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

TL;DR

This paper introduces an $ ext{O}( ext{log} N)$ time algorithm for the generalized egg dropping problem, improving efficiency over previous methods by optimizing search strategies and decision tree reconstruction.

Contribution

It presents a novel $ ext{O}( ext{log} N)$ time algorithm that refines search over the decision domain and provides an explicit $ ext{O}(1)$ space policy for dynamic decision tree reconstruction.

Findings

Achieves $ ext{O}( ext{log} N)$ time complexity for the problem.

Develops an explicit $ ext{O}(1)$ space policy for decision tree reconstruction.

Demonstrates the suboptimality of binary search over the complete domain.

Abstract

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.