pairing function#Cantor pairing function

A pairing function is a bijection

:$\backslash pi:\backslash mathbb\{N\}\; \backslash times\; \backslash mathbb\{N\}\; \backslash to\; \backslash mathbb\{N\}.${{sfn|Pigeon}}{{sfn|Lisi|2007}}{{sfn|Regan|1992}}

More generally, a pairing function on a set $A$ is a function that maps each pair of elements from $A$ into an element of $A$, such that any two pairs of elements of $A$ are associated with different elements of $A$,{{sfn|Szudzik|2006}}{{efn|That is, an injection from $A^2\; \backslash rightarrow\; A$.}} or a bijection from $A^2$ to $A$.{{sfn|Szudzik|2017}}

Instead of abstracting from the domain, the arity of the pairing function can also be generalized: there exists an n-ary generalized Cantor pairing function on $\backslash mathbb\{N\}$.{{sfn|Lisi|2007}}

File:Cantor's Pairing Function.svg

File:Cantor's Pairing Function Plot.svg

The Cantor pairing function is a primitive recursive pairing function

:$\backslash pi:\backslash mathbb\{N\}\; \backslash times\; \backslash mathbb\{N\}\; \backslash to\; \backslash mathbb\{N\}$

defined by

:$\backslash pi(k\_1,k\_2)\; :=\; \backslash frac\{1\}\{2\}(k\_1\; +\; k\_2)(k\_1\; +\; k\_2\; +\; 1)+k\_2$

where $k\_1,\; k\_2\backslash in\backslash \{0,\; 1,\; 2,\; 3,\; \backslash dots\backslash \}$.{{sfn|Pigeon|loc=Equation 8}}{{bsn|date=November 2024}}

It can also be expressed as $\backslash pi(x,\; y)\; :=\; \backslash frac\{x^2\; +\; x\; +\; 2xy\; +\; 3y\; +\; y^2\}\{2\}$.{{sfn|Szudzik|2006}}

It is also strictly monotonic w.r.t. each argument, that is, for all $k\_1,\; k\_1\text{'},\; k\_2,\; k\_2\text{'}\; \backslash in\; \backslash mathbb\{N\}$, if , then ; similarly, if , then .{{Citation needed|date=August 2021}}

The statement that this is the only quadratic pairing function is known as the Fueter–Pólya theorem.{{harvtxt|Stein|1999|pp=448-452}} cited in {{harvtxt|Pigeon}}. Whether this is the only polynomial pairing function is still an open question. When we apply the pairing function to {{math|k₁}} and {{math|k₂}} we often denote the resulting number as {{math|⟨k₁, k₂⟩}}.{{Citation needed|date=August 2021}}

This definition can be inductively generalized to the {{Citation needed span|text=Cantor tuple function|date=August 2021}}

:$\backslash pi^\{(n)\}:\backslash mathbb\{N\}^n\; \backslash to\; \backslash mathbb\{N\}$

for $n\; >\; 2$ as

:$\backslash pi^\{(n)\}(k\_1,\; \backslash ldots,\; k\_\{n-1\},\; k\_n)\; :=\; \backslash pi\; (\; \backslash pi^\{(n-1)\}(k\_1,\; \backslash ldots,\; k\_\{n-1\})\; ,\; k\_n)$

with the base case defined above for a pair: $\backslash pi^\{(2)\}(k\_1,k\_2)\; :=\; \backslash pi(k\_1,k\_2).${{sfn|Pigeon|loc=Equations 13-7}}

Let $z\; \backslash in\; \backslash mathbb\{N\}$ be an arbitrary natural number. We will show that there exist unique values $x,\; y\; \backslash in\; \backslash mathbb\{N\}$ such that

:$z\; =\; \backslash pi(x,\; y)\; =\; \backslash frac\{(x\; +\; y\; +\; 1)(x\; +\; y)\}\{2\}\; +\; y$

and hence that the function {{math|π(x, y)}} is invertible. It is helpful to define some intermediate values in the calculation:

:$w\; =\; x\; +\; y\; \backslash !$

:$t\; =\; \backslash frac\{1\}\{2\}w(w\; +\; 1)\; =\; \backslash frac\{w^2\; +\; w\}\{2\}$

:$z\; =\; t\; +\; y\; \backslash !$

where {{math|t}} is the triangle number of {{math|w}}. If we solve the quadratic equation

