发表机构
University of Maryland, College Park（马里兰大学帕克分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究证明树的并行边排序是P-完全问题，在MPC模型中可通过特定算法求解，强化了NC与MPC可计算性的区分。
AI 中文摘要
在本研究中，我们证明了并行计算树的边排序是P-完全问题。最优树边排序为边分配正整数秩，使得任意两个具有相同秩的边被一个更高秩的边隔开，同时最小化最高秩。树边排序抽象了多个经典问题，如制造业中的并行装配、最小高度树状图、可逆 pebble 游戏以及树上的边查询二分搜索。自de la Torre、Greenlaw和Schäffer在[SODA'93]的开创性工作以来，其并行复杂性问题已悬而未决超过三十年，并被列入Greenlaw、Hoover和Ruzzo所著《并行计算的极限》[1995]一书中的开放问题。我们证明，判定树的边排序是否至多为K是P-完全问题，即使对于直径为6的树亦是如此。我们的归约基于NOR-CVP，模拟了已知顺序方法背后的贪心过程。尽管这排除了NC算法的存在可能，但我们证明，在具有强次线性本地内存的著名大规模并行计算（MPC）模型中，可绕过该困难性障碍，表明O(log n)轮MPC足以求解用于证明P-完全性的困难树边排序实例。具体而言，我们提出一种确定性MPC算法，该算法可为直径为D的n顶点树计算最优边排序，所需轮数为O(log D + log log n)，本地内存为O(n^(3/4)D^(1/4))。我们的算法通过压缩子树合并生成字典序最小排序所需的信息，规避了线性内存障碍。总体而言，该结果强化了NC与MPC中可高效计算内容之间的严格区分。
英文摘要
In this work, we prove that computing the edge ranking of a tree in parallel is P-Complete. 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. Tree edge ranking abstracts several classical problems such as parallel assembly in manufacturing, minimum-height dendrograms, reversible pebble game and edge-query binary search on trees. Its parallel complexity remained open for over thirty years, since the seminal work of de la Torre, Greenlaw, and Sch{ä}ffer [SODA'93], and was listed as an open problem in the book Limits to Parallel Computation by Greenlaw, Hoover, and Ruzzo [1995]. We prove that deciding whether a tree has edge ranking at most $K$ is P-Complete, already for trees of diameter six. Our reduction is from NOR-CVP and simulates a greedy procedure underlying known sequential approaches. Despite ruling out NC algorithms, we prove that this hardness barrier can be bypassed in the well-known model of Massively Parallel Computation (MPC) with strongly sublinear local memory, showing that $O(\log n)$ MPC rounds suffice to solve the hard tree-edge ranking instances used to prove P-Completeness. Specifically, we present a deterministic MPC algorithm that computes an optimal edge ranking of an $n$-vertex tree of diameter $D$ in $O(\log D+\log\log n)$ rounds with $O(n^{3/4}D^{1/4})$ local memory. Our algorithm circumvents the linear-memory barrier by compressing the information required to produce a lexicographically minimal ranking from subtree merges. Overall, this result reinforces the strict separation between NC and what can be computed efficiently in MPC.
