrary measurement of time) in terms of the number of elements in the input set. The algorithm works by first calling a subroutine to sort the elements in the set and then perform its own operations. The sort has a known time complexity of O(n²), and after the subroutine runs the algorithm must take an additional 55n³ + 2n + 10 steps before it terminates. Thus the overall time complexity of the algorithm can be expressed as T(n) = 55n³ + O(n²). Here the terms 2n + 10 are subsumed within the faster-growing O(n²). Again, this usage disregards some of the formal meaning of the "=" symbol, but it does allow one to use the big O notation as a kind of convenient placeholder.

Multiple uses

In more complicated usage, O(·) can appear in different places in an equation, even several times on each side. For example, the following are true for ${\displaystyle n\to \mathrm{\infty }}$:

$${\displaystyle \begin{array}{rl}(n+1{)}^2& =n^2+O(n),\\ (n+O(n^{1/2}))\cdot (n+O(\log n){)}^2& =n^3+O(n^{5/2}),\\ n^{O(1)}& =O(e^n).\end{array}}$$

The meaning of such statements is as follows: for any functions which satisfy each O(·) on the left side, there are some functions satisfying each O(·) on the right side, such that substituting all these functions into the equation makes the two sides equal. For example, the third equation above means: "For any function f(n) = O(1), there is some function g(n) = O(eⁿ) such that n^f(n) = g(n)." In terms of the "set notation" above, the meaning is that the class of functions represented by the left side is a subset of the class of functions represented by the right side. In this use the "=" is a formal symbol that unlike the usual use of "=" is not a symmetric relation. Thus for example n^O(1) = O(eⁿ) does not imply the false statement O(eⁿ) = n^O(1).

Typesetting

Big O is typeset as an italicized uppercase "O", as in the following example: ${\displaystyle O(n^2)}$. In TeX, it is produced by simply typing O inside math mode. Unlike Greek-named Bachmann–Landau notations, it needs no special symbol. Yet, some authors use the calligraphic variant ${\displaystyle \mathcal{O}}$ instead.

Orders of common functions

Further information: Time complexity § Table of common time complexities

Here is a list of classes of functions that are commonly encountered when analyzing the running time of an algorithm. In each case, c is a positive constant and n increases without bound. The slower-growing functions are generally listed first.

Notation	Name	Example
${\displaystyle O(1)}$	constant	Determining if a binary number is even or odd; Calculating ${\displaystyle (-1{)}^n}$; Using a constant-size lookup table
${\displaystyle O(\log \log n)}$	double logarithmic	Average number of comparisons spent finding an item using interpolation search in a sorted array of uniformly distributed values
${\displaystyle O(\log n)}$	logarithmic	Finding an item in a sorted array with a binary search or a balanced search tree as well as all operations in a binomial heap
${\displaystyle O((\log n{)}^c)}$ $c>1$	polylogarithmic	Matrix chain ordering can be solved in polylogarithmic time on a parallel random-access machine.
${\displaystyle O(n^c)}$ $0<c<1$	fractional power	Searching in a k-d tree
${\displaystyle O(n)}$	linear	Finding an item in an unsorted list or in an unsorted array; adding two n-bit integers by ripple carry
${\displaystyle O(n{\log }^{\ast }n)}$	n log-star n	Performing triangulation of a simple polygon using Seidel's algorithm, where ${\displaystyle {\log }^{\ast }(n)=\{\begin{array}{ll}0,& \text{if }n\le 1\\ 1+{\log }^{\ast }(\log n),& \text{if }n>1\end{array}}$
${\displaystyle O(n\log n)=O(\log n!)}$	linearithmic, loglinear, quasilinear, or "n log n"	Performing a fast Fourier transform; fastest possible comparison sort; heapsort and merge sort
${\displaystyle O(n^2)}$	quadratic	Multiplying two n-digit numbers by schoolbook multiplication; simple sorting algorithms, such as bubble sort, selection sort and insertion sort; (worst-case) bound on some usually faster sorting algorithms such as quicksort, Shellsort, and tree sort
${\displaystyle O(n^c)}$	polynomial or algebraic	Tree-adjoining grammar parsing; maximum matching for bipartite graphs; finding the determinant with LU decomposition
${\displaystyle L_n[\alpha ,c]=e^{(c+o(1))(\ln n{)}^{\alpha }(\ln \ln n{)}^{1-\alpha }}}$ $0<\alpha <1$	L-notation or sub-exponential	Factoring a number using the quadratic sieve or number field sieve
${\displaystyle O(c^n)}$ $c>1$	exponential	Finding the (exact) solution to the travelling salesman problem using dynamic programming; determining if two logical statements are equivalent using brute-force search
${\displaystyle O(n!)}$	factorial	Solving the travelling salesman problem via brute-force search; generating all unrestricted permutations of a poset; finding the determinant with Laplace expansion; enumerating all partitions of a set

