H-BRICK: Hierarchical Boundary Reachability Index with Compressed Kleene Closure for Directed Grid Graphs

Frontier-based exploration systems repeatedly evaluate geodesic travel costs between robot poses and candidate viewpoints, and tour-based planners consume the full pairwise distance matrix built from them. Recomputing these matrices with one graph search per source scales with the explored map size, whereas distance caches that refresh only locally affected or newly created frontiers can retain stale costs between surviving pairs when newly discovered free space creates a global shortcut. We present H-TRIP (Hierarchical Tropical Rank-one Incremental Propagation), an exact distance index grounded in Kleene algebra over the min-plus semiring for monotonically growing occupancy grids. H-TRIP partitions the known map into a spatial hierarchy with fixed boundary ports. Each unknown-to-free revelation is decomposed into efficient rank-one min-plus edge insertions at the affected leaf, and only decreased port-to-port summaries are propagated along its ancestor chain. A shared hierarchical lift evaluates the dense many-to-many matrix of shortest distances for all active query points on the occupancy map. We prove that sequential insertions preserve exact shortest-path distances on the grid. Benchmark experiments confirm exact agreement with reference searches on fully built maps and show that dense many-to-many queries at a few dozen viewpoints are up to two orders of magnitude faster than sequential k x BFS. On exploration runs with sixteen robots, in-loop queries on the growing occupancy graph are 2.21x-8.59x faster than co-scheduled k x BFS on maps with side 256, 512, and 1024 cells.

Determining the occupancy status of locations in the environment is a fundamental task for safety-critical robotic applications. Traditional occupancy grid mapping methods subdivide the environment into a grid of voxels, each associated with one of three occupancy states: free, occupied, or unknown. These methods explicitly maintain all voxels within the mapped volume and determine the occupancy state of a location by directly querying the corresponding voxel that the location falls within. However, maintaining all grid voxels in high-resolution and large-scale scenarios requires substantial memory resources. In this paper, we introduce a novel representation that only maintains the boundary of the mapped volume. Specifically, we explicitly represent the boundary voxels, such as the occupied voxels and frontier voxels, while free and unknown voxels are automatically represented by volumes within or outside the boundary, respectively. As our representation maintains only a closed surface in two-dimensional (2D) space, instead of the entire volume in three-dimensional (3D) space, it significantly reduces memory consumption. Then, based on this 2D representation, we propose a method to determine the occupancy state of arbitrary locations in the 3D environment. We term this method as boundary map. Besides, we design a novel data structure for maintaining the boundary map, supporting efficient occupancy state queries. Theoretical analyses of the occupancy state query algorithm are also provided. Furthermore, to enable efficient construction and updates of the boundary map from the real-time sensor measurements, we propose a global-local mapping framework and corresponding update algorithms. Finally, we will make our implementation of the boundary map open-source on GitHub to benefit the community:https://github.com/hku-mars/BDM.

Computing geometric representations of data is a cornerstone of modern machine learning, typically achieved by training dual encoders which map queries and documents into a shared embedding space. Recent work of You et al. [NeurIPS '25] has extended this approach to hierarchical retrieval, where relevance is determined by the ancestor-descendant relationships in a Directed Acyclic Graph (DAG). While previous work has shown that valid embeddings exist when the number of descendants is small, these bounds degrade significantly for deep hierarchies, requiring dimensions as large as the total number of nodes. In this paper, we investigate compact reachability embeddings for more general graph classes and provide theoretical guarantees for representing hierarchies using embeddings whose dimension depends on structural graph parameters. We prove that for any directed tree, there exists a reachability embedding in constant dimension 3, independent of the tree's size or depth. We generalize this result to graphs characterized by treewidth $t$, constructing embeddings of dimension $O(t \log n)$, where $n$ is the number of nodes. Complementing these upper bounds, we provide matching or near-matching lower bounds, showing that dimension $Ω(n)$ is necessary for general DAGs and $Ω(t/\log(n/t))$ is required for graphs of treewidth $t$. We also obtain upper and lower bounds parameterized by the number of cross-edges in the DAG. We additionally show that our embeddings can be constructed on real world datasets, and that they give much smaller dimensions in high recall regimes compared to prior embeddings with theoretical guarantees.