… The primary downside to the Cantor function is that it is inefficient in terms of value packing. the Szudzik pairing function, on two vectors of equal length. This is an implement of Szudzik's function. $$b = \left\{\begin{array}{ll} That fiddle makes note of the following references:$$index = \left\{\begin{array}{ll} \right.$$,$$a = \left\{\begin{array}{ll} %PDF-1.4 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. This means that all one hundred possible variations of ([0-9], [0-9]) would be covered (keeping in mind our values are 0-indexed). Example , the unpairing function is defined as follows (for Cantor's unpairing function please refer to (Wolfram and Gad-el-Hak, 2003) and (Cantor, 1878)): (14) p f S z u d z i k − 1 (a) = a − ⌊ a ⌋ 2 x ⌊ a ⌋ y if x < y, or (15) p f S z u d z i k − 1 (a) = ⌊ a ⌋ x a − ⌊ a ⌋ 2 − ⌊ a ⌋ y. else. 5 0 obj 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 1. ambuj_kumar 16. Wolfram Science Conference NKS 2006. 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. a^2 + a + b & : a \ge b Essentially any time you want to compose a unique identifier from a pair of values. \right.$$, https://en.wikipedia.org/wiki/Pairing_function. A data.frame containing IDs and the computed integer. However, a simple transformation can be applied so that negative input can be used. There, we need to make a distinction between values below the diagonale and those above it. In a more pragmatic way, it may be … 39. A pairing function that maps two values to a unique third value. Additional space can be saved, giving improved packing efficiency, by transferring half to the negative axis. Szudzik, Matthew P. Abstract This article surveys the known results (and not very well-known results) associated with Cantor's pairing function and the Rosenberg-Strong pairing function, including their inverses, their generalizations to higher dimensions, and a discussion of a few of the advantages of the Rosenberg-Strong pairing function over Cantor's pairing function in practical applications. My original idea was to have it tie into a "midasgym.com" website with Google Calendar and .ICS export integration, but when I cut those features it made the website seem redundant, and the whole system was now pretty overengineered. For the Szudzik PF in eq. /// 3- We use the unique number as the key for the entry. \end{array} \right.$$ However, cantor(9, 9) = 200. A pairing function is a function which maps two values to a single, unique value. Matthew P. Szudzik 2019-01-28. Viewed 40 times 0. This can be easily implemented in any language. The performance between Cantor and Szudzik is virtually identical, with Szudzik having a slight advantage. The inverse function is described at the wiki page. In this ramble we will cover two different pairing functions: Cantor and Szudzik. - pelian/pairing b^2 + a & : a < b\\ /// /// 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. 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]) 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. a * a + a + b : a + b * b; where a, b >= 0 x and y have to be non-negative integers. The typical example of a pairing function that encodes two non-negative integers onto a single non-negative integer (therefore a function ) is the Cantor function, instrumental to the demonstration that, for example, the rational can be mapped onto the integers.. Source. /// 2- We use a pairing function to generate a unique number out of two hash codes. Passenger List. Other than that, the same principles apply. The pairing function then combines two integers in [0, 2 26 -2] into a single integer in [0, 2 52). It should be noted that this article was adapted from an earlier jsfiddle of mine. This is useful in a wide variety of applications, and have personally used pairing functions in shaders, map systems, and renderers. 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. Sometimes you have to encode reversibly two (or more) values onto a single one. This function superseeds od_id_order as it is faster on large datasets Like a window into their day-to-day life, Szudzik census records can tell you where and how your ancestors worked, their level of education, veteran status, and more. Pairing library using George Cantor (1891) and Matthew Szudzik (2006) pairing algorithms that reversibly maps Z × Z onto Z*. Parameters [in] _a: First value, must be a non-negative integer. As such, we can calculate the max input pair to Szudzik to be the square root of the maximum integer value. \end{array} This function superseeds od_id_order as it is faster on large datasets See Also. $$index = \left\{\begin{array}{ll} Here are the slides from my presentation, "An elegant pairing function" (in Mathematica format). It returns a vector of ID numbers. Ask Question Asked 1 year, 2 months ago. Abstract This article surveys the known results (and not very well-known re- sults) associated with Cantor’s pairing function and the Rosenberg-Strong pairing function, including their inverses, their generalizations to higher dimensions, and a discussion of a few of the advantages of the Rosenberg- Strong pairing function over Cantor… The full results of the performance comparison can be found on jsperf. \right.$$, $$c(a,b) = \left\{\begin{array}{ll} The pairing function can be understood as an ordering of the points in the plane. They may also differ in their performance. In a perfectly efficient function we would expect the value of pair(9, 9) to be 99. It returns a vector of ID numbers. 