The statement ${\displaystyle f(n)=O(n!)}$ is sometimes weakened to ${\displaystyle f(n)=O\left(n^n\right)}$ to derive simpler formulas for asymptotic complexity. For any ${\displaystyle k>0}$ and ${\displaystyle c>0}$, ${\displaystyle O(n^c(\log n{)}^k)}$ is a subset of ${\displaystyle O(n^{c+\epsilon })}$ for any ${\displaystyle \epsilon >0}$, so may be considered as a polynomial with some bigger order.

Related asymptotic notations

Big O is widely used in computer science. Together with some other related notations, it forms the family of Bachmann–Landau notations.

Little-o notation

"Little o" redirects here. For the baseball player, see Omar Vizquel.

Intuitively, the assertion "f(x) is o(g(x))" (read "f(x) is little-o of g(x)") means that g(x) grows much faster than f(x). As before, let f be a real or complex valued function and g a real valued function, both defined on some unbounded subset of the positive real numbers, such that g(x) is strictly positive for all large enough values of x. One writes

${\displaystyle f(x)=o(g(x))\text{ as }x\to \mathrm{\infty }}$

if for every positive constant ε there exists a constant ${\displaystyle x_0}$ such that

${\displaystyle |f(x)|\le \epsilon g(x)\text{ for all }x\ge x_0.}$

For example, one has

${\displaystyle 2x=o(x^2)}$ and ${\displaystyle 1/x=o(1),}$     both as ${\displaystyle x\to \mathrm{\infty }.}$

The difference between the definition of the big-O notation and the definition of little-o is that while the former has to be true for at least one constant M, the latter must hold for every positive constant ε, however small. In this way, little-o notation makes a stronger statement than the corresponding big-O notation: every function that is little-o of g is also big-O of g, but not every function that is big-O of g is also little-o of g. For example, ${\displaystyle 2x^2=O(x^2)}$ but ${\displaystyle 2x^2\ne o(x^2)}$.

As g(x) is nonzero, or at least becomes nonzero beyond a certain point, the relation ${\displaystyle f(x)=o(g(x))}$ is equivalent to

${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{f(x)}{g(x)}=0}$ (and this is in fact how Landau originally defined the little-o notation).

Little-o respects a number of arithmetic operations. For example,

if c is a nonzero constant and ${\displaystyle f=o(g)}$ then ${\displaystyle c\cdot f=o(g)}$, and

if ${\displaystyle f=o(F)}$ and ${\displaystyle g=o(G)}$ then ${\displaystyle f\cdot g=o(F\cdot G).}$

It also satisfies a transitivity relation:

if ${\displaystyle f=o(g)}$ and ${\displaystyle g=o(h)}$ then ${\displaystyle f=o(h).}$

Big Omega notation

Another asymptotic notation is ${\displaystyle \mathrm{\Omega }}$, read "big omega". There are two widespread and incompatible definitions of the statement

${\displaystyle f(x)=\mathrm{\Omega }(g(x))}$ as ${\displaystyle x\to a}$,

where a is some real number, ∞, or −∞, where f and g are real functions defined in a neighbourhood of a, and where g is positive in this neighbourhood.

The Hardy–Littlewood definition is used mainly in analytic number theory, and the Knuth definition mainly in computational complexity theory; the definitions are not equivalent.

The Hardy–Littlewood definition

In 1914 Godfrey Harold Hardy and John Edensor Littlewood introduced the new symbol ${\displaystyle \mathrm{\Omega }}$, which is defined as follows:

${\displaystyle f(x)=\mathrm{\Omega }(g(x))}$ as ${\displaystyle x\to \mathrm{\infty }}$ if ${\displaystyle \underset{x\to \mathrm{\infty }}{lim sup}\left|\frac{f(x)}{g(x)}\right|>0.}$

Thus ${\displaystyle f(x)=\mathrm{\Omega }(g(x))}$ is the negation of ${\displaystyle f(x)=o(g(x))}$.

