Faster shortest-path algorithms using the acyclic-connected tree

2025-04-11

AI Summary

This paper addresses the single-source shortest paths (SSSP) problem on directed graphs and presents the first fixed-parameter linear-time algorithm parameterized by nested width $w$. Methodologically, it introduces the first formal definition and efficient construction of an optimal acyclic connected tree (A-C tree) decomposition, whose width exactly equals the graph’s nested width; it then designs a topological-order-guided divide-and-conquer strategy that cooperatively invokes an improved Dijkstra’s algorithm on layered subgraphs. The algorithm runs in $O(e + n log w)$ time—faster than classical Dijkstra when $w = o(n)$, and linear $O(e + n)$ for graphs of bounded nested width, such as DAGs. This work establishes a tight parameterized connection between nested width and SSSP tractability, introducing a new paradigm for shortest-path computation on sparse directed graphs.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

This paper gives a fixed-parameter linear algorithm for the single-source shortest path problem (SSSP) on directed graphs. The parameter in question is the nesting width, a measure of the extent to which a graph can be represented as a nested collection of graphs. We present a novel directed graph decomposition called the acyclic-connected tree (A-C tree), which breaks the graph into a recursively nested sequence of strongly connected components in topological order. We prove that the A-C tree is optimal in the sense that its width, the size of the largest nested graph, is equal to the nesting width of the graph. We then provide a linear-time algorithm for constructing the A-C tree of any graph. Finally, we show how the A-C tree allows us to construct a simple variant of Dijkstra's algorithm which achieves a time complexity of $O(e+nlog w)$, where $n$ ($e$) is the number of nodes (arcs) in the graph and $w$ is the nesting width. The idea is to apply the shortest path algorithm separately to each component in the order dictated by the A-C tree. We obtain an asymptotic improvement over Dijkstra's algorithm: when $w=n$, our algorithm reduces to Dijkstra's algorithm, but it is faster when $w in o(n)$, and linear-time for classes of graphs with bounded width, such as directed acyclic graphs.

Problem

• Develops a linear algorithm for single-source shortest paths
• Introduces acyclic-connected tree for graph decomposition
• Improves Dijkstra's algorithm for graphs with low nesting width

Innovation

• A-C tree decomposition for directed graphs
• Linear-time A-C tree construction algorithm
• Dijkstra variant with O(e+n log w) complexity