Permutation Matching Under Parikh Budgets: Linear-Time Detection, Packing, and Disjoint Selection

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.

OBHS: An Optimized Block Huffman Scheme for Real-Time Audio Compression

In this paper, we introduce OBHS (Optimized Block Huffman Scheme), a novel lossless audio compression algorithm tailored for real-time streaming applications. OBHS leverages block-wise Huffman coding with canonical code representation and intelligent fallback mechanisms to achieve high compression ratios while maintaining low computational complexity. Our algorithm partitions audio data into fixed-size blocks, constructs optimal Huffman trees for each block, and employs canonical codes for efficient storage and transmission. Experimental results demonstrate that OBHS attains compression ratios of up to 93.6% for silence-rich audio and maintains competitive performance across various audio types, including pink noise, tones, and real-world recordings. With a linear time complexity of O(n) for n audio samples, OBHS effectively balances compression efficiency and computational demands, making it highly suitable for resource-constrained real-time audio streaming scenarios.