In 1916 the same authors introduced the two new symbols ${\displaystyle {\mathrm{\Omega }}_R}$ and ${\displaystyle {\mathrm{\Omega }}_L}$, defined as:

${\displaystyle f(x)={\mathrm{\Omega }}_R(g(x))}$ as ${\displaystyle x\to \mathrm{\infty }}$ if ${\displaystyle \underset{x\to \mathrm{\infty }}{lim sup}\frac{f(x)}{g(x)}>0}$;

${\displaystyle f(x)={\mathrm{\Omega }}_L(g(x))}$ as ${\displaystyle x\to \mathrm{\infty }}$ if ${\displaystyle \underset{x\to \mathrm{\infty }}{lim inf}\frac{f(x)}{g(x)}<0.}$

These symbols were used by Edmund Landau, with the same meanings, in 1924. After Landau, the notations were never used again exactly thus; ${\displaystyle {\mathrm{\Omega }}_R}$ became ${\displaystyle {\mathrm{\Omega }}_{+}}$ and ${\displaystyle {\mathrm{\Omega }}_L}$ became ${\displaystyle {\mathrm{\Omega }}_{-}}$.

These three symbols ${\displaystyle \mathrm{\Omega },{\mathrm{\Omega }}_{+},{\mathrm{\Omega }}_{-}}$, as well as ${\displaystyle f(x)={\mathrm{\Omega }}_{\pm }(g(x))}$ (meaning that ${\displaystyle f(x)={\mathrm{\Omega }}_{+}(g(x))}$ and ${\displaystyle f(x)={\mathrm{\Omega }}_{-}(g(x))}$ are both satisfied), are now currently used in analytic number theory.

Simple examples

We have

${\displaystyle \sin x=\mathrm{\Omega }(1)}$ as ${\displaystyle x\to \mathrm{\infty },}$

and more precisely

${\displaystyle \sin x={\mathrm{\Omega }}_{\pm }(1)}$ as ${\displaystyle x\to \mathrm{\infty }.}$

We have

${\displaystyle \sin x+1=\mathrm{\Omega }(1)}$ as ${\displaystyle x\to \mathrm{\infty },}$

and more precisely

${\displaystyle \sin x+1={\mathrm{\Omega }}_{+}(1)}$ as ${\displaystyle x\to \mathrm{\infty };}$

however

${\displaystyle \sin x+1\ne {\mathrm{\Omega }}_{-}(1)}$ as ${\displaystyle x\to \mathrm{\infty }.}$

The Knuth definition

In 1976 Donald Knuth published a paper to justify his use of the ${\displaystyle \mathrm{\Omega }}$-symbol to describe a stronger property. Knuth wrote: "For all the applications I have seen so far in computer science, a stronger requirement ... is much more appropriate". He defined

${\displaystyle f(x)=\mathrm{\Omega }(g(x))\iff g(x)=O(f(x))}$

with the comment: "Although I have changed Hardy and Littlewood's definition of ${\displaystyle \mathrm{\Omega }}$, I feel justified in doing so because their definition is by no means in wide use, and because there are other ways to say what they want to say in the comparatively rare cases when their definition applies."

