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An $\mathcal{O}(\log N)$ Time Algorithm for the Generalized Egg Dropping Problem

Kleitos Papadopoulos

TL;DR

This paper demonstrates that the discrete binary search over the decision tree can be bypassed entirely and forms an explicit $\mathcal{O}(1)$ space deterministic policy to dynamically retrace the optimal sequential choices, eliminating classical state-transition matrices completely.

Abstract

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.

An $\mathcal{O}(\log N)$ Time Algorithm for the Generalized Egg Dropping Problem

TL;DR

This paper demonstrates that the discrete binary search over the decision tree can be bypassed entirely and forms an explicit space deterministic policy to dynamically retrace the optimal sequential choices, eliminating classical state-transition matrices completely.

Abstract

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 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 . 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 discrete steps. Because constraints inherently enforce , our algorithm achieves an unconditional worst-case time complexity of . Furthermore, we formulate an explicit space deterministic policy to dynamically retrace the optimal sequential choices, eliminating classical state-transition matrices completely.
Paper Structure (10 sections, 2 theorems, 13 equations, 1 table, 1 algorithm)

This paper contains 10 sections, 2 theorems, 13 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Mathematical Formulation
  4. The Algorithm
  5. Phase 1: Decision-Tree Lower Bound
  6. Phase 2: Continuous Root Approximation
  7. Phases 3 & 4: Initial Evaluation and Incremental Search
  8. Complexity Analysis
  9. Retracing the Optimal Policy
  10. Conclusion

Key Result

Lemma 1

The absolute difference between the continuous estimate $T_{\text{est}}$ and the exact optimal target $T^*$ is bounded by $K$, such that $T_{\text{est}} - K < T^* \le T_{\text{est}} + K$.

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Theorem 1
  • proof