Discrete Structure - Gray Code

1. The text gives a recursive definition for Gray codes, G. (pp. 220,221), for which successive entries in the list of codes differ by only one bit. We can prove that successive entries differ in only one bit using mathematical induction. The base case, for n=1 is trivial, given the definition for Gj. For the induction step, we assume that for an arbitrary, n > 1, successive entries of the gray code, G., differ in only one bit. In order to prove that, under this assumption, successive entries of the gray code, Gn+1, differ in only one bit, we can consider three cases: - Both entries are part of (OGA) - One is the last entry in (OG), the next is the first entry in (16) - Both entries are part of (16) Using the following notation: - For i = 1, ..., 2n+1S, for the i-th entry in G+ - For j = 1, ..., n +1: S.[] for the j-th bit in the i-th entry in Gh+1 - For j, k = 1, .... n + I with j sk: Slj...k] for bits through k of the i-th entry in G.+1 along with the definition given in the text and the notation used in the text, explain: (a) Why, for i - 1. .... (2" - 1): S, and Sr+1 differ only in one bit (b) Why S and Sa+1) differ only in one bit (e) Why, for i = (24 + 1). .... 21+1, S, and S. 1 differ only in one bit V W IS LIUL LO UMIY HALUAI Ulu Ul Lue n-cube, but it is the easiest to describe. The standard way to write the Gray Code is as a column of strings, where the last string is followed by the first string to complete the circuit. Basis for the Gray Code (n = 1): The Gray Code for the 1-cube is G - CHAPTER 9. GRAPH THEORY 1. Note that the edge between 0 and 1 is used twice in this circuit. That doesn't violate any rules for Hamiltonian circuits, but can only happen if a graph has two vertices, Recursive definition of the Gray Code: Given the Gray Code for the 7-eube, 12 1. then G+ is obtained by (1) listing G with each string prefixed with 0. and then (2) reversing the list of strings in G with each string prefixed with 1. Symbolically, the recursion can be expressed as follows, where G N the reverse of list G G 16 The Gray Codes for the 2-cube and 3-cube are 010 will think of these from th Gract 1. The text gives a recursive definition for Gray codes, G. (pp. 220,221), for which successive entries in the list of codes differ by only one bit. We can prove that successive entries differ in only one bit using mathematical induction. The base case, for n=1 is trivial, given the definition for Gj. For the induction step, we assume that for an arbitrary, n > 1, successive entries of the gray code, G., differ in only one bit. In order to prove that, under this assumption, successive entries of the gray code, Gn+1, differ in only one bit, we can consider three cases: - Both entries are part of (OGA) - One is the last entry in (OG), the next is the first entry in (16) - Both entries are part of (16) Using the following notation: - For i = 1, ..., 2n+1S, for the i-th entry in G+ - For j = 1, ..., n +1: S.[] for the j-th bit in the i-th entry in Gh+1 - For j, k = 1, .... n + I with j sk: Slj...k] for bits through k of the i-th entry in G.+1 along with the definition given in the text and the notation used in the text, explain: (a) Why, for i - 1. .... (2" - 1): S, and Sr+1 differ only in one bit (b) Why S and Sa+1) differ only in one bit (e) Why, for i = (24 + 1). .... 21+1, S, and S. 1 differ only in one bit V W IS LIUL LO UMIY HALUAI Ulu Ul Lue n-cube, but it is the easiest to describe. The standard way to write the Gray Code is as a column of strings, where the last string is followed by the first string to complete the circuit. Basis for the Gray Code (n = 1): The Gray Code for the 1-cube is G - CHAPTER 9. GRAPH THEORY 1. Note that the edge between 0 and 1 is used twice in this circuit. That doesn't violate any rules for Hamiltonian circuits, but can only happen if a graph has two vertices, Recursive definition of the Gray Code: Given the Gray Code for the 7-eube, 12 1. then G+ is obtained by (1) listing G with each string prefixed with 0. and then (2) reversing the list of strings in G with each string prefixed with 1. Symbolically, the recursion can be expressed as follows, where G N the reverse of list G G 16 The Gray Codes for the 2-cube and 3-cube are 010 will think of these from th Gract