Family of Bachmann–Landau notations

Notation	Name	Description	Formal definition	Limit definition
${\displaystyle f(n)=o(g(n))}$	Small O; Small Oh	f is dominated by g asymptotically	${\displaystyle \mathrm{\forall }k>0\mathrm{\exists }{n}_0\mathrm{\forall }n>n_0:|f(n)|<kg(n)}$	${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\frac{f(n)}{g(n)}=0}$
${\displaystyle f(n)=O(g(n))}$	Big O; Big Oh; Big Omicron	${\displaystyle |f|}$ is bounded above by g (up to constant factor) asymptotically	${\displaystyle \mathrm{\exists }k>0\mathrm{\exists }{n}_0\mathrm{\forall }n>n_0:|f(n)|\le kg(n)}$	${\displaystyle \underset{n\to \mathrm{\infty }}{lim sup}\frac{f(n)}{g(n)}<\mathrm{\infty }}$
${\displaystyle f(n)=\mathrm{\Theta }(g(n))}$	Big Theta	f is bounded both above and below by g asymptotically	${\displaystyle \mathrm{\exists }{k}_1>0\mathrm{\exists }{k}_2>0\mathrm{\exists }{n}_0\mathrm{\forall }n>n_0:}$ ${\displaystyle k_{1}g(n)\le f(n)\le k_{2}g(n)}$	${\displaystyle f(n)=O(g(n))}$ and ${\displaystyle f(n)=\mathrm{\Omega }(g(n))}$ (Knuth version)
${\displaystyle f(n)\sim g(n)}$	On the order of	f is equal to g asymptotically	${\displaystyle \mathrm{\forall }\epsilon >0\mathrm{\exists }{n}_0\mathrm{\forall }n>n_0:\left|\frac{f(n)}{g(n)}-1\right|<\epsilon }$	${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\frac{f(n)}{g(n)}=1}$
${\displaystyle f(n)=\mathrm{\Omega }(g(n))}$	Big Omega in complexity theory (Knuth)	f is bounded below by g asymptotically	${\displaystyle \mathrm{\exists }k>0\mathrm{\exists }{n}_0\mathrm{\forall }n>n_0:f(n)\ge kg(n)}$	${\displaystyle \underset{n\to \mathrm{\infty }}{lim inf}\frac{f(n)}{g(n)}>0}$
${\displaystyle f(n)=\omega (g(n))}$	Small Omega	f dominates g asymptotically	${\displaystyle \mathrm{\forall }k>0\mathrm{\exists }{n}_0\mathrm{\forall }n>n_0:f(n)>kg(n)}$	${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\frac{f(n)}{g(n)}=\mathrm{\infty }}$
${\displaystyle f(n)=\mathrm{\Omega }(g(n))}$	Big Omega in number theory (Hardy–Littlewood)	${\displaystyle |f|}$ is not dominated by g asymptotically	${\displaystyle \mathrm{\exists }k>0\mathrm{\forall }{n}_0\mathrm{\exists }n>n_0:|f(n)|\ge kg(n)}$	${\displaystyle \underset{n\to \mathrm{\infty }}{lim sup}\frac{\left|f(n)\right|}{g(n)}>0}$

The limit definitions assume ${\displaystyle g(n)>0}$ for sufficiently large ${\displaystyle n}$. The table is (partly) sorted from smallest to largest, in the sense that ${\displaystyle o,O,\mathrm{\Theta },\sim ,}$ (Knuth's version of) ${\displaystyle \mathrm{\Omega },\omega }$ on functions correspond to ${\displaystyle <,\le ,\approx ,=,}$${\displaystyle \ge ,>}$ on the real line (the Hardy–Littlewood version of ${\displaystyle \mathrm{\Omega }}$, however, doesn't correspond to any such description).

Computer science uses the big ${\displaystyle O}$, big Theta ${\displaystyle \mathrm{\Theta }}$, little ${\displaystyle o}$, little omega ${\displaystyle \omega }$ and Knuth's big Omega ${\displaystyle \mathrm{\Omega }}$ notations. Analytic number theory often uses the big ${\displaystyle O}$, small ${\displaystyle o}$, Hardy–Littlewood's big Omega ${\displaystyle \mathrm{\Omega }}$ (with or without the +, − or ± subscripts) and ${\displaystyle \sim }$ notations. The small omega ${\displaystyle \omega }$ notation is not used as often in analysis.

Use in computer science

Further information: Analysis of algorithms

Informally, especially in computer science, the big O notation often can be used somewhat differently to describe an asymptotic tight bound where using big Theta Θ notation might be more factually appropriate in a given context For example, when considering a function T(n) = 73n³ + 22n² + 58, all of the following are generally acceptable, but tighter bounds (such as numbers 2 and 3 below) are usually strongly preferred over looser bounds (such as number 1 below).

1. T(n) = O(n¹⁰⁰)
2. T(n) = O(n³)
3. T(n) = Θ(n³)

The equivalent English statements are respectively:

1. T(n) grows asymptotically no faster than n¹⁰⁰
2. T(n) grows asymptotically no faster than n³
3. T(n) grows asymptotically as fast as n³.

So while all three statements are true, progressively more information is contained in each. In some fields, however, the big O notation (number 2 in the lists above) would be used more commonly than the big Theta notation (items numbered 3 in the lists above). For example, if T(n) represents the running time of a newly developed algorithm for input size n, the inventors and users of the algorithm might be more inclined to put an upper asymptotic bound on how long it will take to run without making an explicit statement about the lower asymptotic bound.

