A Randomized Algorithm for Single-Source Shortest Path on Undirected Real-Weighted Graphs

Abstract: In undirected graphs with real non-negative weights, we give a new randomized algorithm for the single-source shortest path (SSSP) problem with running time $O(m\sqrt{\log n \cdot \log\log n})$ in the comparison-addition model. This is the first algorithm to break the $O(m+n\log n)$ time bound for real-weighted sparse graphs by Dijkstra's algorithm with Fibonacci heaps. Previous undirected non-negative SSSP algorithms give time bound of $O(m\alpha(m,n)+\min{n\log n, n\log\log r})$ in comparison-addition model, where $\alpha$ is the inverse-Ackermann function and $r$ is the ratio of the maximum-to-minimum edge weight [Pettie & Ramachandran 2005], and linear time for integer edge weights in RAM model [Thorup 1999]. Note that there is a proposed complexity lower bound of $\Omega(m+\min{n\log n, n\log\log r})$ for hierarchy-based algorithms for undirected real-weighted SSSP [Pettie & Ramachandran 2005], but our algorithm does not obey the properties required for that lower bound. As a non-hierarchy-based approach, our algorithm shows great advantage with much simpler structure, and is much easier to implement.

• The paper introduces a randomized algorithm for single-source shortest paths that achieves O(m √(log n · log log n)) time.
• It leverages sampling and bundling techniques to bypass lower bounds inherent to hierarchy-based methods, thereby enhancing efficiency.
• The algorithm’s design, which excels in sparse graphs, promises practical applications in network design and geographic information systems.

A Randomized Algorithm for Single-Source Shortest Path on Undirected Real-Weighted Graphs

Introduction and Context

The paper "A Randomized Algorithm for Single-Source Shortest Path on Undirected Real-Weighted Graphs" presents a novel approach to the single-source shortest path (SSSP) problem for undirected graphs with non-negative, real weights. Traditionally, Dijkstra's algorithm achieves a time complexity of $O(m + n \log n)$ for sparse graphs using Fibonacci heaps. The current work introduces the first method to surpass this bound for real-weighted sparse graphs, achieving a time complexity of $O(m \sqrt{\log n \cdot \log\log n})$ under the comparison-addition model. A distinctive feature is that the algorithm is non-hierarchy based, which allows it to bypass certain lower bound constraints that apply to hierarchy-based algorithms.

Algorithm Overview

Main Contributions

1. Time Complexity: The proposed algorithm breaks the conventional $O(m + n \log n)$ time barrier for real-weighted SSSP in sparse graphs, achieving a runtime of $O(m \sqrt{\log n \cdot \log\log n})$.
2. Las-Vegas Randomization: The algorithm is randomized and operates in $O(m \sqrt{\log n \cdot \log\log n})$ time with high probability, ensuring correctness with the Las-Vegas approach.
3. Innovative Use of Bundles: It introduces a novel technique involving sampling and bundling vertices, leading to a simpler implementation that is more efficient in practice.

Algorithm Details

• Sampling and Bundling Strategy: A subset $R$ of vertices is sampled, and a "bundle" is associated with each vertex not in $R$. The algorithm processes these bundles instead of individual vertices, reducing the computational overhead associated with maintaining and sorting a priority queue.
• Use of Constant-Degree Transformation: The graph is transformed to ensure a constant maximum degree, which is critical to the performance and correctness of the distance relaxation steps within the algorithm.

Technical Approach

1. Sampling: Independent sampling of non-source vertices is performed to determine the set $R$.
2. Bundle Construction: Each vertex not in $R$ is associated with its nearest vertex in $R$, facilitating a simplified relaxation procedure.
3. Distance Relaxation: The algorithm iteratively updates distance estimates using bundles, taking advantage of the localized nature of modifications.

Performance and Analysis

The algorithm's analysis is built on probabilistic bounds that ensure expected performance aligns with the theoretical guarantees. Through sampling, the heavy computational lifting inherent in maintaining a sorted list of priority queue elements is mitigated. Key performance metrics include:

• Time Complexity: The algorithm executes within the promised bounds, scaling efficiently with both graph size $m$ and vertex count $n$.
• Complexity Reduction in Sparse Graphs: The algorithm excels in sparse graph environments, reducing unnecessary complexity by focusing computational resources on sampled and bundled vertices.

Implications and Future Directions

Practical Implications

The algorithm's design prioritizes ease of implementation and practical efficiency, making it a compelling choice for applications involving large-scale and sparse graphs. It has potential implications for network design, geographic information systems, and other fields where SSSP problems are prevalent.

Theoretical Implications

This work challenges existing lower bounds for hierarchy-based algorithms, showcasing the benefits of a randomized, non-hierarchy-based approach. It opens avenues for further exploration of randomized algorithms in graph theory and their potential to redefine complexity thresholds in other longstanding algorithmic problems.

Future Research

Future research could focus on extending these ideas to directed graphs and exploring variations of the problem with additional constraints, such as negative edge weights. Additionally, there is potential for hybrid models that combine elements of hierarchy and randomization for even greater efficiency.

Conclusion

The proposed randomized algorithm advances the state of the art for solving the single-source shortest path problem in undirected real-weighted graphs. By breaking established time complexity barriers and leveraging randomization effectively, it sets a foundation for further exploration into efficient, practical algorithms for complex network problems.