The generalized egg dropping problem is a canonical benchmark in sequential decision-making. Standard dynamic programming evaluates the minimum number of tests in the worst case in $\mathcal{O}(K \cdot N^2)$ time. The previous state-of-the-art approach formulates the testable thresholds as a partial sum of binomial coefficients and applies a combinatorial search to reduce the time complexity to $\mathcal{O}(K \log N)$. In this paper, we demonstrate that the discrete binary search over the decision tree can be bypassed entirely. By utilizing a relaxation of the binomial bounds, we compute an approximate root that tightly bounds the optimal value. We mathematically prove that this approximation restricts the remaining search space to exactly $\mathcal{O}(K)$ discrete steps. Because constraints inherently enforce $K < \log_2(N+1)$, our algorithm achieves an unconditional worst-case time complexity of $\mathcal{O}(\min(K, \log N))$. Furthermore, we formulate an explicit $\mathcal{O}(1)$ space deterministic policy to dynamically retrace the optimal sequential choices, eliminating classical state-transition matrices completely.

翻译：广义鸡蛋掉落问题是序贯决策中的一个经典基准问题。标准的动态规划方法在最坏情况下评估所需最少测试次数的复杂度为 $\mathcal{O}(K \cdot N^2)$。先前的最先进方法将可测试阈值表述为二项式系数的部分和，并应用组合搜索将时间复杂度降低至 $\mathcal{O}(K \log N)$。本文证明，可以完全绕过在决策树上进行的离散二分搜索。通过利用二项式边界的松弛，我们计算出一个紧密约束最优值的近似根。我们数学上证明了该近似将剩余搜索空间严格限制在 $\mathcal{O}(K)$ 个离散步骤内。由于约束本身强制 $K < \log_2(N+1)$，我们的算法实现了无条件最坏情况时间复杂度 $\mathcal{O}(\min(K, \log N))$。此外，我们构建了一个显式的 $\mathcal{O}(1)$ 空间确定性策略，以动态回溯最优序贯选择，从而完全消除了经典的状态转移矩阵。