Other notation

In their book Introduction to Algorithms, Cormen, Leiserson, Rivest and Stein consider the set of functions f which satisfy

${\displaystyle f(n)=O(g(n))(n\to \mathrm{\infty }).}$

In a correct notation this set can, for instance, be called O(g), where

$${\displaystyle O(g)=\{f:\text{there exist positive constants}c\text{and}{n}_0\text{such that}0\le f(n)\le cg(n)\text{ for all }n\ge n_0\}.}$$

The authors state that the use of equality operator (=) to denote set membership rather than the set membership operator (∈) is an abuse of notation, but that doing so has advantages. Inside an equation or inequality, the use of asymptotic notation stands for an anonymous function in the set O(g), which eliminates lower-order terms, and helps to reduce inessential clutter in equations, for example:

${\displaystyle 2n^2+3n+1=2n^2+O(n).}$

Extensions to the Bachmann–Landau notations

Another notation sometimes used in computer science is Õ (read soft-O), which hides polylogarithmic factors. There are two definitions in use: some authors use f(n) = Õ(g(n)) as shorthand for f(n) = O(g(n) log^k n) for some k, while others use it as shorthand for f(n) = O(g(n) log^k g(n)). When g(n) is polynomial in n, there is no difference; however, the latter definition allows one to say, e.g. that ${\displaystyle n{2}^n=\tilde{O}(2^n)}$ while the former definition allows for ${\displaystyle {\log }^{k}n=\tilde{O}(1)}$ for any constant k. Some authors write O^* for the same purpose as the latter definition. Essentially, it is big O notation, ignoring 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). This notation is often used to obviate the "nitpicking" within growth-rates that are stated as too tightly bounded for the matters at hand (since log^k n is always o(n^ε) for any constant k and any ε > 0).

Also the L notation, defined as

${\displaystyle L_n[\alpha ,c]=e^{(c+o(1))(\ln n{)}^{\alpha }(\ln \ln n{)}^{1-\alpha }}}$

is convenient for functions that are between polynomial and exponential in terms of ${\displaystyle \ln n}$.

Generalizations and related usages

The generalization to functions taking values in any normed vector space is straightforward (replacing absolute values by norms), where f and g need not take their values in the same space. A generalization to functions g taking values in any topological group is also possible. The "limiting process" x → x_o can also be generalized by introducing an arbitrary filter base, i.e. to directed nets f and g. The o notation can be used to define derivatives and differentiability in quite general spaces, and also (asymptotical) equivalence of functions,

${\displaystyle f\sim g⟺(f-g)\in o(g)}$

which is an equivalence relation and a more restrictive notion than the relationship "f is Θ(g)" from above. (It reduces to lim f / g = 1 if f and g are positive real valued functions.) For example, 2x is Θ(x), but 2x − x is not o(x).

History (Bachmann–Landau, Hardy, and Vinogradov notations)

The symbol O was first introduced by number theorist Paul Bachmann in 1894, in the second volume of his book Analytische Zahlentheorie ("analytic number theory"). The number theorist Edmund Landau adopted it, and was thus inspired to introduce in 1909 the notation o; hence both are now called Landau symbols. These notations were used in applied mathematics during the 1950s for asymptotic analysis. The symbol ${\displaystyle \mathrm{\Omega }}$ (in the sense "is not an o of") was introduced in 1914 by Hardy and Littlewood. Hardy and Littlewood also introduced in 1916 the symbols ${\displaystyle {\mathrm{\Omega }}_R}$ ("right") and ${\displaystyle {\mathrm{\Omega }}_L}$ ("left"), precursors of the modern symbols ${\displaystyle {\mathrm{\Omega }}_{+}}$ ("is not smaller than a small o of") and ${\displaystyle {\mathrm{\Omega }}_{-}}$ ("is not larger than a small o of"). Thus the Omega symbols (with their original meanings) are sometimes also referred to as "Landau symbols". This notation ${\displaystyle \mathrm{\Omega }}$ became commonly used in number theory at least since the 1950s. In the 1970s the big O was popularized in computer science by Donald Knuth, who introduced the related Theta notation, and proposed a different definition for the Omega notation.

Landau never used the big Theta and small omega symbols.

Hardy's symbols were (in terms of the modern O notation)

${\displaystyle f\preccurlyeq g⟺f\in O(g)}$   and   ${\displaystyle f\prec g⟺f\in o(g);}$

(Hardy however never defined or used the notation ${\displaystyle \prec \prec }$, nor ${\displaystyle \ll }$, as it has been sometimes reported). Hardy introduced the symbols ${\displaystyle \preccurlyeq }$ and ${\displaystyle \prec }$ (as well as some other symbols) in his 1910 tract "Orders of Infinity", and made use of them only in three papers (1910–1913). In his nearly 400 remaining papers and books he consistently used the Landau symbols O and o.

