Pairing function

In mathematics, a pairing function is a process to uniquely encode two natural numbers into a single natural number.

Any pairing function can be used in set theory to prove that integers and rational numbers have the same cardinality as natural numbers. ^[1]

Definition

A pairing function is a bijection

${\displaystyle \pi$ :\mathbb {N} \times \mathbb {N} \to \mathbb {N} .} ^{[2] [3] [4]}

Generalization

More generally, a pairing function on a set ${\displaystyle A}$ is a function that maps each pair of elements from ${\displaystyle A}$ into an element of ${\displaystyle A}$, such that any two pairs of elements of ${\displaystyle A}$ are associated with different elements of ${\displaystyle A}$, ^{[5] [a]} or a bijection from ${\displaystyle A^{2}}$ to ${\displaystyle A}$. ^[6]

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 ${\displaystyle \mathbb {N} }$. ^[3]

Hopcroft and Ullman pairing function

Hopcroft and Ullman (1979) define the following pairing function: ${\displaystyle \langle i,j\rangle$ :={\frac {1}{2}}(i+j-2)(i+j-1)+i}, where ${\displaystyle i,j\in \{1,2,3,\dots \}}$. ^[7] This is the same as the Cantor pairing function below, shifted to exclude 0 (i.e., ${\displaystyle i=k_{2}+1}$, ${\displaystyle j=k_{1}+1}$, and ${\displaystyle \langle i,j\rangle -1=\pi (k_{2},k_{1})}$). ^[8]

Cantor pairing function

The Cantor pairing function assigns one natural number to each pair of natural numbers

Graph of the Cantor pairing function

The Cantor pairing function is a primitive recursive pairing function

${\displaystyle \pi$ :\mathbb {N} \times \mathbb {N} \to \mathbb {N} }

defined by

${\displaystyle \pi (k_{1},k_{2}):={\frac {1}{2}}(k_{1}+k_{2})(k_{1}+k_{2}+1)+k_{2}}$

where ${\displaystyle k_{1},k_{2}\in \{0,1,2,3,\dots \}}$. ^{[8] [ better source needed ]}

It can also be expressed as ${\displaystyle \pi (x,y):={\frac {x^{2}+x+2xy+3y+y^{2}}{2}}}$. ^[5]

It is also strictly monotonic w.r.t. each argument, that is, for all ${\displaystyle k_{1},k_{1}',k_{2},k_{2}'\in \mathbb {N} }$, if ${\displaystyle k_{1}<k_{1}'}$, then ${\displaystyle \pi (k_{1},k_{2})<\pi (k_{1}',k_{2})}$; similarly, if ${\displaystyle k_{2}<k_{2}'}$, then ${\displaystyle \pi (k_{1},k_{2})<\pi (k_{1},k_{2}')}$.^{[ citation needed ]}

The statement that this is the only quadratic pairing function is known as the Fueter–Pólya theorem. ^[9] Whether this is the only polynomial pairing function is still an open question. When we apply the pairing function to k₁ and k₂ we often denote the resulting number as ⟨k₁, k₂⟩.^{[ citation needed ]}

This definition can be inductively generalized to the Cantor tuple function^{[ citation needed ]}

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

for ${\displaystyle n>2}$ as

${\displaystyle \pi ^{(n)}(k_{1},\ldots ,k_{n-1},k_{n}):=\pi (\pi ^{(n-1)}(k_{1},\ldots ,k_{n-1}),k_{n})}$

with the base case defined above for a pair: ${\displaystyle \pi ^{(2)}(k_{1},k_{2}):=\pi (k_{1},k_{2}).}$ ^[10]

Inverting the Cantor pairing function

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

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

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

${\displaystyle w=x+y\!}$

${\displaystyle t={\frac {1}{2}}w(w+1)={\frac {w^{2}+w}{2}}}$

${\displaystyle z=t+y\!}$

where t is the triangle number of w. If we solve the quadratic equation

${\displaystyle w^{2}+w-2t=0\!}$

for w as a function of t, we get

${\displaystyle w={\frac {{\sqrt {8t+1}}-1}{2}}}$

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

${\displaystyle t\leq z=t+y<t+(w+1)={\frac {(w+1)^{2}+(w+1)}{2}}}$

we get that

${\displaystyle w\leq {\frac {{\sqrt {8z+1}}-1}{2}}<w+1}$

and thus

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

where ⌊ ⌋ is the floor function. So to calculate x and y from z, we do:

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

${\displaystyle t={\frac {w^{2}+w}{2}}}$

${\displaystyle y=z-t\!}$

${\displaystyle x=w-y.\!}$

Since the Cantor pairing function is invertible, it must be one-to-one and onto. ^{[5] [ additional citation(s) needed ]}

Examples

To calculate π(47, 32):

47 + 32 = 79,

79 + 1 = 80,

79 × 80 = 6320,

6320 ÷ 2 = 3160,

3160 + 32 = 3192,

so π(47, 32) = 3192.

