AI 中文总结
研究单机截止期限问题$1 \mid\mid \sum U_j$，采用仅插入最短作业优先规则，通过平衡增强二叉搜索树实现$O(n\log n)$时间算法，分析其字典序最优性及组合结构，揭示嵌套拟阵和多拟阵秩函数等特性。
AI 中文摘要
我们研究经典单机截止期限问题$1 \mid\mid \sum U_j$，其中每个任务有截止期限和执行需求，目标是选择尽可能多的按时完成任务。标准的Moore-Hodgson算法按截止期限处理任务，可能会删除先前接受的任务。我们研究了Lin和Wang提出的仅插入最短作业优先规则：按执行需求非递减顺序处理任务，且仅在保持可行性时接受任务。我们使用以截止期限为键的平衡增强二叉搜索树给出了直接的$O(n\log n)$时间实现。与之前该SJF规则的$O(n\log n)$实现不同，我们的实现既不需要抢占式调度，也不需要对区间变化进行摊还分析。我们的分析给出了该规则字典序（lex-first）最优性的显式阈值形式：对于每个阈值$e$，其输出最大化执行需求至多为$e$的所选任务数量。分析还揭示了额外的组合结构。在较短任务被贪心固定后，单个执行需求层内的可行选择形成一个嵌套拟阵。这些层拟阵作为直和组装成一个整体分层拟阵，其基恰好是贪心输出。最后，一个编码截止期限前缀约束的流网络给出了基础调度可行性结构的多拟阵秩函数。这种流视角也恢复了控制等执行层的嵌套拟阵。
英文摘要
Scheduling $n$ jobs with deadlines and processing times on a single resource to minimize late jobs equates to finding a maximum-cardinality feasible subset. Moore (1968) proposed a natural greedy algorithm for this: process jobs in nondecreasing order of processing time, adding each if the set remains feasible. While Zhao and Yuan recently matched the $O(n\log n)$ time of the classic Moore-Hodgson algorithm via amortized $O(\log n)$ feasibility queries, our first contribution is a simpler augmented BST data structure that achieves worst-case $O(\log n)$ query time. Moore's greedy algorithm is known to produce a maximum-cardinality feasible subset with minimum total processing time. Though the system of feasible subsets does not form a matroid, the algorithm behaves like a matroid greedy algorithm. Our second contribution explains this phenomenon: if all jobs have equal processing times, the problem corresponds to a nested matroid; with distinct processing time "tiers," Moore's algorithm effectively solves a sequence of nested matroid problems for each tier. Finally, we present an explicit linear-size flow network defining a polymatroid rank. A set is feasible exactly when its rank equals its total processing time, and Moore's algorithm accepts a job precisely when it yields its full processing-time marginal. By contracting shorter accepted jobs and measuring residual rank, we recover the tier matroid's complete rank function, offering a deep structural explanation for the greedy algorithm's correctness.
Comments18 pages