2x & : x \ge 0 Wen W, Zhang Y, Fang Y, Fang Z (2018) Image salient regions encryption for generating visually meaningful ciphertext image. y^2 + x & : x < y\\$$index = {(x + y)(x + y + 1) \over 2} + y$$. Trying to bump up your data type to an unsigned 32-bit integer doesn’t buy you too much more space: cantor(46500, 46500) = 4,324,593,000, another overflow. An example in JavaScript: How Cantor pairing works is that you can imagine traversing a 2D field, where each real number point is given a value based on the order it which it was visited. For example, extracting the n th element from our enumeration of three-tuples indexes about \sqrt[3]{n} elements into each of its components instead of, say, indexing \sqrt[2]{n} into one and \sqrt[4]{n} into the other two, as you would if a three-tuple were built out of nested pairs. I used Matthew Szudzik's pairing function and got this: (p - \lfloor\sqrt{p}\rfloor^2)\cdot\lfloor\sqrt{p}\rfloor = n A pairing function is a mathematical function taking two numbers as an argument and returning a third number, which uniquely identifies the pair of input arguments. In[13]:= PairOrderedQ@8u_,v_<,8x_,y_ The proc gen seed is the gym number listed at the top of the board, which feeds into a Szudzik pairing function. The function is commutative. Slides for Matthew Szudzik's Presentation. Enter Szudzik's function: a >= b ? Use a pairing function for prime factorization. If you want to have all paris x, y < 2 15, then you can go with the Szudzik's function: σ (x, y) = { x 2 + x + y if x ≥ y x + y 2 otherwise Like Cantor, the Szudzik function can be easily implemented anywhere. Value. Matthew P. Szudzik. \right.$$, $$index = {(a + b)(a + b + 1) \over 2} + b$$, $$index(a,b) = \left\{\begin{array}{ll} F{����+��j#,��{"1Ji��+p@{�ax�/q+M��B�H��р��� DQ�P�����K�����o��� �u��Z��x��>� �-_��2B�����;�� �u֑. On Linux/OSX this value is uint32_t. (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 … -2y - 1 & : y < 0\\ Special NKS 2006 Wolfram Science Conference, pp 1–12. c & : (a < 0 \cap b < 0) \cup (a \ge 0 \cap b \ge 0)\\ On Windows this value is uint16_t. \end{array} So for a 32-bit signed return value, we have the maximum input value without an overflow being 46,340. a^2 + a + b & : a \ge b Yes, the Szudzik function has 100% packing efficiency. This is an implement of Szudzik's function. Pairing functions with square shells, such as the Rosenberg-Strong pairing function, are binary perfect. Szudzik's Elegant Pairing Function. One nice feature about using the Szudzik pairing function is that all values below the diagonale are actually subsequent numbers. \end{array} A pairing function that maps two values to a unique third value. the Szudzik pairing function, on two vectors of equal length. Active 1 year, 2 months ago. b^2 + a & : a < b\\ The Rosenberg-Strong Pairing Function. September 17, 2019 2:47 AM. \end{array} 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. 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. %�쏢 -2x - 1 & : x < 0\\ 2y & : y \ge 0 So for a 32-bit signed return value, we have the maximum input value without an overflow being 46,340. Szudzik pairing function accepts optional boolean argument to map Z x Z to Z. We quickly start to brush up against the limits of 32-bit signed integers with input values that really aren’t that large. There are 14 immigration records available for the last name Szudzik. Java : 97% speed and 66.67% memory : using Szudzik's Pairing Function and HashSet. The algorithms have been modified to allow negative integers for tuple inputs (x, y). The function outputs a single non-negative integer that is uniquely associated with that unordered pair. Download ElegantPairing.nb. ��� ^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! The elegant-pair module implements the pairing function created by Matthew Szudzik from Wolfram Research, Inc. API Module Functions pair(x, y) Generate the elegant pair of the non-negative integers x and y. unpair(z) Inverts an elegant pair into its component integers x and y. [in] _b: … 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�4hd���b�pq��^['���Lm<5D'�����"�U�'�� As such, we can calculate the max input pair to Szudzik to be the square root of the maximum integer value. For a 32-bit unsigned return value the maximum input value for Szudzik is 65,535. Neither Cantor nor Szudzik pairing functions work natively with negative input values. For the Szudzik pairing function, the situation is only slightly more complicated. x^2 + x + y & : x \ge y Szudzik M (2006) An elegant pairing function. For the Cantor function, this graph is traversed in a diagonal function is illustrated in the graphic below. \end{array} The foundation of efficient enumeration lies in both direct access to encoded structures, achieved with well-known and new pairing functions, and dependent enumeration, which embeds constraints into the enumeration to avoid backtracking. od_oneway 148 VIEWS. 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. -c - 1 & : (a < 0 \cap b \ge 0) \cup (a \ge 0 \cap b < 0) It should be noted though that all returned pair values are still positive, as such the packing efficiency for both functions will degrade. So we use 200 pair values for the first 100 combinations, an efficiency of 50%. Different pairing functions known from the literature differ in their scrambling behavior, which may impact the hashing functionality mentioned in the question. Another JavaScript example: Szudzik can also be visualized as traversing a 2D field, but it covers it in a box-like pattern. Comparing against Cantor we see: Yes, the Szudzik function has 100% packing efficiency. It is always possible to re-compute the pair of arguments from the output value. For example, cantor(33000, 33000) = 2,178,066,000 which would result in an overflow. 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. stream In: Wolfram Research (ed.) \right.$$ GitHub Gist: instantly share code, notes, and snippets. 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. 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