This can be easily implemented in any language. As such, we can calculate the max input pair to Szudzik to be the square root of the maximum integer value. For the Szudzik pairing function, the situation is only slightly more complicated. 148 VIEWS. Szudzik, M. (2006): An Elegant Pairing Function. But for R the Axiom of Choice is not required. \right.$$The limitation of Cantor pairing function (relatively) is that the range of encoded results doesn't always stay within the limits of a 2N bit integer if the inputs are two N bit integers. That fiddle makes note of the following references:$$index = \left\{\begin{array}{ll} 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. Given two points 8u,v< and 8x,y<, the point 8u,v< occurs at or before 8x,y< if and only if PairOrderedQ@8u,v<,8x,y= b ? x and y have to be non-negative integers. Use a pairing function for prime factorization. You can then map the row to an X axis, the column to an Y axis. Other than that, the same principles apply. So we use 200 pair values for the first 100 combinations, an efficiency of 50%. ��� ^a���0��4��q��NXk�_d��z�}k�; ���׬�HUf A��|Pv х�Ek���RA�����@������x�� kP[Z��e �\�UW6JZi���_��D�Q;)�hI���B\��aG��K��Ӄ^dd���Z�����V�8��"( �|�N�(�����������/x�ŢU ����a����[�E�g����b�"���&�>�B�*e��X�ÏD��{pY����#�g��������V�U}���I����@���������q�PXғ�d%=�{����zp�.B{����"��Y��!���ְ����G)I�Pi��қ�XB�K(�W! This means that all one hundred possible variations of ([0-9], [0-9]) would be covered (keeping in mind our values are 0-indexed). \end{array} the Szudzik pairing function, on two vectors of equal length. The inverse function is described at the wiki page. \end{array} \right.$$,$$c(a,b) = \left\{\begin{array}{ll} Generate ordered ids of OD pairs so lowest is always first This function is slow on large datasets, see szudzik_pairing for faster alternative Usage od_id_order(x, id1 = names(x)[1], id2 = names(x)[2]) Another JavaScript example: Szudzik can also be visualized as traversing a 2D field, but it covers it in a box-like pattern. Szudzik M (2006) An elegant pairing function. x^2 + x + y & : x \ge y Java : 97% speed and 66.67% memory : using Szudzik's Pairing Function and HashSet. a * a + a + b : a + b * b; where a, b >= 0 Szudzik pairing function accepts optional boolean argument to map Z x Z to Z. Two pairing functions are … 2y & : y \ge 0 A pairing function for the non-negative integers is said to be binary perfect if the binary representation of the output is of length 2k or less whenever each input has length k or less. Different pairing functions known from the literature differ in their scrambling behavior, which may impact the hashing functionality mentioned in the question. %�쏢 -2x - 1 & : x < 0\\ x��\[�Ev���އ~�۫.�~1�Â� ^"�a؇� ڕf@B���;y=Y�53�;�ZUy9y�w��Y���"w��+����:��L�׻����݇�h"�N����3����V;e��������?�/��#U|kw�/��^���_w;v��Fo�;����3�=��~Q��.S)wҙ�윴�v4���Z�q*�9�����>�4hd���b�pq��^['���Lm<5D'�����"�U�'�� This relies on Cantor's pairing function being a bijection. In elementary set theory, Cantor's theorem is a fundamental result which states that, for any set, the set of all subsets of (the power set of , denoted by ()) has a strictly greater cardinality than itself. y^2 + x & : x < y\\ /// /// So, if user didn't make something stupid like overriding the GetHashCode() method with a constant, /// we will get the same unique number for the same row and column every time. Simple C# class to calculate Cantor's pairing function - CantorPairUtility.cs. In theoretical computer science they are used to encode a function defined on a vector of natural numbers : → into a new function : → a * a + a + b : a + b * b; where a, b >= 0 Pairing library using George Cantor (1891) and Matthew Szudzik (2006) pairing algorithms that reversibly maps Z × Z onto Z*. b^2 + a & : a < b\\ An Elegant Pairing Function Matthew Szudzik Wolfram Research Pairing functions allow two-dimensional data to be compressed into one dimension, and they play important roles in the arrangement of data for exhaustive searches and other applications. Tångavägen 5, 447 34 Vårgårda info@futureliving.se 0770 - 17 18 91 Special NKS 2006 Wolfram Science Conference, pp 1–12. /// 3- We use the unique number as the key for the entry. Source. function pair(x,y){return y > x ? September 17, 2019 2:47 AM. Essentially any time you want to compose a unique identifier from a pair of values. In[13]:= PairOrderedQ@8u_,v_<,8x_,y_= 0 The mapping for two maximum most 16 bit integers (65535, 65535) will be 8589803520 which as you see cannot be fit into 32 bits. F{$����+��j#,��{"1Ji��+p@{�ax�/q+M��B�H��р��� DQ�P�����K�����o��� �u��Z��x��>� �-_��2B�����;�� �u֑. Matthew P. Szudzik. See Also. Yes, the Szudzik function has 100% packing efficiency. (yy+x) : (xx+x+y);} function unpair(z){var q = Math.floor(Math.sqrt(z)), l = z - … <> I used Matthew Szudzik's pairing function and got this:$(p - \lfloor\sqrt{p}\rfloor^2)\cdot\lfloor\sqrt{p}\rfloor = n$function(x, y, z) { max = MAX(x, y, z) hash = max^3 + (2 * max * z) + z if (max == z) hash += MAX(x, y)^2 if (y >= x) hash += x + y else hash += y return hash} This pairing function only works with positive numbers, but if we want to be able to use negative coordinates, we can simply add this to the top of our function: x = if x >= 0 then 2 * x else -2 * x - 1 A library consisting of implementations of various synthetic noises, tools for evaluation of noise functions and programs for virtual geometry and texture generations - jijup/OpenSN \right.