We analyze the absolute spectral gap of Markov chains on graphs obtained from a cycle of $n$ vertices and perturbed only at approximately $n^{1/ρ}$ random locations with an appropriate, possibly sparse, interconnection structure. Together with a strong asymmetry along the cycle, the gap of the resulting chain can be bounded inversely proportionally by the longest arc length (up to logarithmic factors) with high probability, providing a significant mixing speedup compared to the reversible version.