[Paper Review] Shortest Paths in Planar Graphs with Real Lengths in $O(n\log^2n/\log\log n)$ Time

Shay Mozes, Christian Wulff‐Nilsen|ArXiv.org|Nov 25, 2009

Computational Geometry and Mesh Generation7 references4 citations

TL;DR

This paper presents an $O(n\log^2 n / \log\log n)$-time, linear-space algorithm for computing single-source shortest paths in planar directed graphs with real edge lengths and no negative cycles. The approach improves upon prior work by optimizing Bellman-Ford-style relaxation within regions of an $r$-division using a divide-and-conquer strategy with efficient shortest path computations on subgraphs bounded by external and internal cycles, leveraging monotone matrix searching and reduced cost functions to achieve the improved time bound.

ABSTRACT

Given an $n$-vertex planar directed graph with real edge lengths and with no negative cycles, we show how to compute single-source shortest path distances in the graph in $O(n\log^2n/\log\log n)$ time with O(n) space. This is an improvement of a recent time bound of $O(n\log^2n)$ by Klein et al.

Motivation & Objective

• To develop a faster algorithm for single-source shortest paths (SSSP) in planar directed graphs with arbitrary real edge lengths and no negative cycles.
• To reduce the time complexity of the state-of-the-art $O(n\log^2 n)$ algorithm by a $\log\log n$ factor while maintaining linear space usage.
• To extend the applicability of efficient SSSP computation to problems such as bipartite planar perfect matching, feasible flow, and feasible circulation in planar graphs.
• To optimize the intra- and inter-region shortest path computations in an $r$-division framework using efficient data structures and reduced cost functions.

Proposed method

• Construct an $r$-division of the planar graph using the cycle separator theorem, dividing the graph into $O(n/r)$ regions with $O(\sqrt{r})$ boundary vertices each.
• Use a feasible price function to transform edge lengths into non-negative reduced costs, enabling the use of Dijkstra’s algorithm on subgraphs.
• Apply Klein’s algorithm to compute all-pairs shortest paths between boundary vertices of each region in $O(|V_R|\log|V_R|)$ time.
• Use a Bellman-Ford variant with dynamic programming over layers to relax edges between boundary cycles, exploiting the structure of rightmost and leftmost paths in regions with holes.
• Leverage the total monotonicity of distance matrices to accelerate the computation of shortest paths between boundary vertices using the SMAWK algorithm.
• Combine results from subgraphs recursively, using the reduced cost function to recover original distances in constant time per query.

Experimental results

Research questions

• RQ1 Can the $O(n\log^2 n)$ time bound for SSSP in planar graphs with real edge lengths be improved by a $\log\log n$ factor?
• RQ2 Is it possible to maintain linear space while achieving this improved time complexity?
• RQ3 Can the improved SSSP algorithm be used to solve other planar graph problems such as perfect matching and feasible flow in the same time bound?
• RQ4 How can the structure of regions with holes and boundary cycles be exploited to optimize shortest path relaxation?
• RQ5 What is the role of monotone matrix searching and reduced cost functions in accelerating dynamic programming steps in the algorithm?

Key findings

• The algorithm achieves a time complexity of $O(n\log^2 n / \log\log n)$ for single-source shortest paths in planar directed graphs with real edge lengths and no negative cycles.
• The space complexity remains linear, $O(n)$, by carefully managing data structures and reusing memory across recursive and iterative phases.
• The improvement over the prior $O(n\log^2 n)$ bound is a factor of $\log\log n$, achieved through optimized use of monotone matrix searching and path decomposition.
• The algorithm enables solving bipartite planar perfect matching, feasible flow, and feasible circulation in $O(n\log^2 n / \log\log n)$ time as corollaries.
• The use of reduced cost functions and feasible price functions allows the application of Dijkstra’s algorithm on subgraphs while preserving correctness of original distances.
• The construction of rightmost and leftmost paths in regions with holes ensures that all relevant edges between boundary cycles are relaxed without redundancy.