A Construction for Balancing Non-Binary Sequences Based on Gray Code Prefixes Elie N

Total Page:16

File Type:pdf, Size:1020Kb

A Construction for Balancing Non-Binary Sequences Based on Gray Code Prefixes Elie N 1 A Construction for Balancing Non-Binary Sequences Based on Gray Code Prefixes Elie N. Mambou and Theo G. Swart, Senior Member, IEEE Abstract—We introduce a new construction for the balancing simply has to sequentially complement the bits until this of non-binary sequences that make use of Gray codes for prefix weight is attained. coding. Our construction provides full encoding and decoding Al-Bassam [3] presented a generalization of Knuth’s algo- of sequences, including the prefix. This construction is based on rithm for binary codes, non-binary codes and semi-balanced a generalization of Knuth’s parallel balancing approach, which codes (the latter occur where the number of 0’s and 1’s differs can handle very long information sequences. However, the overall by at most a certain value in each sequence of the code). The sequence—composed of the information sequence, together with the prefix—must be balanced. This is reminiscent of Knuth’s balancing of binary codes with low DC level is based on DC- serial algorithm. The encoding of our construction does not make free coset codes. For the design of non-binary balanced codes, use of lookup tables, while the decoding process is simple and symbols in the information sequence are q-ary complemented can be done in parallel. from one side, but because this process does not guarantee balancing, an extra redundant symbol is added to enforce Keywords—Balanced sequence, DC-free codes, Gray code prefix. the balancing (similar to our approach later on). Information regarding how many symbols to complement is sent by using a balanced prefix. I. INTRODUCTION Capocelli et al. [4] proposed using two functions that must HE use of balanced codes is crucial for some information satisfy certain properties to encode any q-ary sequence into T transmission systems. Errors can occur in the process of balanced sequences. The first function is similar to Knuth’s storing data onto optical devices due to the low frequency of serial scheme: it outputs a prefix sequence depending on operation between structures of the servo and the data written the original sequence’s weight. Additionally, all the q-ary on the disc. This can be avoided by using encoded balanced sequences are partitioned into disjointed chains, where each codes, as no low frequencies are observed. In such systems, chain’s sequences have unique weights. The second function balanced codes are also useful for tracking the data on the is then used to select an alternate sequence in the chain disc. Balanced codes are also used for countering cut-off at low containing the original information sequence, such that the frequencies in digital transmission through capacitive coupling chosen prefix and the alternate sequence together are balanced. or transformers. This cut-off is caused by multiple same-charge Tallini and Vaccaro [8] presented another construction for bits, and results in a DC level that charges the capacitor in the balanced q-ary sequences that makes use of balancing and AC coupler [1]. In general, the suppression of low-frequency compression. Sequences that are close to being balanced are spectrum can be done with balanced codes. encoded with a generalization of Knuth’s serial scheme. Based A large body of work on balanced codes is derived from on the weight of the information sequence, a prefix is chosen. the simple algorithm for balancing sequences proposed by Symbols are then “complemented in stages”, one at a time, Knuth [2]. According to Knuth’s parallel algorithm, a binary until the weight that balances the sequence and prefix together sequence, x, of even length k, can always be balanced by is attained. Other sequences are compressed with a uniquely complementing its first or last i bits, where 0 i k. The decodable variable length code and balanced using the saved index i is then encoded as a balanced prefix that≤ is≤ appended space. to the data. The decoder can easily recover i from the prefix, Swart and Weber [5] extended Knuth’s parallel balancing and then again complementing the first or last i bits to obtain scheme to q-ary sequences with parallel decoding. However, arXiv:1706.00852v1 [cs.IT] 2 Jun 2017 the original information. For Knuth’s serial (or sequential) this technique does not provide a prefix code implementation, algorithm, the prefix is used to provide information regarding with the assumption that small lookup tables can be