This paper addresses the following problem: given a planar embedding graph, compute a representation of the shortest-path trees rooted at all the boundary nodes of the graph. Klein gave an $O(n \log n)$ algorithm for this problem; the algorithm subsequently became an essential ingredient in dozens of algorithms, addressing problems ranging from distance oracles to edit distance. However, the correctness and analysis in that original paper is complicated and messy and hard to understand. In this paper, we give a simple and clear analysis.

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