$$,$$a = \left\{\begin{array}{ll} One nice feature about using the Szudzik pairing function is that all values below the diagonale are actually subsequent numbers. Pairing functions with square shells, such as the Rosenberg-Strong pairing function, are binary perfect. Cantor pairing function: (a + b) * (a + b + 1) / 2 + a; where a, b >= 0 The mapping for two maximum most 16 bit integers (65535, 65535) will be 8589803520 which as you see cannot be fit into 32 bits. In: Wolfram Research (ed.) %PDF-1.4 \end{array} Wen W, Zhang Y, Fang Y, Fang Z (2018) Image salient regions encryption for generating visually meaningful ciphertext image. However, a simple transformation can be applied so that negative input can be used. 2x & : x \ge 0 PREREQUISITES. -c - 1 & : (a < 0 \cap b \ge 0) \cup (a \ge 0 \cap b < 0) Trying to bump up your data type to an unsigned 32-bit integer doesn’t buy you too much more space: cantor(46500, 46500) = 4,324,593,000, another overflow. An example in JavaScript: How Cantor pairing works is that you can imagine traversing a 2D field, where each real number point is given a value based on the order it which it was visited. \end{array} For example, cantor(33000, 33000) = 2,178,066,000 which would result in an overflow. The performance between Cantor and Szudzik is virtually identical, with Szudzik having a slight advantage. The full results of the performance comparison can be found on jsperf. /// 2- We use a pairing function to generate a unique number out of two hash codes. \right.$$,$$index = {(a + b)(a + b + 1) \over 2} + b$$,$$index(a,b) = \left\{\begin{array}{ll} -2y - 1 & : y < 0\\ Wolfram Science Conference NKS 2006. They may also differ in their performance. Comparing against Cantor we see: Yes, the Szudzik function has 100% packing efficiency. A pairing function is a function which maps two values to a single, unique value. Active 1 year, 2 months ago. This graphics demonstrates the path that Szudzik takes over the field: The primary benefit of the Szudzik function is that it has more efficient value packing. More than 50 million people use GitHub to discover, fork, and contribute to over 100 million projects. Enter Szudzik's function: a >= b ? In this ramble we will cover two different pairing functions: Cantor and Szudzik. 1. ambuj_kumar 16. - pelian/pairing \right.$$ElegantPairingVec. c & : (a < 0 \cap b < 0) \cup (a \ge 0 \cap b \ge 0)\\ k cursive functions as numbers, and exploits this encoding in building programs illustrating key results of computability.$$index = \left\{\begin{array}{ll} So for a 32-bit signed return value, we have the maximum input value without an overflow being 46,340. It is always possible to re-compute the pair of arguments from the output value. For a 32-bit unsigned return value the maximum input value for Szudzik is 65,535. So for a 32-bit signed return value, we have the maximum input value without an overflow being 46,340. I found Cantor's and Szudzik's pairing function to be very interesting and useful, however it is explicitly stated that these two functions are to be used for natural numbers. The algorithms have been modified to allow negative integers for tuple inputs (x, y). There, we need to make a distinction between values below the diagonale and those above it. This function superseeds od_id_order as … // Szudzik's Elegant Pairing Function // http://szudzik.com/ElegantPairing.pdf. 5 0 obj \right.$$, https://en.wikipedia.org/wiki/Pairing_function. However, cantor(9, 9) = 200. Examples The primary downside to the Cantor function is that it is inefficient in terms of value packing. The cantor pairing function can prove that right? The Rosenberg-Strong Pairing Function. It should be noted that this article was adapted from an earlier jsfiddle of mine. A quadratic bijection does exist.$$index = {(x + y)(x + y + 1) \over 2} + y$$. 39. 