Efficient Oracles for Generating Binary Bubble Languages

Given a string α = 10γ in a binary language L, the bubble lower bound is the smallest non-negative integer j such that 1s−10t−j10jγ ∈ L. We present an efficient oracle, which is a function that returns the bubble lower bound, for a number of fixed-density languages including: k-ary Dyck words, connected unit interval graphs, linear-extensions ofB-posets, those lexicographically larger than ω, those avoiding 10 for fixed k, reversible strings and feasible solutions to knapsack problems. Using these oracles, we apply the cool-lex framework to generate each language in constant amortized time. The algorithms generate the strings in either co-lex order or the cyclic cool-lex Gray code and can output each string’s sequence of bits, its run-length block encoding, or the operation (shift or swap) used in the Gray code. Additionally, we prove that the cool-lex Gray code for L can be generated inO(m) amortized time given anO(m) time membership tester for L.