To find x and y such that π(x, y) = 1432:

8 × 1432 = 11456,

11456 + 1 = 11457,

√11457 = 107.037,

107.037 − 1 = 106.037,

106.037 ÷ 2 = 53.019,

⌊53.019⌋ = 53,

so w = 53;

53 + 1 = 54,

53 × 54 = 2862,

2862 ÷ 2 = 1431,

so t = 1431;

1432 − 1431 = 1,

so y = 1;

53 − 1 = 52,

so x = 52; thus π(52, 1) = 1432.^{[ citation needed ]}

Derivation

A diagonally incrementing "snaking" function, from same principles as Cantor's pairing function, is often used to demonstrate the countability of the rational numbers.

The graphical shape of Cantor's pairing function, a diagonal progression, is a standard trick in working with infinite sequences and countability. ^[b] 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 nth pair, what is the (n+1)th pair? The way Cantor's function progresses diagonally across the plane can be expressed as

${\displaystyle \pi (x,y)+1=\pi (x-1,y+1)}$.

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:

${\displaystyle \pi (0,k)+1=\pi (k+1,0)}$.

Also we need to define the starting point, what will be the initial step in our induction method: 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

${\displaystyle \pi (x,y)=ax^{2}+by^{2}+cxy+dx+ey+f}$.

Plug in our initial and boundary conditions to get f = 0 and:

${\displaystyle bk^{2}+ek+1=a(k+1)^{2}+d(k+1)}$,

so we can match our k terms to get

b = a

d = 1-a

e = 1+a.

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

${\displaystyle {\begin{aligned}\pi (x,y)+1&=a(x^{2}+y^{2})+cxy+(1-a)x+(1+a)y+1\\&=a((x-1)^{2}+(y+1)^{2})+c(x-1)(y+1)+(1-a)(x-1)+(1+a)(y+1).\end{aligned}}}$

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

a = ⁠1/2⁠ = b = d

c = 1

e = ⁠3/2⁠

f = 0.

Therefore

${\displaystyle {\begin{aligned}\pi (x,y)&={\frac {1}{2}}(x^{2}+y^{2})+xy+{\frac {1}{2}}x+{\frac {3}{2}}y\\&={\frac {1}{2}}(x+y)(x+y+1)+y,\end{aligned}}}$

is the Cantor pairing function, and we also demonstrated through the derivation that this satisfies all the conditions of induction.^{[ citation needed ]}

Other pairing functions

The function ${\displaystyle P_{2}(x,y):=2^{x}(2y+1)-1}$ 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.^{[ clarification needed ]} 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. ^{[4] [ clarification needed ]}

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

${\displaystyle \langle i,j\rangle _{P}={\begin{cases}T&{\text{if}}\ i=j=0;\\\langle \lfloor i/2\rfloor ,\lfloor j/2\rfloor \rangle _{P}:i_{0}:j_{0}&{\text{otherwise,}}\end{cases}}}$

where ${\displaystyle i_{0}}$ and ${\displaystyle j_{0}}$ are the least significant bits of i and j respectively. ^{[11] [ better source needed ]}

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

${\displaystyle \operatorname {ElegantPair} [x,y]:={\begin{cases}y^{2}+x&{\text{if}}\ x<y,\\x^{2}+x+y&{\text{if}}\ x\geq y.\\\end{cases}}}$

Which can be unpaired using the expression:

${\displaystyle \operatorname {ElegantUnpair} [z]:={\begin{cases}\left\{z-\lfloor {\sqrt {z}}\rfloor ^{2},\lfloor {\sqrt {z}}\rfloor \right\}&{\text{if }}z-\lfloor {\sqrt {z}}\rfloor ^{2}<\lfloor {\sqrt {z}}\rfloor ,\\\left\{\lfloor {\sqrt {z}}\rfloor ,z-\lfloor {\sqrt {z}}\rfloor ^{2}-\lfloor {\sqrt {z}}\rfloor \right\}&{\text{if }}z-\lfloor {\sqrt {z}}\rfloor ^{2}\geq \lfloor {\sqrt {z}}\rfloor .\end{cases}}}$

