Girth of a Planar Digraph with Real Edge Weights in O(n(log n)^3) Time

The girth of a graph is the length of its shortest cycle. We give an algorithm that computes in O(n(log n)^3) time and O(n) space the (weighted) girth of an n-vertex planar digraph with arbitrary real edge weights. This is an improvement of a previous time bound of O(n^(3/2)), a bound which was only valid for non-negative edge-weights. Our algorithm can be modified to output a shortest cycle within the same time and space bounds if such a cycle exists.

💡 Research Summary

The paper addresses the problem of computing the girth—the minimum‑weight cycle—of a planar directed graph whose edges may have arbitrary real weights. While previous work achieved an O(n³⁄²) running time for non‑negative weights, no sub‑quadratic algorithm was known that could handle negative or mixed‑sign weights. The authors present a deterministic algorithm that runs in O(n·(log n)³) time and uses O(n) space, and they also show how to output an actual shortest cycle within the same bounds.

The algorithm builds on three classical tools: the planar separator theorem, dense distance graphs (DDGs), and potential‑based reweighting. First, a planar separator is computed, recursively partitioning the input graph into a binary decomposition tree of depth O(log n). Each node of the tree corresponds to a subgraph whose boundary consists of O(√n) vertices (the separator). For every subgraph the authors construct a DDG that stores, for each pair of boundary vertices, the length of the shortest path that stays inside the subgraph. Because only boundary vertices are represented, each DDG has size linear in the number of boundary vertices, and the total size over all levels remains O(n).

Negative edge weights prevent the direct use of Dijkstra‑type procedures. To overcome this, the authors compute a global potential function π that reweights every edge e = (u, v) to w′(e) = w(e) + π(u) – π(v). The potential is chosen so that the reweighted graph contains no negative cycles, yet the order of cycle weights is preserved; consequently the girth of the original graph equals the girth of the reweighted one. The potential can be obtained by a single linear‑time planar shortest‑path computation because the graph is planar.

With non‑negative reweighted edges, the algorithm employs the FR‑Dijkstra (Fakcharoenphol‑Rao) technique, which answers shortest‑path queries among boundary vertices in O(log n) time per query after a preprocessing phase that costs O(|DDG|·log |DDG|). Because each level of the separator tree contributes O(n·log n) work, the total preprocessing time across all levels is O(n·(log n)²).

The girth is then found by examining each directed edge e = (u, v). For that edge the algorithm queries the DDG of the lowest‑level subgraph that contains both u and v, obtaining the shortest distance d(u, v) that does not use e. The candidate cycle weight is d(u, v) + w′(e). Taking the minimum over all edges yields the reweighted girth; the original girth follows by subtracting the potentials. Since each edge participates in O(log n) levels, the total number of distance queries is O(n·log n), and each query costs O(log n), giving the final O(n·(log n)³) bound.

Correctness is proved in two parts. First, the potential transformation is shown to be cycle‑preserving and to eliminate negative cycles, guaranteeing that any shortest cycle in the original graph corresponds to a shortest cycle in the reweighted graph. Second, the DDG representation is proved to be exact for paths whose interior vertices lie inside the subgraph, using the planar property that any such path can be “pushed” onto the boundary without increasing its length. Consequently the distance returned by the DDG equals the true shortest‑path distance in the original graph.

To output an actual shortest cycle, the algorithm records, for the edge that achieves the minimum, the predecessor information inside the DDG. By back‑tracking through the DDG’s stored paths and concatenating the edge itself, a concrete cycle is reconstructed in O(n·log n) additional time, still within the overall asymptotic budget.

Space usage remains linear because each DDG is stored only for the current level of recursion, and the separator tree itself occupies O(n) space. The paper concludes with a discussion of possible extensions: reducing the logarithmic factors by employing more advanced planar shortest‑path data structures, handling dynamic updates to the graph, and generalising the technique to graphs of bounded genus or small treewidth, where similar separator‑based decompositions exist.

In summary, the authors deliver the first near‑linear‑time algorithm for the weighted girth of planar digraphs with arbitrary real edge weights, improving the best known bound by a factor of roughly n¹⁄² and opening the door to efficient cycle‑based analyses in a broad range of planar network applications.