The McNaughton-Zielonka divide et impera algorithm is the simplest and most flexible approach available in the literature for determining the winner in a parity game. Despite its theoretical exponential worst-case complexity and the negative reputation as a poorly effective algorithm in practice, it has been shown to rank among the best techniques for solving such games. Also, it proved to be resistant to a lower bound attack, even more than the strategy improvements approaches, but finally Friedmann provided a family of games on which the algorithm requires exponential time. An easy analysis of this family shows that a simple memoization technique can help the algorithm solve the family in polynomial time. The same result can also be achieved by exploiting an approach based on the dominion-decomposition techniques proposed in the literature. These observations raise the question whether a suitable combination of dynamic programming and game-decomposition techniques can improve on the exponential worst case of the original algorithm. In this paper we answer this question negatively, by providing a robustly exponential worst case, showing that no possible intertwining of the above mentioned techniques can help mitigating the exponential nature of the divide et impera approaches. The resulting worst case is even more robust than that, since it serves as a lower bound for progress measures based algorithms as well, such as Small Progress Measure and its quasi-polynomial variant recently proposed by Jurdziński and Lazic.

麦克诺顿-齐隆卡分裂与黑斑病算法是文献中确定平价游戏赢家的最简单，最灵活的方法。尽管其理论上的指数级最坏情况复杂性和在实践中效果不佳的算法的声誉不佳，但它已被证明是解决此类游戏的最佳技术。而且，它被证明可以抵抗下限攻击，甚至比策略改进方法更能抵抗攻击，但是弗里德曼最终提供了一系列算法需要算法花费时间的游戏。对该族的简单分析表明，简单的记忆技术可以帮助算法在多项式时间内求解该族。通过利用基于文献中提出的支配分解技术的方法，也可以实现相同的结果。这些发现提出了一个问题，即动态编程和游戏分解技术的适当组合是否可以改善原始算法的指数最坏情况。在本文中，我们通过提供健壮的指数最坏情况来否定性地回答该问题，表明上述技术不可能相互缠绕，可以帮助缓解风险的指数性质。分而治之的方法。由此产生的最坏情况甚至比以前更鲁棒，因为它也可以作为基于进度测量的算法的下限，例如Jurdziński和Lazic最近提出的小进度测量及其准多项式变体。