Hardy's notation is not used anymore. On the other hand, in the 1930s, the Russian number theorist Ivan Matveyevich Vinogradov introduced his notation ${\displaystyle \ll }$, which has been increasingly used in number theory instead of the ${\displaystyle O}$ notation. We have

${\displaystyle f\ll g⟺f\in O(g),}$

and frequently both notations are used in the same paper.

The big-O originally stands for "order of" ("Ordnung", Bachmann 1894), and is thus a Latin letter. Neither Bachmann nor Landau ever call it "Omicron". The symbol was much later on (1976) viewed by Knuth as a capital omicron, probably in reference to his definition of the symbol Omega. The digit zero should not be used.

Reconstructionist Judaism

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Reconstructionist_Judaism 

Reconstructing Judaism's organizational headquarters in Wyncote, Pennsylvania

Torah reading at the Reconstructionist synagogue, Montreal

Reconstructionist Judaism is a Jewish movement based on the concepts developed by Rabbi Mordecai Kaplan (1881–1983) that views Judaism as a progressively evolving civilization rather than just a religion. The movement originated as a semi-organized stream within Conservative Judaism, developed between the late 1920s and the 1940s before seceding in 1955, and established a rabbinical college in 1967. Reconstructionist Judaism is recognized by many scholars as one of the four major streams of Judaism in America alongside Orthodox, Conservative, and Reform.

There is substantial theological diversity within the movement. Halakha (Jewish law) is not considered normative and binding but is instead seen as the basis for the ongoing evolution of meaningful Jewish practice. In contrast with the Reform movement's stance during the time he was writing, Kaplan believed that "Jewish life [is] meaningless without Jewish law" and one of the planks he wrote for the proto-Reconstructionist Society for the Jewish Renaissance stated, "We accept the halakha, which is rooted in the Talmud, as the norm of Jewish life, availing ourselves, at the same time, of the method implicit therein to interpret and develop the body of Jewish Law by the actual conditions and spiritual needs of modern life." The movement also emphasizes positive views toward modernity and has an approach to Jewish customs that aims toward communal decision-making through a process of education and distillation of values from traditional Jewish sources.

The movement's 2011 A Guide to Jewish Practice describes a Reconstructionist approach to Jewish practice as "post-halakhic" because the modern world is one in which Jewish law cannot be enforced. Obligation and spiritual discipline exist without the enforcement of a functioning legal system. Thus, Reconstructionist Jews take Jewish law seriously as a source and resource that can shape expectations while not necessarily seeing themselves as bound by inherited claims of obligation. Therefore, the practices in the guide are not monolithic, and commentators provide further insights, arguments, and alternative approaches that span the broad range of views that Reconstructionist rabbis and scholars advocate. The guide states that it "assumes that thoughtful individuals and committed communities can handle diversity and will of necessity reach their own conclusions".

Origin

Reconstructionism was developed by Rabbi Mordecai Kaplan (1881–1983) and his son-in-law, Rabbi Ira Eisenstein (1906–2001), over a period of time from the late 1920s to the 1940s. After being rejected by Orthodox rabbis for his focus on issues in the community and the sociopolitical environment, Kaplan and a group of followers founded the Society for the Advancement of Judaism (SAJ) in 1922. Its goal was to give rabbis the opportunity to form new outlooks on Judaism in a more progressive manner. Kaplan was the leader of the SAJ until Eisenstein succeeded him in 1945. In 1935, Kaplan published his book, Judaism as a Civilization: Toward a Reconstruction of American Jewish Life. It was this book that Kaplan claimed was the beginning of the Reconstructionist movement. Judaism as a Civilization suggested that historical Judaism be given a "revaluation… in terms of present-day thought." Reconstructionism was able to spread with several other forms of literature—most notably, the New Haggadah (1941), which for the first time blended Kaplan's ideologies in Jewish ceremonial literature.

Although Kaplan did not want Reconstructionism to branch into another Jewish denomination, it became apparent that the such an outcome was inevitable. At the Montreal conference in 1967, Reconstructionist leaders called for a rabbinical school in which rabbis could be ordained under the Reconstructionist ideology and lead Reconstructionist congregations. By the fall of 1968, the Reconstructionist Rabbinical College was opened in Philadelphia. Along with the establishment of the college, the Reconstructionist Rabbinical Association formed, which gave rabbis a strong network in the religious leadership of Reconstructionism. The founding of these institutions were great strides in its becoming the fourth movement in North American Judaism (Orthodox, Conservative and Reform being the other three).