:$w^2\; +\; w\; -\; 2t\; =\; 0\; \backslash !$

for {{math|w}} as a function of {{math|t}}, we get

:$w\; =\; \backslash frac\{\backslash sqrt\{8t\; +\; 1\}\; -\; 1\}\{2\}$

which is a strictly increasing and continuous function when {{math|t}} is non-negative real. Since

:

we get that

:

and thus

:$w\; =\; \backslash left\backslash lfloor\; \backslash frac\{\backslash sqrt\{8z\; +\; 1\}\; -\; 1\}\{2\}\; \backslash right\backslash rfloor.$

where {{math|⌊ ⌋}} is the floor function.

So to calculate {{math|x}} and {{math|y}} from {{math|z}}, we do:

:$w\; =\; \backslash left\backslash lfloor\; \backslash frac\{\backslash sqrt\{8z\; +\; 1\}\; -\; 1\}\{2\}\; \backslash right\backslash rfloor$

:$t\; =\; \backslash frac\{w^2\; +\; w\}\{2\}$

:$y\; =\; z\; -\; t\; \backslash !$

:$x\; =\; w\; -\; y.\; \backslash !$

Since the Cantor pairing function is invertible, it must be one-to-one and onto.{{sfn|Szudzik|2006}}{{Additional citation needed|date=August 2021}}

To calculate {{math|π(47, 32)}}:

:{{math|47 + 32 {{=}} 79}},

:{{math|79 + 1 {{=}} 80}},

:{{math|79 × 80 {{=}} 6320}},

:{{math|6320 ÷ 2 {{=}} 3160}},

:{{math|3160 + 32 {{=}} 3192}},

so {{math|π(47, 32) {{=}} 3192}}.

To find {{math|x}} and {{math|y}} such that {{math|π(x, y) {{=}} 1432}}:

:{{math|8 × 1432 {{=}} 11456}},

:{{math|11456 + 1 {{=}} 11457}},

:{{math|{{radic|11457}} {{=}} 107.037}},

:{{math|107.037 − 1 {{=}} 106.037}},

:{{math|106.037 ÷ 2 {{=}} 53.019}},

:{{math|⌊53.019⌋ {{=}} 53}},

so {{math|w {{=}} 53}};

:{{math|53 + 1 {{=}} 54}},

:{{math|53 × 54 {{=}} 2862}},

:{{math|2862 ÷ 2 {{=}} 1431}},

so {{math|t {{=}} 1431}};

:{{math|1432 − 1431 {{=}} 1}},

so {{math|y {{=}} 1}};

:{{math|53 − 1 {{=}} 52}},

so {{math|x {{=}} 52}}; thus {{math|π(52, 1) {{=}} 1432}}.{{Citation needed|date=August 2021}}

File:Diagonal argument.svg

The graphical shape of Cantor's pairing function, a diagonal progression, is a standard trick in working with infinite sequences and countability.{{efn|The term "diagonal argument" is sometimes used to refer to this type of enumeration, but it is not directly related to Cantor's diagonal argument.{{citation needed|date=August 2021}}}} The algebraic rules of this diagonal-shaped function can verify its validity for a range of polynomials, of which a quadratic will turn out to be the simplest, using the method of induction. Indeed, this same technique can also be followed to try and derive any number of other functions for any variety of schemes for enumerating the plane.

A pairing function can usually be defined inductively – that is, given the {{math|n}}th pair, what is the {{math|(n+1)}}th pair? The way Cantor's function progresses diagonally across the plane can be expressed as

:$\backslash pi(x,y)+1\; =\; \backslash pi(x-1,y+1)$.

The function must also define what to do when it hits the boundaries of the 1st quadrant – Cantor's pairing function resets back to the x-axis to resume its diagonal progression one step further out, or algebraically:

:$\backslash pi(0,k)+1\; =\; \backslash pi(k+1,0)$.

Also we need to define the starting point, what will be the initial step in our induction method: {{math|π(0, 0) {{=}} 0}}.

Assume that there is a quadratic 2-dimensional polynomial that can fit these conditions (if there were not, one could just repeat by trying a higher-degree polynomial). The general form is then

:$\backslash pi(x,y)\; =\; ax^2+by^2+cxy+dx+ey+f$.

