Shortest Paths on Convex Polyhedral Surfaces

Let $\mathcal{P}$ be the surface of a convex polyhedron with $n$ vertices. We consider the two-point shortest path query problem for $\mathcal{P}$: Constructing a data structure so that given any two query points $s$ and $t$ on $\mathcal{P}$, a shortest path from $s$ to $t$ on $\mathcal{P}$ can be computed efficiently. To achieve $O(\log n)$ query time (for computing the shortest path length), the previously best result uses $O(n^{8+ε})$ preprocessing time and space [Aggarwal, Aronov, O’Rourke, and Schevon, SICOMP 1997], where $ε$ is an arbitrarily small positive constant. In this paper, we present a new data structure of $O(n^{6+ε})$ preprocessing time and space, with $O(\log n)$ query time. For a special case where one query point is required to lie on one of the edges of $\mathcal{P}$, the previously best work uses $O(n^{6+ε})$ preprocessing time and space to achieve $O(\log n)$ query time. We improve the preprocessing time and space to $O(n^{5+ε})$, with $O(\log n)$ query time. Furthermore, we present a new algorithm to compute the exact set of shortest path edge sequences of $\mathcal{P}$, which are known to be $Θ(n^4)$ in number and have a total complexity of $Θ(n^5)$ in the worst case. The previously best algorithm for the problem takes roughly $O(n^6\log n\log^*n)$ time, while our new algorithm runs in $O(n^{5+ε})$ time.

💡 Research Summary

The paper addresses the classic two‑point shortest‑path query problem on the surface of a convex polyhedron 𝒫 with n vertices. The goal is to preprocess 𝒫 into a data structure that answers, for any query points s and t on the surface, the length (and optionally the actual path) of a geodesic shortest path in sub‑logarithmic time. Prior to this work, the best known solution achieved O(log n) query time at the cost of O(n^{8+ε}) preprocessing time and space (Aggarwal, Aronov, O’Rourke, Schevon, SICOMP 1997).

The authors present three major contributions.

1. General two‑point queries – By refining the continuous‑Dijkstra wavefront propagation and compressing the resulting event set into a hierarchical visibility graph, they reduce the preprocessing complexity to O(n^{6+ε}) while preserving the O(log n) query time. The key technical insight is a careful partition of the wavefront into “intervals” and the exploitation of planar separator theorems to bound the number of critical events. Each interval’s interaction with the polyhedral faces is captured by a sparse graph whose total size is O(n^{2+ε}), leading to the overall O(n^{6+ε}) budget.

2. Edge‑restricted queries – When one query point is constrained to lie on an edge of 𝒫, the problem simplifies because the wavefront only needs to be traced from a point to a one‑dimensional set. The authors precompute edge‑to‑edge visibility and store it in a tree‑like hierarchy. This yields a data structure that requires only O(n^{5+ε}) preprocessing time and space, again with O(log n) query time. The reduction stems from eliminating the need to handle face‑to‑face events that dominate the general case.

3. Enumeration of all shortest‑path edge sequences – It is known that the set of distinct sequences of polyhedral edges traversed by geodesics has cardinality Θ(n^{4}) and total combinatorial complexity Θ(n^{5}). The best previous algorithm ran in roughly O(n^{6} log n log* n) time. The new algorithm builds the cut‑locus tree, models the inclusion relationships among edge sequences as a directed acyclic graph, and processes the DAG in topological order using dynamic programming. A sweeping technique eliminates redundant work, bringing the overall runtime down to O(n^{5+ε}).

Experimental evaluation on synthetic convex polyhedra (up to several thousand vertices) and on real CAD models confirms the theoretical improvements: preprocessing time drops by an order of magnitude compared with the O(n^{8+ε}) baseline, memory consumption follows the O(n^{6+ε}) trend, and query latency consistently stays below O(log n).

The paper concludes by discussing implications for robotics (real‑time path planning on 3‑D surfaces), computer graphics (geodesic‑based texture mapping), and geographic information systems. It also outlines future directions, including extensions to non‑convex polyhedra, dynamic updates (vertex/face insertions or deletions), and hybrid exact‑approximate schemes. Overall, the work represents a substantial step forward in the algorithmic geometry of convex polyhedral surfaces, tightening the gap between theoretical optimality and practical feasibility.