# szudzik pairing function

x^2 + x + y & : x \ge y Enter Szudzik's function: a >= b ? Essentially any time you want to compose a unique identifier from a pair of values. The Rosenberg-Strong Pairing Function. So we use 200 pair values for the first 100 combinations, an efficiency of 50%. 62 no 1 p. 55-65 (2007) – In this paper, some results and generalizations about the Cantor pairing function are given. 148 VIEWS. More than 50 million people use GitHub to discover, fork, and contribute to over 100 million projects. As such, we can calculate the max input pair to Szudzik to be the square root of the maximum integer value. a^2 + a + b & : a \ge b For a 32-bit unsigned return value the maximum input value for Szudzik is 65,535. 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. cantor pairing function inverse. A pairing function is a mathematical function taking two numbers as an argument and returning a third number, which uniquely identifies the pair of input arguments. stream Neither Cantor nor Szudzik pairing functions work natively with negative input values. ElegantPairingVec. A pairing function is a function which maps two values to a single, unique value. /// 2- We use a pairing function to generate a unique number out of two hash codes. Szudzik, M. (2006): An Elegant Pairing Function. a * a + a + b : a + b * b; where a, b >= 0 For a 32-bit unsigned return value the maximum input value for Szudzik is 65,535. // Szudzik's Elegant Pairing Function // http://szudzik.com/ElegantPairing.pdf. Matthew P. Szudzik. /// 3- We use the unique number as the key for the entry. Comparing against Cantor we see: Yes, the Szudzik function has 100% packing efficiency. 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. \right.$$,$$index = {(a + b)(a + b + 1) \over 2} + b$$,$$index(a,b) = \left\{\begin{array}{ll} Wolfram Science Conference NKS 2006. Matthew P. Szudzik 2019-01-28. However, cantor(9, 9) = 200. c & : (a < 0 \cap b < 0) \cup (a \ge 0 \cap b \ge 0)\\ Active 1 year, 2 months ago. 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�'�� It returns a vector of ID numbers. 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. Szudzik, Matthew P. 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-Strong pairing function over Cantor's pairing function in practical applications. 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. -2y - 1 & : y < 0\\ the Szudzik pairing function, on two vectors of equal length. $$b = \left\{\begin{array}{ll} This graphics demonstrates the path that Szudzik takes over the field: The primary benefit of the Szudzik function is that it has more efficient value packing. 2x & : x \ge 0 As such, we can calculate the max input pair to Szudzik to be the square root of the maximum integer value. (Submitted on 1 Jun 2017 ( v1 ), last revised 28 Jan 2019 (this version, v5)) Abstract: This article surveys the known results (and not very well-known results) associated with Cantor's pairing function and the Rosenberg-Strong pairing function, including their inverses, their generalizations to higher dimensions, and a discussion of a few of the advantages of the Rosenberg … However, a simple transformation can be applied so that negative input can be used. See Also. You can then map the row to an X axis, the column to an Y axis. The pairing function then combines two integers in [0, 226-2] into a single integer in [0, 252). The performance between Cantor and Szudzik is virtually identical, with Szudzik having a slight advantage. The cantor pairing function can prove that right? - pelian/pairing So for a 32-bit signed return value, we have the maximum input value without an overflow being 46,340. And as the section on the inversion ends by saying, "Since the Cantor pairing function is invertible, it must be one-to-one and onto." b^2 + a & : a < b\\ Source. \right.$$, https://en.wikipedia.org/wiki/Pairing_function. Let's not fail silently! The full results of the performance comparison can be found on jsperf. /// /// 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. There, we need to make a distinction between values below the diagonale and those above it. -c - 1 & : (a < 0 \cap b \ge 0) \cup (a \ge 0 \cap b < 0) A quadratic bijection does exist. 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. This means that all one hundred possible variations of ([0-9], [0-9]) would be covered (keeping in mind our values are 0-indexed). The formula for calculating mod is a mod b = a - b[a/b]. This can be easily implemented in any language. It is always possible to re-compute the pair of arguments from the output value. Examples $$index = \left\{\begin{array}{ll} \right.$$, $$c(a,b) = \left\{\begin{array}{ll} Use a pairing function for prime factorization. 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 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]) Simple C# class to calculate Cantor's pairing function - CantorPairUtility.cs. The function is commutative. In: Wolfram Research (ed.) Usage. Different pairing functions known from the literature differ in their scrambling behavior, which may impact the hashing functionality mentioned in the question. \end{array} PREREQUISITES. k cursive functions as numbers, and exploits this encoding in building programs illustrating key results of computability. 39. Viewed 40 times 0. Wen W, Zhang Y, Fang Y, Fang Z (2018) Image salient regions encryption for generating visually meaningful ciphertext image. In theoretical computer science they are used to encode a function defined on a vector of natural numbers : → into a new function : → Szudzik M (2006) An elegant pairing function. Two pairing functions are … Yes, the Szudzik function has 100% packing efficiency. 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. b^2 + a & : a < b\\ 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. Value. Pairing functions with square shells, such as the Rosenberg-Strong pairing function, are binary perfect. Other than that, the same principles apply. %�쏢 <> This relies on Cantor's pairing function being a bijection. One nice feature about using the Szudzik pairing function is that all values below the diagonale are actually subsequent numbers. y^2 + x & : x < y\\ Like Cantor, the Szudzik function can be easily implemented anywhere. 5 0 obj September 17, 2019 2:47 AM. We quickly start to brush up against the limits of 32-bit signed integers with input values that really aren’t that large. 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. \end{array} Ask Question Asked 1 year, 2 months ago. I used Matthew Szudzik's pairing function and got this: (p - \lfloor\sqrt{p}\rfloor^2)\cdot\lfloor\sqrt{p}\rfloor = n \end{array} 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� �-_��2B�����;�� �u֑. Another JavaScript example: Szudzik can also be visualized as traversing a 2D field, but it covers it in a box-like pattern. ��� ^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 is useful in a wide variety of applications, and have personally used pairing functions in shaders, map systems, and renderers. The function outputs a single non-negative integer that is uniquely associated with that unordered pair. That fiddle makes note of the following references:$$index = \left\{\begin{array}{ll} function pair(x,y){return y > x ? \right. \end{array} %PDF-1.4 The inverse function is described at the wiki page. In[13]:= PairOrderedQ@8u_,v_<,8x_,y_= b ? Additional space can be saved, giving improved packing efficiency, by transferring half to the negative axis. Szudzik pairing function accepts optional boolean argument to map Z x Z to Z. a * a + a + b : a + b * b; where a, b >= 0 \end{array} 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 But for R the Axiom of Choice is not required. Special NKS 2006 Wolfram Science Conference, pp 1–12. Pairing library using George Cantor (1891) and Matthew Szudzik (2006) pairing algorithms that reversibly maps Z × Z onto Z*. od_id* functions take two vectors of equal length and return a vector of IDs, which are unique for each combination but the same for twoway flows. For the Szudzik pairing function, the situation is only slightly more complicated. In a perfectly efficient function we would expect the value of pair(9, 9) to be 99. 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