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.

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