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

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 nonnegative 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-hierarchybased approach, our algorithm shows great advantage with much simpler structure, and is much easier to implement.