Breaking the Sorting Barrier for Shortest Paths
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Breaking the Sorting Barrier for Shortest Paths
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This document presents a deterministic algorithm for the single-source shortest path (SSSP) problem on directed graphs with non-negative edge weights, achieving a time complexity of O(m log^(2/3) n). This groundbreaking result surpasses the long-standing O(m + n log n) barrier of Dijkstra's algorithm, demonstrating that Dijkstra's is not optimal for SSSP on sparse graphs when the vertex ordering by distance is not strictly required. The approach ingeniously merges concepts from Dijkstra's and Bellman-Ford algorithms using a recursive partitioning technique to manage the "frontier" of uncertain distances more efficiently, avoiding the sorting bottleneck inherent in traditional methods. It introduces a "FindPivots" procedure and a specialized data structure to limit the size of the set of vertices that need active consideration, thereby reducing computational overhead and improving performance.