Reinforcement learning with the help of neural-guided search consumes huge computational resources to achieve remarkable performance.

Hirschberg's algorithm (1975) reduces the space complexity for the longest common subsequence problem from $O(N^2)$ to $O(N)$ via recursive midpoint bisection on a grid dynamic program (DP). We show that the underlying idea generalizes to a broad class of dynamic programs with local dependencies on directed acyclic graphs (DP DAGs). Modeling a DP as deterministic time evolution over a topologically ordered DAG with frontier width $ω$ and bounded in-degree, and assuming a max-type semiring with deterministic tie breaking, we prove that in a standard offline random-access model any such DP admits deterministic traceback in space $O(ω\log T + (\log T)^{O(1)})$ cells over a fixed finite alphabet, where $T$ is the number of states. Our construction replaces backward dynamic programs by forward-only recomputation and organizes the time order into a height-compressed recursion tree whose nodes expose small "middle frontiers'' across which every optimal path must pass. The framework yields near-optimal traceback bounds for asymmetric and banded sequence alignment, one-dimensional recurrences, and dynamic-programming formulations on graphs of bounded pathwidth. We also show that an $Ω(ω)$ space term (in bits) is unavoidable in forward single-pass models and discuss conjectured $\sqrt{T}$-type barriers in streaming settings, supporting the view that space-efficient traceback is a structural property of width-bounded DP DAGs rather than a peculiarity of grid-based algorithms.

Path planning in grid maps, arising from various applications, has garnered significant attention. Existing methods, such as A*, Dijkstra, and their variants, work well for small-scale maps but fail to address large-scale ones due to high search time and memory consumption. Recently, Large Language Models (LLMs) have shown remarkable performance in path planning but still suffer from spatial illusion and poor planning performance. Among all the works, LLM-A* \cite{meng2024llm} leverages LLM to generate a series of waypoints and then uses A* to plan the paths between the neighboring waypoints. In this way, the complete path is constructed. However, LLM-A* still suffers from high computational time for large-scale maps. To fill this gap, we conducted a deep investigation into LLM-A* and found its bottleneck, resulting in limited performance. Accordingly, we design an innovative LLM-enhanced algorithm, abbr. as iLLM-A*. iLLM-A* includes 3 carefully designed mechanisms, including the optimization of A*, an incremental learning method for LLM to generate high-quality waypoints, and the selection of the appropriate waypoints for A* for path planning. Finally, a comprehensive evaluation on various grid maps shows that, compared with LLM-A*, iLLM-A* \textbf{1) achieves more than $1000\times$ speedup on average, and up to $2349.5\times$ speedup in the extreme case, 2) saves up to $58.6\%$ of the memory cost, 3) achieves both obviously shorter path length and lower path length standard deviation.}

Three sampling techniques are presented for comparative investigations.

Thus, it is natural to endow each state with a distribution over the tokens it may emit.

These works optimize the heuristic function to solve regression problems .

The overall network architecture is shown in Figure 1. This work was done when the author was with Rutgers University. The overall network architecture is shown in Figure 1. We also apply the ReLU activation after its first and second layers. Empirical evaluations show that NHE exhibits admissibility and consistency.

Sampling-based motion planners (SBMPs) are widely used to compute dynamically feasible robot paths. However, their reliance on uniform sampling often leads to poor efficiency and slow planning in complex environments. We introduce a novel non-uniform sampling strategy that integrates into existing SBMPs by biasing sampling toward `certified' regions. These regions are constructed by (i) generating an initial, possibly infeasible, path using any heuristic path predictor (e.g., A* or vision-language models) and (ii) applying conformal prediction to quantify the predictor's uncertainty. This process yields prediction sets around the initial-guess path that are guaranteed, with user-specified probability, to contain the optimal solution. To our knowledge, this is the first non-uniform sampling approach for SBMPs that provides such probabilistically correct guarantees on the sampling regions. Extensive evaluations demonstrate that our method consistently finds feasible paths faster and generalizes better to unseen environments than existing baselines.