🤖 AI Summary
This paper investigates how dynamic programming (DP) degenerates into classical greedy algorithms under specific structural constraints. Focusing on three canonical problems—unit-weight interval scheduling, unit-value knapsack, and shortest paths with nonnegative edge weights—it proposes a systematic DP-to-greedy transformation framework. The method involves expanding the Bellman equation and leveraging monotonicity of the objective function together with optimal substructure to rigorously identify and eliminate redundant state transitions, thereby deriving the greedy-choice property. This approach unifies the theoretical justification for well-known greedy rules (e.g., “earliest finishing time”) and exposes an intrinsic continuity between DP and greedy paradigms within the algorithm design spectrum. The resulting derivation paradigm provides a transferable methodology for proving greedy correctness, offering both theoretical insight and pedagogical utility in algorithm design and analysis. (149 words)
📝 Abstract
We show for several computational problems how classical greedy algorithms for special cases can be derived in a simple way from dynamic programs for the general case: interval scheduling (restricted to unit weights), knapsack (restricted to unit values), and shortest paths (restricted to nonnegative edge lengths). Conceptually, we repeatedly expand the Bellman equations underlying the dynamic program and use straightforward monotonicity properties to figure out which terms yield the optimal value under the respective restrictions. The approach offers an alternative for developing these greedy algorithms in undergraduate algorithms courses and/or for arguing their correctness. In the setting of interval scheduling, it elucidates the change in order from earliest start time first for the memoized dynamic program to earliest finish time first for the greedy algorithm.