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.

翻译：我们研究一般字母表$Σ$上的置换（乱序/阿贝尔）模式匹配问题。给定长度为$m$的模式$P$和长度为$n$的文本$T$，经典任务是判断$T$是否包含一个长度为$m$的子串，其帕里赫向量与$P$相等。虽然该存在性问题可通过线性时间滑动窗口算法求解，但许多实际应用需要超越单纯检测的优化与打包变体。我们提出一种基于维护$P$与$T$当前窗口间帕里赫向量差的统一滑动窗口框架，可在$O(n + σ)$时间和$O(σ)$空间内实现置换匹配，其中$σ = |Σ|$。在此基础上，我们引入一种组合优化变体——模式供给下的最大可行子串问题：寻找$T$中最长的子串$S$，其符号计数在分量意义上不超过$P$的对应计数。我们证明MFSP问题同样可通过双指针可行性维护算法在$O(n + σ)$时间内求解，为$P$提供了作为资源预算的精确打包解释。最后，我们通过将每个置换匹配建模为等长区间，并证明贪心最早结束策略可得到最大基数的不相交匹配集合，从而解决非重叠出现选择问题，该集合可在枚举所有匹配后线性时间内计算。我们的结果提供了具有紧致界限的简洁可证明正确算法，并将基于频率的字符串匹配与打包式优化原语联系起来。