used the information sequence’s initial weight. Bits are sequentially for this. Our approach aims to implement these prefixes via complemented from one side of the overall sequence, until the Gray codes. Swart and Weber’s scheme will be expanded on information sequence and prefix together are balanced. Since in Section II-A, as it also forms the basis of our proposed the original weight is indicated by the prefix, the decoder algorithm. Swart and Immink [6] described a prefixless algorithm for This paper was presented in part at the IEEE International Symposium on balancing of q-ary sequences. By using the scheme from [5] Information Theory, Barcelona, Spain, July, 2016. and applying precoding to a very specific error correction code, The authors are with the Department of Electrical and Electronic Engineer- it was shown that balancing can be achieved without the need ing Science, University of Johannesburg, P. O. Box 524, Auckland Park, 2006, South Africa (e-mails: emambou, tgswart @uj.ac.za). for a prefix. This work is based onf research supportedg in part by the National Research Pelusi et al. [7] presented a refined implementation of Foundation of South Africa (UID 77596). Knuth’s algorithm for parallel decoding of q-ary balanced 2 6 codes, similar to [5]. This method significantly improved [4] and [8] in terms of complexity. 5 The rest of this paper is structured as follows. In Section II, 4 β4,3 we present the background for our work, which includes Swart ) y and Weber’s balancing scheme for q-ary sequences [5] and ( 3 w non-binary Gray code theory [10]. In Section III, a construction 2 is presented for sequences where k = qt. Section IV extends on our proposed construction to sequences with k = qt. 1 Finally, Section V deals with the redundancy and complexity6 0 1 2 3 4 5 6 7 8 9 10 11 of our construction compared to prior art constructions, our z conclusions are presented in Section VI. Fig. 1. Random walk graph of w(y) for Example 1 II. PRELIMINARIES balanced. Let x = ( ) be a -ary information sequence of x1x2 : : : xk q z x b = y w(y) length , where 0 1 1 is from a non-binary q z k xi ; ; : : : q 0 (2101) ⊕ (0000) = (2101) 4 alphabet. A prefix of2 length f is− appendedg to x. The prefix 3 r 1 (2101) ⊕ (1000) = (0101) 2 and information together are denoted by c = ( ) of 3 c1c2 : : : cn 2 (2101) ⊕ (1100) = (0201) 3 length = + , where 0 1 1 . Let (c) refer 3 n k r ci ; ; : : : q w 3 (2101) ⊕ (1110) = (0211) 4 to the weight of c, that is the2 f algebraic sum− g of symbols in c. 3 4 (2101) ⊕ (1111) = (0212) 5 The sequence c is said to be balanced if 3 5 (2101) ⊕ (2111) = (1212) 6 ⊕3 n 6 (2101) 3 (2211) = (1012) 4 X n(q 1) ⊕ w(c) = c = − : 7 (2101) 3 (2221) = (1022) 5 i 2 ⊕ i=1 8 (2101) 3 (2222) = (1020) 3 9 (2101) ⊕ (0222) = (2020) 4 ⊕3 Let β represent this value obtained at the balancing state. 10 (2101) 3 (0022) = (2120) 5 n;q ⊕ For the rest of the paper, the parameters k, n, q and r are 11 (2101) 3 (0002) = (2100) 3 ⊕ chosen in such a way that the balancing value, β = n(q n;q − For this example, there are four occurrences of balanced 1)=2, leads to a positive integer. sequences. A (γ; τ)-random walk refers to a path with random increases of γ and decreases of τ. In our case, a random walk graph is the A. Balancing of q-ary Sequences plot of the function of w(y) versus z. In general, the random walk graph of w(y) always forms a (1; q 1)-random walk Any information sequence, x of length k and alphabet size − q, can always be balanced by adding (modulo q) to that [5]. Fig. 1 presents the (1; 2)-random walk for Example 1. The sequence one sequence from a set of balancing sequences [5]. dashed line indicates the balancing value β4;3 = 4. The balancing sequence, b = (b b : : : b ) is derived as This method, as presented in [5], assumed that the z indices s;p 1 2 k can be sent using balanced prefixes, but the actual encoding of these was not taken into account. For instance, in Example 1 s; i > p; b = indices z = 0, 3, 6 and 9 must be encoded into balanced i s + 1 (mod q); i p; ≤ prefixes, in order to send overall balanced sequences. where s and p are positive integers with 0 s q 1 and 0 p k 1. Let z be the iterator through≤ all≤ possible− B. Non-binary Gray Codes ≤ ≤ − balancing sequences, such that z = sn+p and 0 z kq 1. Binary Gray codes were first proposed by Gray [9] for ≤ ≤ − Let y refer to the resulting sequence when adding (modulo solving problems in pulse code communication, and have q) the balancing sequence to the information sequence, y = been extended to various other applications. The assumption x q bs;p, where q denotes modulo q addition.
