在Parikh预算下进行排列匹配：线性时间的检测、打包和不相交选择
Permutation Matching Under Parikh Budgets: Linear-Time Detection, Packing, and Disjoint Selection
AI总结：
本文提出了一种在Parikh预算下进行排列匹配的线性时间算法，解决检测、打包和不相交选择问题。
摘要：
我们研究了在一般字母表Σ上的一般排列（混乱/阿贝尔）模式匹配问题。给定一个长度为m的模式P和一个长度为n的文本T，经典任务是决定T中是否存在一个长度为m的子字符串，其Parikh向量等于P的Parikh向量。虽然这个存在性问题可以采用线性时间的滑动窗口方案解决，但许多实际应用需要优化和打包变种，而不仅仅是检测。我们提出了一种基于维护P与当前T窗口的Parikh向量差的统一滑动窗口框架，使排列匹配可以在O(n + σ)时间内完成，空间复杂度为O(σ)，其中σ = |Σ|。在此基础上，我们引入了一个组合优化变种，称为在模式供应下的最长可行子字符串（MFSP）：找到T中最长的子字符串S，其符号计数在每个组件上都小于等于P的符号计数。我们证明MFSP也可以通过双指针可行性维护算法在O(n + σ)时间内解决，为P作为资源预算提供了精确的打包解释。最后，我们通过将每个排列匹配建模为一个等长区间，证明了贪心最早完成策略可以得到一个最大基数的不相交匹配集，一旦所有匹配都被枚举，该集合可以在线性时间内计算出来。我们的结果提供了简洁且可证明正确的算法，具有紧致的界限，并将基于频率的字符串匹配与打包式优化原语联系起来。
英文摘要：
We study permutation (jumbled/Abelian) pattern matching over a general alphabet $Σ$. Given a pattern P of length m and a text T of length n, the classical task is to decide whether T contains a length-m substring whose Parikh vector equals that of P . While this existence problem admits a linear-time sliding-window solution, many practical applications require optimization and packing variants beyond mere detection. We present a unified sliding-window framework based on maintaining the Parikh-vector difference between P and the current window of T , enabling permutation matching in O(n + σ) time and O(σ) space, where σ = |Σ|. Building on this foundation, we introduce a combinatorial-optimization variant that we call Maximum Feasible Substring under Pattern Supply (MFSP): find the longest substring S of T whose symbol counts are component-wise bounded by those of P . We show that MFSP can also be solved in O(n + σ) time via a two-pointer feasibility maintenance algorithm, providing an exact packing interpretation of P as a resource budget. Finally, we address non-overlapping occurrence selection by modeling each permutation match as an equal-length interval and proving that a greedy earliest-finishing strategy yields a maximum-cardinality set of disjoint matches, computable in linear time once all matches are enumerated. Our results provide concise, provably correct algorithms with tight bounds, and connect frequency-based string matching to packing-style optimization primitives.
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