Optimization problems such as Viterbi decoding and V-optimal histogram construction seek a path of exact length L through a state space that minimizes a cost function. These problems are traditionally solved using dynamic programming (DP). A best-first-search (BFS) solution is also applicable, yet requires maintaining a priority queue. In all cases, memory usage grows linearly with both state space size and path length. In this paper, we propose CompactBFS, a framework that limits the growth of the BFS priority queue to space-efficiently determine the exact optimal cost for a fixed-length path and then constructs such a path by a divide-and-conquer strategy that eliminates the memory overhead. We apply CompactBFS to Viterbi decoding, which remains relevant to speech recognition, and V-optimal histogram construction. Our experimental results demonstrate significant gains over state-of-the-art solutions in runtime and memory consumption.