Recommended publications
  • Final-Del-Lab-Manual-2018-19
    LAB MANUAL DIGITAL ELECTRONICS LABORATORY Subject Code: 210246 2018 - 19 INDEX Batch : - Page Date of Date of Signature of Sr.No Title No Conduction Submission Staff GROUP - A Realize Full Adder and Subtractor using 1 a) Basic Gates and b) Universal Gates Design and implement Code converters-Binary to Gray and BCD to 2 Excess-3 Design of n-bit Carry Save Adder (CSA) and Carry Propagation Adder (CPA). Design and Realization of BCD 3 Adder using 4-bit Binary Adder (IC 7483). Realization of Boolean Expression for suitable combination logic using MUX 4 74151 / DMUX 74154 Verify the truth table of one bit and two bit comparators using logic gates and 5 comparator IC Design & Implement Parity Generator 6 using EX-OR. GROUP - B Flip Flop Conversion: Design and 7 Realization Design of Ripple Counter using suitable 8 Flip Flops a. Realization of 3 bit Up/Down Counter using MS JK Flip Flop / D-FF 9 b. Realization of Mod -N counter using ( 7490 and 74193 ) Design and Realization of Ring Counter 10 and Johnson Ring counter Design and implement Sequence 11 generator using JK flip-flop INDEX Batch : - Design and implement pseudo random 12 sequence generator. Design and implement Sequence 13 detector using JK flip-flop Design of ASM chart using MUX 14 controller Method. GROUP - C Design and Implementation of 15 Combinational Logic using PLAs. Design and simulation of - Full adder , 16 Flip flop, MUX using VHDL (Any 2) Use different modeling styles. Design & simulate asynchronous 3- bit 17 counter using VHDL. Design and Implementation of 18 Combinational Logic using PALs GROUP - D Study of Shift Registers ( SISO,SIPO, 19 PISO,PIPO ) Study of TTL Logic Family: Feature, 20 Characteristics and Comparison with CMOS Family Study of Microcontroller 8051 : 21 Features, Architecture and Programming Model ` Digital Electronics Lab (Pattern 2015) GROUP - A Digital Electronics Lab (Pattern 2015) Assignment: 1 Title: Adder and Subtractor Objective: 1.