Plug in our initial and boundary conditions to get {{math|f {{=}} 0}} and:

:$bk^2+ek+1\; =\; a(k+1)^2+d(k+1)$,

so we can match our {{math|k}} terms to get

:{{math|b {{=}} a}}

:{{math|d {{=}} 1-a}}

:{{math|e {{=}} 1+a}}.

So every parameter can be written in terms of {{math|a}} except for {{math|c}}, and we have a final equation, our diagonal step, that will relate them:

:$\backslash begin\{align\}$

\pi(x,y)+1 &= a(x^2+y^2) + cxy + (1-a)x + (1+a)y + 1 \\

&= a((x-1)^2+(y+1)^2) + c(x-1)(y+1) + (1-a)(x-1) + (1+a)(y+1).

\end{align}

Expand and match terms again to get fixed values for {{math|a}} and {{math|c}}, and thus all parameters:

:{{math|a {{=}} {{sfrac|1|2}} {{=}} b {{=}} d}}

:{{math|c {{=}} 1}}

:{{math|e {{=}} {{sfrac|3|2}}}}

:{{math|f {{=}} 0}}.

Therefore

:$\backslash begin\{align\}$

\pi(x,y) &= \frac{1}{2}(x^2+y^2) + xy + \frac{1}{2}x + \frac{3}{2}y \\

&= \frac{1}{2}(x+y)(x+y+1) + y,

\end{align}

is the Cantor pairing function, and we also demonstrated through the derivation that this satisfies all the conditions of induction.{{Citation needed|date=August 2021}}

The following pairing function: $\backslash langle\; i,\; j\backslash rangle\; :=\; \backslash frac\{1\}\{2\}(i+j-2)(i+j-1)\; +\; i$, where $i,\; j\backslash in\backslash \{1,\; 2,\; 3,\; \backslash dots\; \backslash \}$.{{harvtxt|Hopcroft|Ullman|1979|p=169}} cited in {{harv|Pigeon|loc=Equations 2, 3}}. is the same as the Cantor pairing function, but shifted to exclude 0 (i.e., $i=k\_2+1$, $j=k\_1+1$, and $\backslash langle\; i,\; j\backslash rangle\; -\; 1\; =\; \backslash pi(k\_2,k\_1)$).{{sfn|Pigeon|loc=Equation 8}} It was used in the popular computer textbook of Hopcroft and Ullman (1979).

The function $P\_2(x,\; y):=\; 2^x(2y\; +\; 1)\; -\; 1$ is a pairing function.

In 1990, Regan proposed the first known pairing function that is computable in linear time and with constant space (as the previously known examples can only be computed in linear time if multiplication can be too, which is doubtful). In fact, both this pairing function and its inverse can be computed with finite-state transducers that run in real time.{{Clarify|date=August 2021}} In the same paper, the author proposed two more monotone pairing functions that can be computed online in linear time and with logarithmic space; the first can also be computed offline with zero space.{{sfn|Regan|1992}}{{Clarify|reason=What is "zero space"?|date=August 2021}}

In 2001, Pigeon proposed a pairing function based on bit-interleaving, defined recursively as:

:$\backslash langle\; i,j\backslash rangle\_\{P\}=\backslash begin\{cases\}$

T & \text{if}\ i=j=0;\\

\langle\lfloor i/2\rfloor,\lfloor j/2\rfloor\rangle_{P}:i_0:j_0&\text{otherwise,}

\end{cases}

where $i\_0$ and $j\_0$ are the least significant bits of i and j respectively.{{sfn|Pigeon|loc=Equation 12}}{{bsn|date=November 2024}}

In 2006, Szudzik proposed a "more elegant" pairing function defined by the expression:

:$\backslash operatorname\{ElegantPair\}[x,\; y]\; :=\; \backslash begin\{cases\}$

y^2 + x&\text{if}\ x < y,\\

x^2 + x + y&\text{if}\ x \ge y.\\

\end{cases}

Which can be unpaired using the expression:

:$\backslash operatorname\{ElegantUnpair\}[z]\; :=\; \backslash begin\{cases\}$

\left\{ z - \lfloor\sqrt{z}\rfloor^2, \lfloor\sqrt{z}\rfloor \right\}

& \text{if }z - \lfloor\sqrt{z}\rfloor^2 < \lfloor\sqrt{z}\rfloor,

\\

\left\{ \lfloor\sqrt{z}\rfloor, z - \lfloor\sqrt{z}\rfloor^2 - \lfloor\sqrt{z}\rfloor \right\}

& \text{if }z - \lfloor\sqrt{z}\rfloor^2\geq\lfloor\sqrt{z}\rfloor.

