Linear-Time Algorithms for Geometric Graphs with Sublinearly Many Edge Crossings

We provide linear-time algorithms for geometric graphs with sublinearly many crossings. That is, we provide algorithms running in O(n) time on connected geometric graphs having n vertices and k crossings, where k is smaller than n by an iterated logarithmic factor. Specific problems we study include Voronoi diagrams and single-source shortest paths. Our algorithms all run in linear time in the standard comparison-based computational model; hence, we make no assumptions about the distribution or bit complexities of edge weights, nor do we utilize unusual bit-level operations on memory words. Instead, our algorithms are based on a planarization method that “zeroes in” on edge crossings, together with methods for extending planar separator decompositions to geometric graphs with sublinearly many crossings. Incidentally, our planarization algorithm also solves an open computational geometry problem of Chazelle for triangulating a self-intersecting polygonal chain having n segments and k crossings in linear time, for the case when k is sublinear in n by an iterated logarithmic factor.

💡 Research Summary

The paper introduces a new class of geometric graphs called “restrained graphs,” defined by having at most k edge crossings where k = O(n / log⁽ᶜ⁾ n) for some constant c. Under this sub‑linear crossing assumption, the authors develop a suite of linear‑time algorithms that work in the standard comparison‑based model, i.e., they make no assumptions about edge‑weight distributions or require bit‑level operations.

The core technical contribution is a randomized planarization procedure. Starting from a connected geometric graph G with n vertices and k crossings, the algorithm samples edges at a rate inversely proportional to an iterated logarithm, builds a trapezoidal decomposition of the sampled edges using the Bentley‑Ottmann line‑segment intersection algorithm, and records for each trapezoid the set C(t) of original edges intersecting it. By repeatedly refining trapezoids whose conflict lists are too large (αₜ = |C(t)|·r/n > 1), and sampling again inside those cells, the method constructs a (1/r)-cutting where each cell meets at most n/r edges of G. The expected size of the cutting and the total work are bounded by O(r + (r/n)k) with r = Θ(n / log⁽ˢ⁾ n). Because k is sub‑linear, the overall expected time for planarization becomes O(n).

Once planarized, the graph G′ has n + k vertices and O(n + k) edges, making it a planar graph. The authors then apply classic planar separator theory: Lipton‑Tarjan’s O(√n) vertex separators and Goodrich’s linear‑time recursive separator construction. By recursively separating G′ into two subgraphs of size at most 2n/3, they obtain a full separator decomposition in O(n) time.

The separator hierarchy is leveraged to solve two fundamental problems in linear time:

1. Voronoi Diagram (Post‑office problem) – Given a set P of sites, the algorithm computes for every vertex the nearest site with respect to the graph metric. The divide‑and‑conquer approach uses the separator tree to localize the nearest‑site computation to subgraphs, merging results across separators without any geometric distance calculations beyond edge‑weight comparisons.

2. Single‑Source Shortest Paths (SSSP) – Instead of Dijkstra’s heap‑based method, the algorithm uses the separator hierarchy to propagate distance labels level by level. Each separator separates the graph into independent regions, allowing linear‑time propagation of distance updates. The method works for arbitrary comparison‑based edge weights, including non‑numeric or user‑defined metrics.

A notable side result solves an open problem posed by Chazelle: triangulating a non‑simple polygonal chain (i.e., a self‑intersecting polygon) with n segments and k crossings in expected O(n + k) time, provided k is sub‑linear by an iterated logarithm. By treating the polygon as a geometric graph and applying the same planarization technique, the arrangement of all edges is obtained in linear time, after which a standard linear‑time triangulation of the resulting planar subdivision yields the desired triangulation. Combined with the Clarkson‑Cole‑Tarjan algorithm for the general case, this gives an optimal O(n + k) expected‑time algorithm for all values of k except the narrow range