    [Show full text]
  • A New Approach to the Snake-In-The-Box Problem
    462 ECAI 2012 Luc De Raedt et al. (Eds.) © 2012 The Author(s). This article is published online with Open Access by IOS Press and distributed under the terms of the Creative Commons Attribution Non-Commercial License. doi:10.3233/978-1-61499-098-7-462 A New Approach to the Snake-In-The-Box Problem David Kinny1 Abstract. The “Snake-In-The-Box” problem, first described more Research on the SIB problem initially focused on applications in than 50 years ago, is a hard combinatorial search problem whose coding [14]. Coils are spread-k circuit codes for k =2, in which solutions have many practical applications. Until recently, techniques n-words k or more positions apart in the code differ in at least k based on Evolutionary Computation have been considered the state- bit positions [12]. (The well-known Gray codes are spread-1 circuit of-the-art for solving this deterministic maximization problem, and codes.) Longest snakes and coils provide the maximum number of held most significant records. This paper reviews the problem and code words for a given word size (i.e., hypercube dimension). prior solution techniques, then presents a new technique, based on A related application is in encoding schemes for analogue-to- Monte-Carlo Tree Search, which finds significantly better solutions digital converters including shaft (rotation) encoders. Longest SIB than prior techniques, is considerably faster, and requires no tuning. codes provide greatest resolution, and single bit errors are either recognised as such or map to an adjacent codeword causing an off- 1 INTRODUCTION by-one error.
    [Show full text]
  • NOTE on GRAY CODES for PERMUTATION LISTS Seymour
    PUBLICATIONS DE L’INSTITUT MATHEMATIQUE´ Nouvelle s´erie,tome 78(92) (2005), 87–92 NOTE ON GRAY CODES FOR PERMUTATION LISTS Seymour Lipschutz, Jie Gao, and Dianjun Wang Communicated by Slobodan Simi´c Abstract. Robert Sedgewick [5] lists various Gray codes for the permutations in Sn including the classical algorithm by Johnson and Trotter. Here we give an algorithm which constructs many families of Gray codes for Sn, which closely follows the construction of the Binary Reflexive Gray Code for the n-cube Qn. 1. Introduction There are (n!)! ways to list the n! permutations in Sn. Some such permutation lists have been generated by a computer, and some such permutation lists are Gray codes (where successive permutations differ by a transposition). One such famous Gray code for Sn is by Johnson [4] and Trotter [6]. In fact, each new permutation in the Johnson–Trotter (JT ) list for Sn is obtained from the preceding permutation by an adjacent transposition. Sedgewick [5] gave a survey of various Gray codes for Sn in 1977. Subsequently, Conway, Sloane and Wilks [1] gave a new Gray code CSW for Sn in 1989 while proving the existence of Gray codes for the reflection groups. Recently, Gao and Wang [2] gave simple algorithms, each with an efficient implementation, for two new permutation lists GW1 and GW2 for Sn, where the second such list is a Gray code for Sn. The four lists, JT , CSW , GW1 and GW2, for n = 4, are pictured in Figure 1. This paper gives an algorithm for constructing many families of Gray codes for Sn which uses an idea from the construction of the Binary Reflected Gray Code (BRGC) of the n-cube Qn.
    [Show full text]
  • Other Binary Codes (3A)
    Other Binary Codes (3A) Young Won Lim 1/3/14 Copyright (c) 2011-2013 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to [email protected]. This document was produced by using OpenOffice and Octave. Young Won Lim 1/3/14 Coding A code is a rule for converting a piece of information (for example, a letter, word, phrase, or gesture) into another form or representation (one sign into another sign), not necessarily of the same type. In communications and information processing, encoding is the process by which information from a source is converted into symbols to be communicated. Decoding is the reverse process, converting these code symbols back into information understandable by a receiver. Young Won Lim Other Binary Codes (4A) 3 1/3/14 Character Coding ASCII code BCD (Binary Coded Decimal) definitions for 128 characters: Number characters (0-9) 33 non-printing control characters (many now obsolete) 95 printable charactersi Young Won Lim Other Binary Codes (4A) 4 1/3/14 Representation of Numbers Fixed Point Number representation ● 2's complement +1234 ● 1's complement 0 -582978 coding ● sign-magnitude Floating Point Number representation +23.84380 ● IEEE 754 -1.388E+08
    [Show full text]
  • Loopless Gray Code Enumeration and the Tower of Bucharest