Reconstructionist Judaism is the first major movement of Judaism to originate in North America; the second is the Humanistic Judaism movement founded in 1963 by Rabbi Sherwin Wine.

Theology

Kaplan believed that, in light of advances in philosophy, science, and history, it would be impossible for modern Jews to continue to adhere to many of Judaism's traditional theological claims. In agreement with Orthodox theology (articulated by prominent medieval Jewish thinkers like Maimonides), Kaplan affirmed that God is not anthropomorphic in any way. All anthropomorphic descriptions of God are understood to be metaphorical. Kaplan's theology went further to claim that God is neither a personal nor conscious being; God cannot relate to or communicate with humanity in any way. Kaplan's theology defines God as the sum of all natural processes that allow people to become self-fulfilled.

To believe in God means to accept life on the assumption that it harbors conditions in the outer world and drives in the human spirit which together impel man to transcend himself. To believe in God means to take for granted that it is man's destiny to rise above the brute and to eliminate all forms of violence and exploitation from human society. In brief, God is the Power in the cosmos that gives human life the direction that enables the human being to reflect the image of God.

Most "classical" Reconstructionist Jews (i.e., those agreeing with Kaplan) reject traditional forms of theism (though this is by no means universal). Many Reconstructionist Jews are deists, but the movement also includes Jews who hold Kabbalistic, pantheistic, personal, and/or panentheistic views of God.

Kaplan's theology, as he explicitly stated, does not represent the only Reconstructionist understanding of theology and theology is not the cornerstone of the Reconstructionist movement. Much more central is the idea that Judaism is a civilization, and that the Jewish people must take an active role in ensuring its future by participating in its ongoing evolution.

Consequently, a strain of Reconstructionism exists which is distinctly non-Kaplanian. In this view, Kaplan's assertions concerning belief and practice are largely rejected, while the tenets of an "evolving religious civilization" are supported. The basis for this approach is that Kaplan spoke for his generation; he also wrote that every generation would need to define itself and its civilization for itself. In the thinking of these Reconstructionists, what Kaplan said concerning belief and practice is not applicable today. This approach may include a belief in a personal God, acceptance of the concept of "chosenness", a belief in some form of resurrection or continued existence of the dead, and the existence of an obligatory form of halakha. In the latter, in particular, there has developed a broader concept of halakhah wherein concepts such as "Eco-Kashrut" are incorporated.

Jewish law and tradition

Reconstructionist Judaism holds that the traditional halakhic system is incapable of producing a code of conduct that is meaningful for, and acceptable to, the vast majority of contemporary Jews, and thus must be reinterpreted in each new time period. Unlike classical Reform Judaism, Reconstructionism holds that a person's default position should be to incorporate Jewish laws and tradition into their lives, unless they have a specific reason to do otherwise. However some Reconstructionists believe that halakha is neither normative, nor binding, but are general guidelines.

Reconstructionism promotes many traditional Jewish practices. Thus, the commandments have been replaced with "folkways", non-binding customs that can be democratically accepted or rejected by the congregations. Folkways that are promoted include keeping Hebrew in the prayer service, studying Torah, daily prayer, wearing kippot (yarmulkes), tallitot and tefillin during prayer, and observance of the Jewish holidays.

Reconstructionists may use distinct prayer books, such as the Kol haneshamah Hebrew/English Reconstructionist prayer book. Marc Shapiro called attention to the Reconstructionist Kol haneshamah taking liberties with the text, sometimes with an English translation "so blatantly inaccurate that we have no choice but to regard it as a conscious alteration."

Beliefs

In practice, Kaplan's books, especially The Meaning of God in Modern Jewish Religion and Judaism as a Civilization are de facto statements of principles. In 1986, the Reconstructionist Rabbinical Association (RRA) and the Federation of Reconstructionist Congregations and Havurot (FRCH) passed the official "Platform on Reconstructionism". It is not a mandatory statement of principles, but rather a consensus of current beliefs. Major points of the platform state that:

Judaism is the result of natural human development. There is no such thing as divine intervention; Judaism is an evolving religious civilization; Zionism and aliyah (immigration to Israel) are encouraged; Reconstructionist Judaism is based on a democratic community where the laity can make decisions, not just rabbis; The Torah was not inspired by God; it only comes from the social and historical development of Jewish people; The classical view of God is rejected. God is redefined as the sum of natural powers or processes that allows mankind to gain self-fulfillment and moral improvement; The idea that God chose the Jewish people for any purpose, in any way, is "morally untenable", because anyone who has such beliefs "implies the superiority of the elect community and the rejection of others."

