Pathfinding with Lazy Successor Generation

Abstract: We study a pathfinding problem where only locations (i.e., vertices) are given, and edges are implicitly defined by an oracle answering the connectivity of two locations. Despite its simple structure, this problem becomes non-trivial with a massive number of locations, due to posing a huge branching factor for search algorithms. Limiting the number of successors, such as with nearest neighbors, can reduce search efforts but compromises completeness. Instead, we propose a novel LaCAS* algorithm, which does not generate successors all at once but gradually generates successors as the search progresses. This scheme is implemented with k-nearest neighbors search on a k-d tree. LaCAS* is a complete and anytime algorithm that eventually converges to the optima. Extensive evaluations demonstrate the efficacy of LaCAS*, e.g., solving complex pathfinding instances quickly, where conventional methods falter.

• The paper presents the LACAS algorithm, which incrementally generates successors via a two-level search to address vast implicit graphs.
• It leverages k-d tree based k-nearest neighbor searches to lazily generate successor nodes, significantly reducing computational overhead.
• Experimental results demonstrate that LACAS achieves high completeness and improved efficiency, even in large-scale, high branching-factor environments.

Pathfinding with Lazy Successor Generation

The paper "Pathfinding with Lazy Successor Generation" by Keisuke Okumura from the University of Cambridge and the National Institute of Advanced Industrial Science and Technology (AIST) presents a novel approach to solving pathfinding problems in graphs with a massive number of vertices and an implicitly defined set of edges. The method, termed \lacas (Lazy Constraints Addition Search), provides a comprehensive solution to these challenges by employing lazy successor generation through a two-level search mechanism.

Problem Definition and Challenges

The pathfinding problem addressed in this paper involves determining a sequence of locations from a given start vertex to a goal vertex within a 2D workspace. The edges between vertices are not explicitly listed but are determined via an oracle answering the connectivity between them. This problem, while structurally simple, introduces significant complexity due to the immense branching factor when the number of vertices is large. Traditional search algorithms struggle with the computational overhead required for such expansive branching factors, and methods that limit the number of successors often sacrifice completeness and optimality.

Proposed Solution: \lacas

The core contribution of this paper is the \lacas algorithm, designed to handle the above challenges efficiently. \lacas does not generate all successors of a node simultaneously. Instead, it progressively generates subsets of successors based on their proximity, utilizing $k$-nearest neighbors search on a k-d tree.

Key features of \lacas are:

• Incremental Successor Generation: Instead of generating all possible successors for a given node, \lacas uses a k-d tree to dynamically generate a limited batch of nearest successor nodes. This lazy generation of successors reduces computational overhead.
• Two-Level Search: \lacas employs a high-level search to handle the original pathfinding problem and a low-level search to manage the extraction of the nearest neighbors incrementally.
• Theoretical Guarantees: The algorithm is both complete and anytime, meaning it will always find a solution if one exists and continues to refine the solution towards optimality over time.

Comparative Analysis and Performance

The paper conducts extensive evaluations of \lacas, comparing it against various baseline methods, including traditional \astar, greedy best-first search (GBFS), depth-first search (DFS), and related pathfinding and sampling-based motion planning methods, such as RRT and its variants.

Numerical Results

The experimental results highlight the efficiency of \lacas in solving complex pathfinding instances where conventional methods falter:

• In the \scen{scatter-1k} scenario with 1,000 randomly placed vertices, \lacas solved 79% of instances within the time limit, with a median solve time of 1.11 seconds and a median path cost of 4.25, requiring 6,695 connectivity checks.
• For the \scen{scatter-10k} scenario with 10,000 vertices, \lacas demonstrated its capability to handle significantly larger graphs, showing notable improvements in the solution quality and computational effort.

Advanced Techniques and Enhancements

Several advanced techniques were applied to \lacas to further improve its performance:

• Order within Batch: Sorting the batch of successor nodes by their distance to the goal improved initial solution discovery speed.
• Node Reinsert: Reinserting nodes representing already known locations prioritized crucial search areas, significantly reducing search effort.
• Node Rolling: Utilizing a double-ended queue for the \open list and periodically rolling nodes within \open prevented the method from getting stuck in local minima.
• Grandparent Check (\lacat): Checking possible connections with the grandparent nodes led to finding paths with lower costs more efficiently, albeit at the cost of additional computational overhead.

Practical and Theoretical Implications

The practical implications of the \lacas algorithm are substantial. It can be applied to various fields where pathfinding is a central problem, including robot motion planning, video games, and navigation systems. The lazy generation of successors and the two-level search algorithm can handle scenarios with a large number of vertices and dynamically defined edges, ensuring completeness and the gradual convergence to optimal solutions.

Theoretically, \lacas embodies a robust framework for incremental problem-solving in large-scale graphs, advancing the state-of-the-art in both single-agent and multi-agent pathfinding techniques. Its design principles can inspire further research in efficient search algorithms capable of managing vast solution spaces without sacrificing optimality.

Future Directions

Future research could explore several directions based on the findings of this study:

• Scalability Improvements: Enhancing the scalability of \lacas by optimizing the underlying data structures or integrating parallel processing techniques.
• Adaptive Parameterization: Developing methods to automatically adjust batch sizes and other parameters based on the complexity of the specific pathfinding instance.
• Application to Dynamic Environments: Extending the lazy generation framework to dynamically changing graphs where the connectivity between vertices can evolve over time.

Conclusion

The paper "Pathfinding with Lazy Successor Generation" presents a comprehensive and innovative approach to address the challenges of pathfinding in large-scale graphs with implicit edge definitions. Through the lazy generation of successors and a two-level search mechanism, \lacas achieves significant computational efficiency while maintaining completeness and eventual optimality. This work not only advances theoretical research in algorithmic pathfinding but also holds substantial practical applications across various domains requiring efficient navigation and path planning.