    Loopless Gray Code Enumeration and the Tower of Bucharest Felix Herter1 and Günter Rote1 1 Institut für Informatik, Freie Universität Berlin Takustr. 9, 14195 Berlin, Germany [email protected], [email protected] Abstract We give new algorithms for generating all n-tuples over an alphabet of m letters, changing only one letter at a time (Gray codes). These algorithms are based on the connection with variations of the Towers of Hanoi game. Our algorithms are loopless, in the sense that the next change can be determined in a constant number of steps, and they can be implemented in hardware. We also give another family of loopless algorithms that is based on the idea of working ahead and saving the work in a buffer. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems Keywords and phrases Tower of Hanoi, Gray code, enumeration, loopless generation Contents 1 Introduction: the binary reflected Gray code and the Towers of Hanoi 2 1.1 The Gray code . .2 1.2 Loopless algorithms . .2 1.3 The Tower of Hanoi . .3 1.4 Connections between the Towers of Hanoi and Gray codes . .4 1.5 Loopless Tower of Hanoi and binary Gray code . .4 1.6 Overview . .4 2 Ternary Gray codes and the Towers of Bucharest 5 3 Gray codes with general radixes 7 4 Generating the m-ary Gray code with odd m 7 5 Generating the m-ary Gray code with even m 9 arXiv:1604.06707v1 [cs.DM] 22 Apr 2016 6 The Towers of Bucharest++ 9 7 Simulation 11 8 Working ahead 11 8.1 An alternative STEP procedure .
    [Show full text]
  • Algorithms for Generating Binary Reflected Gray Code Sequence: Time Efficient Approaches
    2009 International Conference on Future Computer and Communication Algorithms for Generating Binary Reflected Gray Code Sequence: Time Efficient Approaches Md. Mohsin Ali, Muhammad Nazrul Islam, and A. B. M. Foysal Dept. of Computer Science and Engineering Khulna University of Engineering & Technology Khulna – 9203, Bangladesh e-mail: [email protected], [email protected], and [email protected] Abstract—In this paper, we propose two algorithms for Coxeter groups [4], capanology (the study of bell-ringing) generating a complete n-bit binary reflected Gray code [5], continuous space-filling curves [6], classification of sequence. The first one is called Backtracking. It generates a Venn diagrams [7]. Today, Gray codes are widely used to complete n-bit binary reflected Gray code sequence by facilitate error correction in digital communications such as generating only a sub-tree instead of the complete tree. In this digital terrestrial television and some cable TV systems. approach, both the sequence and its reflection for n-bit is In this paper, we present the derivation and generated by concatenating a “0” and a “1” to the most implementation of two algorithms for generating the significant bit position of the n-1 bit result generated at each complete binary reflected Gray code sequence leaf node of the sub-tree. The second one is called MOptimal. It () ≤ ≤ n − is the modification of a Space and time optimal approach [8] G i ,0 i 2 1 , for a given number of bits n. We by considering the minimization of the total number of outer evaluate the performance of these algorithms and other and inner loop execution for this purpose.
    [Show full text]
  • GRAPHS INDUCED by GRAY CODES 1. Introduction an N-Bit
    GRAPHS INDUCED BY GRAY CODES ELIZABETH L. WILMER AND MICHAEL D. ERNST Abstract. We disprove a conjecture of Bultena and Ruskey [1], that all trees which are cyclic graphs of cyclic Gray codes have diameter 2 or 4, by producing codes whose cyclic graphs are trees of arbitrarily large diameter. We answer affirmatively two other questions from [1], showing that strongly Pn Pn- compatible codes exist and that it is possible for a cyclic code to induce× a cyclic digraph with no bidirectional edge. A major tool in these proofs is our introduction of supercomposite Gray codes; these generalize the standard reflected Gray code by allowing shifts. We find supercomposite Gray codes which induce large diameter trees, but also show that many trees are not induced by supercomposite Gray codes. We also find the first infinite family of connected graphs known not to be induced by any Gray code | trees of diameter 3 with no vertices of degree 2. 1. Introduction n An n-bit Gray code B = (b1; b2;:::; bN ), N = 2 , lists all the binary n-tuples (\codewords") so that consecutive n-tuples differ in one bit. In a cyclic code, the first and last n-tuples also differ in one bit. Gray codes can be viewed as Hamiltonian paths on the hypercube graph; cyclic codes correspond to Hamiltonian cycles. Two Gray codes are isomorphic when one is carried to the other by a hypercube isomorphism. The transition sequence τ(B) = (τ1; τ2; : : : ; τN 1) of an n-bit Gray code B lists − the bit positions τi [n] = 1; 2; : : : ; n where bi and bi+1 differ.
