A Forward-Backward Single-Source Shortest Paths Algorithm

We describe a new forward-backward variant of Dijkstra's and Spira's Single-Source Shortest Paths (SSSP) algorithms. While essentially all SSSP algorithm only scan edges forward, the new algorithm scans some edges backward. The new algorithm assumes that edges in the out-going and incoming adjacency lists of the vertices appear in nondecreasing order of weight. (Spira's algorithm makes the same assumption about the out-going adjacency lists, but does not use incoming adjacency lists.) The running time of the algorithm on a complete directed graph on n vertices with independent exponential edge weights is O(n), with very high probability. This improves on the previously best result of O(n log n), which is best possible if only forward scans are allowed, exhibiting an interesting separation between forward-only and forward-backward SSSP algorithms. As a consequence, we also get a new all-pairs shortest paths algorithm. The expected running time of the algorithm on complete graphs with independent exponential edge weights is O(n 2 ), matching a recent result of Peres et al. Furthermore, the probability that the new algorithm requires more than O(n 2 ) time is exponentially small, improving on the polynomially small probability of Peres et al.

Authors

• David Bruce Wilson
• Uri Zwick

Keywords

• Heuristic algorithms
• Probabilistic logic
• Algorithm design and analysis
• Arrays
• Random variables
• Computational modeling
• Shortest Path
• Forward-backward Algorithm
• Single Source Shortest Path
• High Probability
• Running Time
• Edge Weights
• Directed Graph
• Dijkstra’s Algorithm
• Forward Scan
• Exponential Weighting
• Probabilistic Model
• Undirected
• Exponential Distribution
• Correction Algorithm
• Vertices
• Problem In Time
• Operational Algorithm
• Backward Induction
• Behavior Of Algorithm
• Input Graph
• Priority Queue
• Edge List
• Outgoing Edges
• Verification Algorithm
• Source Vertex
• Technical Lemma
• Edge Path
• Shortest Path Problem
• graph algorithms
• shortest paths