Parallel Self-Avoiding Walks for a Low-Autocorrelation Binary Sequences Problem

TL;DR

This work tackles the low-autocorrelation binary sequences (LABS) problem by introducing sokol$_{skew}$, a GPU-based stochastic solver that searches the skew-symmetric subspace using parallel contiguous self-avoiding walks. By reducing the search space to $D=\frac{L+1}{2}$ and carefully orchestrating independent SAWs with efficient energy evaluation, the method achieves substantial speedups over prior solvers and yields new best-known sequences up to $L=247$ with merit factors near 9.36. A predictive exponential stopping model calibrated on small instances enables high-probability optimality claims (up to 99% for $L\le 223$) within fixed runtimes, while larger instances provide strong improvements despite lower optimality probabilities. The results demonstrate the practical impact of GPU-accelerated, memory-efficient stochastic search for challenging combinatorial problems and push the frontier of best-known LABS sequences, highlighting open challenges such as achieving $F\ge 10$ for longer lengths.

Abstract

A low-autocorrelation binary sequences problem with a high figure of merit factor represents a formidable computational challenge. An efficient parallel computing algorithm is required to reach the new best-known solutions for this problem. Therefore, we developed the $\mathit{sokol}_{\mathit{skew}}$ solver for the skew-symmetric search space. The developed solver takes the advantage of parallel computing on graphics processing units. The solver organized the search process as a sequence of parallel and contiguous self-avoiding walks and achieved a speedup factor of 387 compared with $\mathit{lssOrel}$, its predecessor. The $\mathit{sokol}_{\mathit{skew}}$ solver belongs to stochastic solvers and can not guarantee the optimality of solutions. To mitigate this problem, we established the predictive model of stopping conditions according to the small instances for which the optimal skew-symmetric solutions are known. With its help and 99% probability, the $\mathit{sokol}_{\mathit{skew}}$ solver found all the known and seven new best-known skew-symmetric sequences for odd instances from $L=121$ to $L=223$. For larger instances, the solver can not reach 99% probability within our limitations, but it still found several new best-known binary sequences. We also analyzed the trend of the best merit factor values, and it shows that as sequence size increases, the value of the merit factor also increases, and this trend is flatter for larger instances.

Figures (5)

• Figure 1: Architecture of the sokol$_{\mathit{skew}}$ solver.
• Figure 2: The ratio between variable $speed$ and $blocks \in \{54, 108, ..., 6912\}$ for $L=245$.
• Figure 3: The distribution of $\mathit{NSEs}$ that are needed to reach the optimal solution and the model ($R^2=0.9429$) of the exponential distribution parameter $\lambda$ according to the results for $L \in \{71, 73, ..., 119\}$.
• Figure 4: The best merit factors.
• Figure 5: The $speed$ ratio between the sokol$_{\mathit{skew}}$ with 6912 GPU threads, parallel lssOrel with 6 CPU threads, and sequential lssOrel solver.