    [Show full text]
  • Lab Manual Exp Code Converter.Pdf
    Hybrid Electronics Laboratory Design and Simulation of Various Code Converters Aim: To Design and Simulate Binary to Gray, Gray to Binary , BCD to Excess 3, Excess 3 to BCD code converters. Objectives: 1. To understand different codes 2. To design various Code converters using logic gates 3. To simulate various code converters 4. To understand the importance of code converters in real life applications Theory Binary Codes A symbolic representation of data/ information is called code. The base or radix of the binary number is 2. Hence, it has two independent symbols. The symbols used are 0 and 1. A binary digit is called as a bit. A binary number consists of sequence of bits, each of which is either a 0 or 1. Each bit carries a weight based on its position relative to the binary point. The weight of each bit position is one power of 2 greater than the weight of the position to its immediate right. e. g. of binary number is 100011 which is equivalent to decimal number 35. BCD Codes Numeric codes represent numeric information i.e. only numbers as a series of 0’s and 1’s. Numeric codes used to represent decimal digits are called Binary Coded Decimal (BCD) codes. A BCD code is one, in which the digits of a decimal number are encoded-one at a time into group of four binary digits. There are a large number of BCD codes in order to represent decimal digits0, 1, 2 …9, it is necessary to use a sequence of at least four binary digits.
    [Show full text]
  • A Review of Binary to Gray and Gray to Binary Code Conversion
    Volume 3, Issue 10, October – 2018 International Journal of Innovative Science and Research Technology ISSN No:-2456-2165 A Review of Binary to Gray and Gray to Binary Code Conversion Pratibha Soni Vimal Kumar Agarwal M. TECH. Scholar Associate Professor Apex Institute of Engineering and Technology Apex Institute of Engineering and Technology Jaipur, Rajasthan, India Jaipur, Rajasthan, India Abstract:- In today’s world of electronics industries low II. LITERATURE REVIEW power has emerged as a principal theme. Power dissipation has become an essential consideration as area One of the promising computing technologies assuring for VLSI Chip design and performance. With shrinking zero power dissipation is the Reversible Logic. Reversible technology reducing overall power management and Logic has a wide spectrum of applications like power consumption on chip are the main challenges Nanotechnology based systems, Bio Informatics optical below 100nm as a result of increased complexity. When Circuits, quantum computing, and Low Power VLSI. it comes to many designs available, power optimization is Therefore, this paper represents the 4 bit reversible essential as timing as a result to the demand to extend comparator based on classical logic circuit which uses the battery life and reduce package cost. Asto reduce the existing reversible gates. The design simply aims at power consumption we have many techniques from those reducing optimization parameters like quantum cost, techniques, lets we implement on Reverse Logic. In garbage outputs, and the number of constant inputs. The reversible logic it as greater advantage and executing in results obtained in this research work indicate that the high speed, with less power dissipation which is proposed comparator has one constant input and 4 quantum implemented by using basic logic gates in less area.