\end{cases}

(Qualitatively, it assigns consecutive numbers to pairs along the edges of squares.) This pairing function orders SK combinator calculus expressions by depth.{{sfn|Szudzik|2006}}{{Clarify|date=August 2021}}

This method is the mere application to $\backslash N$ of the idea, found in most textbooks on Set Theory,See for instance {{harvtxt|Jech|2006|p=30}}.

used to establish $\backslash kappa^2=\backslash kappa$ for any infinite cardinal $\backslash kappa$ in ZFC.

Define on $\backslash kappa\backslash times\backslash kappa$ the binary relation

:$(\backslash alpha,\backslash beta)\backslash preccurlyeq(\backslash gamma,\backslash delta)\; \backslash text\{\; if\; either\; \}\; \backslash begin\{cases\}$

(\alpha,\beta) = (\gamma,\delta),\\[4pt]

\max(\alpha,\beta) < \max(\gamma,\delta),\\[4pt]

\max(\alpha,\beta) = \max(\gamma,\delta)\ \text{and}\ \alpha<\gamma,\text{ or}\\[4pt]

\max(\alpha,\beta) = \max(\gamma,\delta)\ \text{and}\ \alpha=\gamma\ \text{and}\ \beta<\delta.

\end{cases}

$\backslash preccurlyeq$ is then shown to be a well-ordering such that every element has predecessors, which implies that $\backslash kappa^2=\backslash kappa$.

It follows that $(\backslash N\backslash times\backslash N,\backslash preccurlyeq)$ is isomorphic to $(\backslash N,\backslash leqslant)$ and the pairing function above is nothing more than the enumeration of integer couples in increasing order.{{efn|See also Talk:Tarski's theorem about choice#Proof of the converse.}}

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{{reflist}}

• {{MathWorld|urlname=PairingFunction|author=Steven Pigeon|title=Pairing Function|ref={{SfnRef|Pigeon}}}}{{Sfn whitelist|CITEREFPigeon}}
• {{cite journal|journal=Le Matematiche |first=Meri |last=Lisi |title=Some Remarks on the Cantor Pairing Function |volume=LXII |year=2007 |pp=55-65 |url=https://lematematiche.dmi.unict.it/index.php/lematematiche/article/view/14/13}}
• {{Cite journal|date=December 1992 |title=Minimum-Complexity Pairing Functions|journal=Journal of Computer and System Sciences|language=en|volume=45|issue=3|pages=285–295|doi=10.1016/0022-0000(92)90027-G|issn=0022-0000|doi-access=free|last1=Regan|first1=Kenneth W.}}
• {{Cite web|last=Szudzik|first=Matthew|year=2006|title=An Elegant Pairing Function|url=http://szudzik.com/ElegantPairing.pdf|url-status=live|archive-url=https://web.archive.org/web/20111125104326/http://szudzik.com/ElegantPairing.pdf|archive-date=25 November 2011|access-date=16 August 2021|website=szudzik.com}}
• {{cite arXiv|last=Szudzik|first=Matthew P.|date=1 June 2017|title=The Rosenberg-Strong Pairing Function|class=cs.DM|eprint=1706.04129}}
• {{cite book |first=Thomas |last=Jech |date=2006 |title=Set Theory |edition=The Third Millennium |series=Springer Monographs in Mathematics |doi=10.1007/3-540-44761-X |publisher=Springer-Verlag |isbn=3-540-44085-2}}
• {{cite book

| last1 = Hopcroft

| first1 = John E.

| last2 = Ullman

| first2 = Jeffrey D.

| title = Introduction to Automata Theory, Languages, and Computation

| publisher = Addison-Wesley

| edition = 1st

| year = 1979

| isbn = 0-201-02988-X

}}

• {{cite book|title=Mathematics: The Man-Made Universe |last=Stein |first=Sherman K. |edition=3rd |publisher=Dover |year=1999 |isbn=9780486404509}}

Category:Set theory

Category:Georg Cantor

Category:Functions and mappings