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原文摘要

In this work, we prove that computing the edge ranking of a tree in parallel is P-Complete, and complement this hardness with new algorithmic results for both PRAM and the MPC model. An optimal tree edge ranking assigns positive integer ranks to the edges such that any two edges with the same rank are separated by an edge of higher rank, while minimizing the highest rank. Although an optimal edge ranking, which minimizes the maximum rank, can be computed in polynomial time sequentially, its parallel complexity has remained open for almost 35 years [de la Torre, Greenlaw, Schäffer, SODA'93] and was listed as an open problem in Limits to Parallel Computation [Greenlaw, Hoover, Ruzzo, 1995]. Specifically, we show that deciding whether a tree has edge ranking at most $K$ is P-complete, already for trees of diameter at most six. Our reduction encodes monotone circuit evaluation into a greedy allocation procedure underlying known sequential algorithms. Using a rooted product of trees, we further prove that computing an edge ranking within constant additive error remains P-hard. On the algorithmic side, we break the long-standing 2-approximation barrier by designing a deterministic NC algorithm that computes an $(1+o(1))$-approximate edge ranking. Second, we show that P-completeness can be bypassed in Massively Parallel Computation (MPC) with strongly sublinear local memory: we give a deterministic algorithm that computes an optimal edge ranking of an $n$-vertex, diameter-$D$ tree in $O(\log D + \frac{n^{3/4}D^{1/4}}{S})$ rounds with $S = Θ(n^{3/4+δ})$ local memory, for $δ\in (0, 1/4)$, requiring only $O(1)$ rounds on the hard diameter-six instances. This result reinforces the separation between $\mathsf{NC}$ and what can be computed efficiently in MPC.