    [Show full text]
  • Gray Codes and Efficient Exhaustive Generation for Several Classes Of
    Gray codes and efficient exhaustive generation for several classes of restricted words Ahmad Sabri To cite this version: Ahmad Sabri. Gray codes and efficient exhaustive generation for several classes of restricted words. Other [cs.OH]. Université de Bourgogne, 2015. English. NNT : 2015DIJOS007. tel-01203339 HAL Id: tel-01203339 https://tel.archives-ouvertes.fr/tel-01203339 Submitted on 22 Sep 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Thèse de Doctorat école doctorale sciences pour l’ingénieur et microtechniques UNIVERSITÉ DE BOURGOGNE Gray codes and efficient exhaustive generation for several classes of restricted words n AHMAD SABRI Thèse de Doctorat école doctorale sciences pour l’ingénieur et microtechniques UNIVERSITÉ DE BOURGOGNE THESE` present´ ee´ par AHMAD SABRI pour obtenir le Grade de Docteur de l’Universite´ de Bourgogne Specialit´ e´ : Informatique Gray codes and efficient exhaustive generation for several classes of restricted words Unite´ de Recherche : Laboratoire Electronique, Informatique et Image (LE2I) Soutenue publiquement
    [Show full text]
  • A Gray Code for Combinations of a Multiset
    AGray Co de for Combinations of a Multiset y Frank Ruskey Carla Savage Department of Computer Science Department of Computer Science University of Victoria North Carolina State University Victoria BC VW P Canada Raleigh North Carolina Abstract Let Ck n n n denote the set of all k elementcombinations of the t multiset consisting of n o ccurrences of i for i tEach combination i is itself a multiset For example C ff g f g f g f gg Weshowthatmultiset combinations can b e listed so that successive combina ultaneously generalize tions dier by one element Multiset combinations sim for which minimal change listings called Gray co des are known comp ositions and combinations Research supp orted by NSERC grantA y Research supp orted by National Science Foundation GrantNo CCR and National Security Agency Grant No MDAH Intro duction By Ck n n n we denote the set of all ordered t tuples x x x t t satisfying x x x k where x n for i t t i i We assume throughout the pap er that n is a p ositiveinteger for i t i Solutions to are referred to as combinations of a multiset b ecause they can b e regarded as k subsets of the multiset consisting of n copies of i for i t i Solution x x x corresp onds to the k subset consisting of x copies of i for t i i t For example C corresp onds to ff g f g f g f gg When each n is the set Ck n n n is simply the set of all combinations of i t t elements chosen k at a time Alternatively the elements of Ck n n n can b e regarded as placements t of k identical balls into
    [Show full text]
  • Gray Codes and Puzzles on Iterated Complete Graphs
    GRAY CODES AND PUZZLES ON ITERATED COMPLETE GRAPHS ELIZABETH WEAVER ADVISOR: PAUL CULL OREGON STATE UNIVERSITY ABSTRACT. There exists an infinite family of puzzles, the SF Puzzle, that correspond to labelings on iterated complete graphs of odd dimensions, the simplest being the well-known Towers of Hanoi puzzle. There is also a puzzle, knownas Spin-Out that correspondsto the reflected binary Gray code; however, thus far there has been no family of puzzles defined that can be represented by labelings on all iterated complete graphs of even dimensions. This paper explores the properties that labelings of known puzzles possess, tries to determine the necessary labeling properties for a new puzzle, and n defines a puzzle on K4 . Unfortunately, this puzzle does not extend to labelings on all even dimension iterated complete graphs. 1. INTRODUCTION Mathematics forms the basis for many well-known and interesting puzzles. For example, the popular game the Towers of Hanoi can be represented graphically and has several applications in graph theory and in error correcting codes [2]. By mathematically generalizing the rules of the Towers of Hanoi, a new family of puzzles was created that can be easily represented by graphs [5]. In these puzzles there are an odd number of possible positions for each piece and an easily defined rule to determine the possible moves for each piece. It seems logical that a family of puzzles should exist where there are an even number d possible moves for each piece, but with the exception of d = 2 no one rule seems to describe a puzzle like that of the Towers of Hanoi.
    [Show full text]