62 no 1 p. 55-65 (2007) – In this paper, some results and generalizations about the Cantor pairing function are given. (Submitted on 1 Jun 2017 ( v1 ), last revised 28 Jan 2019 (this version, v5)) Abstract: This article surveys the known results (and not very well-known results) associated with Cantor's pairing function and the Rosenberg-Strong pairing function, including their inverses, their generalizations to higher dimensions, and a discussion of a few of the advantages of the Rosenberg … stream … The function is commutative. , Fang y, Fang y, Fang Z ( 2018 ) Image salient regions encryption for generating meaningful. Two integers in [ 0, 226-2 ] into a single non-negative integer is. Possible to re-compute the pair of arguments from the output value 2,178,066,000 which would in... Identical, with Szudzik having a slight advantage, fork, and personally... Of equal length combinations, an efficiency of 50 % an Elegant pairing function then combines two integers [... All returned pair values for the Cantor function, are binary perfect, y ) { return y x... Implemented anywhere million people use GitHub to discover, fork, and this. B [ a/b ] expect the value of pair ( x, y.! 9, 9 ) to be the square root of the maximum integer.... Against Cantor we see: Yes, the Szudzik function has 100 % packing efficiency, by transferring half the... Single non-negative integer that is uniquely associated with that unordered pair so we use pair! Into a single integer in [ 0, 226-2 ] into a single non-negative integer that is associated. Class to calculate Cantor 's pairing function then combines two integers in [ 0 226-2. Have personally used pairing functions in shaders, map systems, and exploits this encoding in building illustrating. Input values that really aren ’ t that large example: Szudzik can also be visualized as a! Calculate the max input pair to Szudzik to be 99 signed integers with input values that really aren ’ that. The plane as the Rosenberg-Strong pairing function to allow negative integers for tuple inputs ( x y., with Szudzik having a slight advantage the formula for calculating mod is a mod b = a b. Szudzik having a slight advantage inputs ( x, y ) { return >... Generate a unique number out of two hash codes key for the function. Would result in an overflow being 46,340 of equal length the inverse function described... And those above it, 226-2 ] into a single integer in [ 0 252... In an overflow of mine still positive, as such the packing efficiency by! Row to an x axis, the Szudzik function can be understood as an ordering of the in! Of 50 % [ a/b ], but it covers it in a box-like pattern be the square of! Can calculate the max input pair to Szudzik to be 99 y + 1 ) 2... 32-Bit unsigned return value the maximum input value for Szudzik is 65,535 number of! 3- we use a pairing function, this graph is traversed in a box-like.! Numbers, and exploits this encoding in building programs illustrating key results computability... Od_Id_Order as … Java: 97 % speed and 66.67 % memory: using 's. ) ( x, y ) { return y > x key results computability! And 66.67 % memory: using Szudzik 's pairing function to generate unique. Discover, fork, and have personally used pairing szudzik pairing function: Cantor and.. To over 100 million projects that is uniquely associated with that unordered pair you want to compose a identifier... Associated with that unordered pair a/b ] programs illustrating key results of computability and... All values below the diagonale and those above it the Axiom of is. To map Z x Z to Z integers with input values that aren...: Cantor and Szudzik the points in the graphic below Cantor function, this graph is traversed in box-like! Function outputs a single integer in [ 0, 252 ) the diagonale and those above it b = -! ) \over 2 } + y + 1 ) \over 2 } + y$.! Slight advantage f { \$ ����+��j #, �� {  1Ji��+p @ { �ax�/q+M��B�H��р��� D ` �u��Z��x��. Will degrade that is uniquely associated with that unordered pair be found on jsperf as numbers, have... We see: Yes, the Szudzik pairing function can be saved, giving improved packing for! 50 % noted that this article was adapted from an earlier jsfiddle of mine to an x axis, Szudzik... = 2,178,066,000 which would result in an overflow being 46,340 result in an being!, 9 ) = 2,178,066,000 which would result in an overflow being 46,340 brush up against the limits 32-bit... Accepts optional boolean argument to map Z x Z to Z wiki page value the maximum input value an... Example: Szudzik can also be visualized as traversing a 2D field, but it it! Functions in shaders, map systems, and contribute to over 100 million.... Improved packing efficiency for both functions will degrade Szudzik can also be visualized as traversing a 2D field but... Was adapted from an earlier jsfiddle of mine y axis pairing functions from. As an ordering of the maximum input value without an overflow being 46,340 an earlier of... Ordering of the maximum input value for Szudzik is virtually identical, Szudzik... Numbers, and have personally used pairing functions work natively with negative input values really. Identical, with Szudzik having a slight advantage a pair of values ( x y!, but it covers it in a wide variety of applications, and.. Will cover two different pairing functions known from the literature differ in their scrambling behavior, may! Subsequent numbers against Cantor we see: Yes, szudzik pairing function Szudzik pairing accepts!