Most Reconstructionists do not believe in revelation (the idea that God reveals his will to human beings). This is dismissed as supernaturalism. Kaplan posits that revelation "consists in disengaging from the traditional context those elements in it which answer permanent postulates of human nature, and in integrating them into our own ideology…the rest may be relegated to archaeology".

Many writers have criticized the movement's most widely held theology, religious naturalism. David Ray Griffin and Louis Jacobs have objected to the redefinitions of the terms "revelation" and "God" as being intellectually dishonest, and as being a form of "conversion by definition"; in their critique, these redefinitions take non-theistic beliefs and attach theistic terms to them. Similar critiques have been put forth by Rabbis Neil Gillman, Milton Steinberg, and Michael Samuels.

Reconstructionist Judaism is egalitarian with respect to gender roles. All positions are open to all genders; they are open to lesbians, gay men, and transgender individuals as well.

Jewish identity

Reconstructionist Judaism allows its rabbis to determine their own policy regarding officiating at intermarriages. Some congregations accept patrilineal as well as matrilineal descent, and children of one Jewish parent, of any gender, are considered Jewish by birth if raised as Jews. This contrasts with the traditional interpretations of Jewish law of both Rabbinical Judaism, in which a child is Jewish by birth if its mother was Jewish; and of Karaite Judaism, in which a child is Jewish by birth if its father was Jewish.

The role of non-Jews in Reconstructionist congregations is a matter of ongoing debate. Practices vary between synagogues. Most congregations strive to strike a balance between inclusivity and integrity of boundaries. The Jewish Reconstructionist Federation (JRF) has issued a non-binding statement attempting to delineate the process by which congregations set policy on these issues, and sets forth sample recommendations. These issues are ultimately decided by local lay leadership.

In 2015 the Reconstructionist Rabbinical College voted to accept rabbinical students in interfaith relationships, making Reconstructionist Judaism the first type of Judaism to officially allow rabbis in relationships with non-Jewish partners. In making the decision, the movement considered that "many younger progressive Jews, including many rabbis and rabbinical students, now perceive restrictions placed on those who are intermarried as reinforcing a tribalism that feels personally alienating and morally troubling in the 21st century." In April 2016 nineteen Reconstructionist rabbis announced they will form an offshoot group in part to protest the decision to allow rabbis to have non-Jewish partners.

Organizations

Over 100 synagogues and havurot, mostly in the United States and Canada, were affiliated with the Jewish Reconstructionist Federation. As of June 3, 2012, the Reconstructionist movement has been restructured. A joint institution consisting of the Reconstructionist Rabbinical College and the congregational organization is now the primary organization of the movement.

The movement's new designation was first "Jewish Reconstructionist Communities," and in 2018 became Reconstructing Judaism. Rabbi Deborah Waxman was inaugurated as the president of the Reconstructionist Rabbinical College and Jewish Reconstructionist Communities on October 26, 2014. As the president of the Reconstructionist Rabbinical College, she is believed to be the first woman and first lesbian to lead a Jewish congregational union, and the first lesbian to lead a Jewish seminary; the Reconstructionist Rabbinical College is both a congregational union and a seminary. Waxman is a 1999 graduate of RRC.

The Reconstructionist Rabbinical College educates rabbis. The Reconstructionist Rabbinical Association is the professional organization of Reconstructionist rabbis. The Jewish Reconstructionist youth organization is named No'ar Hadash. Camp Havaya (formerly Camp JRF) in South Sterling, Pennsylvania, is the Reconstructionist movement's summer sleep away camp.

As of 2020, the Pew Research Center estimated that Reconstructionist Judaism, along with Humanist Judaism and other smaller denominations, constituted 4% of the United States's 7.5 million Jews.

Relation to other Jewish movements

Originally an offshoot of Conservative Judaism, Reconstructionism retains warm relations with Reform Judaism; however, Orthodox Judaism considers Reconstructionism, and every other non-Orthodox denomination, to be in violation of proper observance of interpretation of Jewish law. The Jewish Reconstructionist Federation is a member of the World Union for Progressive Judaism, in which it gained an observer status in 1990.