Optimal Quasi-Gray Codes: Does the Alphabet Matter?

12/05/2017
by   Diptarka Chakraborty, et al.
0

A quasi-Gray code of dimension n and length ℓ over an alphabet Σ is a sequence of distinct words w_1,w_2,...,w_ℓ from Σ^n such that any two consecutive words differ in at most c coordinates, for some fixed constant c>0. In this paper we are interested in the read and write complexity of quasi-Gray codes in the bit-probe model, where we measure the number of symbols read and written in order to transform any word w_i into its successor w_i+1. We present construction of quasi-Gray codes of dimension n and length 3^n over the ternary alphabet {0,1,2} with worst-case read complexity O( n) and write complexity 2. This generalizes to arbitrary odd-size alphabets. For the binary alphabet, we present quasi-Gray codes of dimension n and length at least 2^n - 20n with worst-case read complexity 6+ n and write complexity 2. Our results significantly improve on previously known constructions and for the odd-size alphabets we break the Ω(n) worst-case barrier for space-optimal (non-redundant) quasi-Gray codes with constant number of writes. We obtain our results via a novel application of algebraic tools together with the principles of catalytic computation [Buhrman et al. '14, Ben-Or and Cleve '92, Barrington '89, Coppersmith and Grossman '75]. We also establish certain limits of our technique in the binary case. Although our techniques cannot give space-optimal quasi-Gray codes with small read complexity over the binary alphabet, our results strongly indicate that such codes do exist.

READ FULL TEXT

Please sign up or login with your details

Forgot password? Click here to reset