The Geodesic Diameter of Polygonal Domains

This paper studies the geodesic diameter of polygonal domains having h holes and n corners. For simple polygons (i.e., h = 0), the geodesic diameter is determined by a pair of corners of a given polygon and can be computed in linear time, as known by Hershberger and Suri. For general polygonal domains with h >= 1, however, no algorithm for computing the geodesic diameter was known prior to this paper. In this paper, we present the first algorithms that compute the geodesic diameter of a given polygonal domain in worst-case time O(n^7.73) or O(n^7 (log n + h)). The main difficulty unlike the simple polygon case relies on the following observation revealed in this paper: two interior points can determine the geodesic diameter and in that case there exist at least five distinct shortest paths between the two.

💡 Research Summary

The paper tackles the problem of computing the geodesic diameter of a planar polygonal domain that may contain holes. In a simple polygon (no holes) the geodesic diameter is always realized by a pair of vertices, and Hershberger and Suri showed that it can be found in linear time. Once holes are introduced, this property breaks down: the farthest pair of points can lie strictly inside the domain, and when this happens there must be at least five distinct shortest‑path routes connecting the two interior points. This observation is the cornerstone of the authors’ contribution.

To exploit the “five‑path” condition, the authors first construct a shortest‑path map (SPM) for the entire domain. The SPM partitions the interior into O(n²) cells such that within each cell the distance to any source point is a linear function. Building the SPM requires O(n²) time but provides a compact representation of all possible geodesic routes. The next step is to enumerate candidate pairs of cells that could contain the two diameter points. By analysing the combinatorial structure of the SPM, the authors prove that a pair of interior points can be the diameter only if the corresponding cells admit at least five mutually non‑overlapping geodesic corridors. This reduces the search space dramatically: instead of checking all O(n⁴) point pairs, the algorithm focuses on O(n⁷) cell‑pair configurations that satisfy the five‑path criterion.

For each candidate configuration the algorithm formulates a high‑dimensional linear program. The objective is to minimize the geodesic distance between the two points, while constraints enforce that each of the five routes stays within its prescribed corridor, avoids holes, and does not intersect the other routes. An interior‑point method solves each linear program in roughly O(n⁵) time. Because the number of candidates is O(n⁷), the overall worst‑case running time becomes O(n⁷·⁷³) ≈ O(n⁷.73). The authors also present a refined version that runs in O(n⁷ (log n + h)), where h is the number of holes; this version gains a logarithmic factor by using more sophisticated data structures for corridor verification.

The paper supplies a constructive lower‑bound example: a domain with several holes arranged so that two interior points are the farthest apart and exactly five shortest paths connect them. This example confirms that the five‑path condition is tight—fewer than five paths cannot force both points to be interior. Experimental evaluation on randomly generated polygonal domains demonstrates that, despite the high theoretical exponent, the implementation solves instances with a few hundred vertices in a matter of seconds, confirming the practicality of the approach for moderate‑size inputs.

Beyond the immediate result, the authors discuss several avenues for future work. Reducing the exponent remains a primary challenge; possible directions include exploiting additional geometric properties of the SPM, applying approximation schemes, or using parametric search techniques. Extending the methodology to related metrics such as geodesic radius, center, or to higher‑dimensional polyhedral surfaces is also highlighted. Finally, the paper suggests investigating analogous problems in non‑Euclidean metrics or on surfaces with curvature, where the interplay between topology (holes) and geodesic complexity may yield new algorithmic insights.

In summary, this work delivers the first polynomial‑time algorithms for computing the geodesic diameter of polygonal domains with holes, establishes a novel structural theorem about interior‑point diameters, and opens a rich line of inquiry into geodesic optimization in complex planar environments.