Understand
We present the first $m\,\text{polylog}(n)$ work, $\text{polylog}(n)$ time algorithm in the PRAM model that computes $(1+\epsilon)$-approximate single-source shortest paths on weighted, undirected graphs.
- This improves upon the breakthrough result of Cohen~[JACM'00] that achieves $O(m^{1+\epsilon_0})$ work and $\text{polylog}(n)$ time.
- While most previous approaches, including Cohen's, leveraged the power of hopsets, our algorithm builds upon the recent developments in \emph{continuous optimization}, studying the shortest path problem from the lens of the closely-related \emph{minimum transshipment} problem.
- To obtain our algorithm, we demonstrate a series of near-linear work, polylogarithmic-time reductions between the problems of approximate shortest path, approximate transshipment, and $\ell_1$-embeddings, and establish a recursive algorithm that cycles through the three problems and reduces the graph size on each cycle.