(Qualitatively, it assigns consecutive numbers to pairs along the edges of squares.) This pairing function orders SK combinator calculus expressions by depth. ^{[5] [ clarification needed ]} This method is the mere application to ${\displaystyle \mathbb {N} }$ of the idea, found in most textbooks on Set Theory, ^[12] used to establish ${\displaystyle \kappa ^{2}=\kappa }$ for any infinite cardinal ${\displaystyle \kappa }$ in ZFC. Define on ${\displaystyle \kappa \times \kappa }$ the binary relation

${\displaystyle (\alpha ,\beta )\preccurlyeq (\gamma ,\delta ){\text{ if either }}{\begin{cases}(\alpha ,\beta )=(\gamma ,\delta ),\\[4pt]\max(\alpha ,\beta )<\max(\gamma ,\delta ),\\[4pt]\max(\alpha ,\beta )=\max(\gamma ,\delta )\ {\text{and}}\ \alpha <\gamma ,{\text{ or}}\\[4pt]\max(\alpha ,\beta )=\max(\gamma ,\delta )\ {\text{and}}\ \alpha =\gamma \ {\text{and}}\ \beta <\delta .\end{cases}}}$

${\displaystyle \preccurlyeq }$ is then shown to be a well-ordering such that every element has ${\displaystyle {}<\kappa }$ predecessors, which implies that ${\displaystyle \kappa ^{2}=\kappa }$. It follows that ${\displaystyle (\mathbb {N} \times \mathbb {N} ,\preccurlyeq )}$ is isomorphic to ${\displaystyle (\mathbb {N} ,\leqslant )}$ and the pairing function above is nothing more than the enumeration of integer couples in increasing order. ^[c]

Citations

Notes

1. That is, an injection from ${\displaystyle A^{2}\rightarrow A}$.
2. 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 ]}
3. See also Talk:Tarski's theorem about choice#Proof of the converse.

Footnotes

1. Pigeon harvnb error: no target: CITEREFPigeon (help):

   "Pairing functions arise naturally in the demonstration that the cardinalities of the rationals ${\displaystyle \mathbb {Q} }$ and the nonnegative integers ${\displaystyle \mathbb {Z} _{\geq 0}}$ are the same, i.e., ${\displaystyle |\mathbb {Q} |=|\mathbb {Z} _{\geq 0}|=\aleph _{0}}$, originally due to Cantor."

2. Pigeon. sfn error: no target: CITEREFPigeon (help)
3. Lisi 2007.
4. Regan 1992.
5. Szudzik 2006.
6. Szudzik 2017.
7. Hopcroft & Ullman (1979 , p. 169) cited in ( Pigeon , Equations 2, 3) harv error: no target: CITEREFPigeon (help).
8. Pigeon, Equation 8. sfn error: no target: CITEREFPigeon (help)
9. Stein (1999 , pp. 448–452) cited in Pigeon harvtxt error: no target: CITEREFPigeon (help).
10. Pigeon, Equations 13-7. sfn error: no target: CITEREFPigeon (help)
11. Pigeon, Equation 12. sfn error: no target: CITEREFPigeon (help)
12. See for instance Jech (2006 , p. 30).

References

• Steven Pigeon. "Pairing Function". MathWorld .
• Lisi, Meri (2007). "Some Remarks on the Cantor Pairing Function". Le Matematiche. LXII: 55–65.
• Regan, Kenneth W. (December 1992). "Minimum-Complexity Pairing Functions". Journal of Computer and System Sciences. 45 (3): 285–295. doi: 10.1016/0022-0000(92)90027-G . ISSN   0022-0000.
• Szudzik, Matthew (2006). "An Elegant Pairing Function" (PDF). szudzik.com. Archived (PDF) from the original on 25 November 2011. Retrieved 16 August 2021.
• Szudzik, Matthew P. (1 June 2017). "The Rosenberg-Strong Pairing Function". arXiv: 1706.04129 [cs.DM].
• Jech, Thomas (2006). Set Theory. Springer Monographs in Mathematics (The Third Millennium ed.). Springer-Verlag. doi:10.1007/3-540-44761-X. ISBN   3-540-44085-2.
• Hopcroft, John E.; Ullman, Jeffrey D. (1979). Introduction to Automata Theory, Languages, and Computation (1st ed.). Addison-Wesley. ISBN   0-201-02988-X.
• Stein, Sherman K. (1999). Mathematics: The Man-Made Universe (3rd ed.). Dover. ISBN   9780486404509.