Optimal Quasi-Gray Codes: The Alphabet Matters

Abstract: A quasi-Gray code of dimension $n$ and length $\ell$ over an alphabet $\Sigma$ is a sequence of distinct words $w_1,w_2,\dots,w_\ell$ from $\Sigma^n$ 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 $O(\log n)$ 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 $6+\log n$ and write complexity $2$. This complements a recent result by Raskin [Raskin '17] who shows that any quasi-Gray code over binary alphabet of length $2^n$ has read complexity $\Omega(n)$. Our results significantly improve on previously known constructions and for the odd-size alphabets we break the $\Omega(n)$ 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].