The generalized egg dropping problem is a classic challenge in sequential decision-making. Standard dynamic programming evaluates the minimax minimum number of tests in $\mathcal{O}(K \cdot N^2)$ time. A known approach formulates the testable thresholds as a partial sum of binomial coefficients and applies binary search to reduce the time complexity to $\mathcal{O}(K \log N)$. In this paper, we demonstrate that binary search over the complete sequential test domain is suboptimal. By restricting a binary search over multiples of $K$, we isolate a dynamic structural envelope that guarantees convergence. We prove that this boundary balances the search depth against the combinatorial evaluation cost, cancelling the $K$ variable to strictly bound the search phase to $\mathcal{O}(\log N)$. Combined with an incremental traversal, our algorithm eliminates the standard bottlenecks. Furthermore, we formulate an explicit $\mathcal{O}(1)$ space policy to dynamically reconstruct the optimal decision tree.

翻译：广义鸡蛋掉落问题是序贯决策中的经典难题。标准动态规划方法求解极小化极大测试次数的时间复杂度为 $\mathcal{O}(K \cdot N^2)$。现有一种方法将可检测阈值表述为二项式系数的部分和，并通过二分查找将时间复杂度降至 $\mathcal{O}(K \log N)$。本文证明，在整个序贯测试域上进行二分查找并非最优方案。通过限制在 $K$ 的倍数区间内进行二分查找，我们分离出一个保证收敛性的动态结构包络。我们证明，该边界能够平衡搜索深度与组合评估开销，从而消去变量 $K$，将搜索阶段严格限制为 $\mathcal{O}(\log N)$。结合增量遍历方法，本算法消除了标准瓶颈。此外，我们提出了一种显式的 $\mathcal{O}(1)$ 空间策略，用于动态重构最优决策树。