Efficient Parity Determination in PP Algorithm: Real-time Tracking and Inverse Mapping

[Yusheng-Hu]

Efficient Parity Determination in PP Algorithm: Real-time Tracking and Inverse Mapping

1. Incremental Mode: Real-time Parity Tracking

During the iterative generation of permutations, the Position-Pure (PP) algorithm can synchronize the parity state with negligible overhead.

• Core Logic: Based on the stepping rule of the PP algorithm, parity is determined by the iteration index i.
  • If the mapping value val == i, no positional swap occurs.
  • If val != i, a transposition is triggered, flipping the parity state.
• Complexity: $O(1)$ per permutation.
• Use Case: Ideal for high-performance scenarios requiring simultaneous parity (e.g., determinant expansion). This logic can be seamlessly embedded directly into the iter function.

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2. Inverse Mapping Mode: Static Parity Analysis

For pre-generated or isolated permutations, parity can be determined by leveraging the inverse logic of the PP Ranking process.

• Core Logic: An auxiliary array M is used to track the current positions of elements. By executing the PP inverse operations, the permutation is restored to the identity sequence (1, 2, ..., n). Every swap performed during this restoration is recorded.
• Conclusion: The parity of the permutation matches the parity of the total swap count.
• Key Advantages:
  • Consistency: Fully self-consistent with the PP Ranking system; no need for external inversion-counting algorithms.
  • Efficiency: $O(N)$ time complexity.
  • Non-destructive: Introducing the auxiliary array M avoids any modifications to the original data array D.

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Note: The specific implementation code for these methods hasn't been added to the repository yet. Please let me know if you would like to see the reference code.