Paper
Universal Hirschberg for Width Bounded Dynamic Programs
Authors
Logan Nye
Abstract
Hirschberg's algorithm (1975) reduces the space complexity for the longest common subsequence problem from to via recursive midpoint bisection on a grid dynamic program (DP). We show that the underlying idea generalizes to a broad class of dynamic programs with local dependencies on directed acyclic graphs (DP DAGs). Modeling a DP as deterministic time evolution over a topologically ordered DAG with frontier width and bounded in-degree, and assuming a max-type semiring with deterministic tie breaking, we prove that in a standard offline random-access model any such DP admits deterministic traceback in space cells over a fixed finite alphabet, where is the number of states. Our construction replaces backward dynamic programs by forward-only recomputation and organizes the time order into a height-compressed recursion tree whose nodes expose small "middle frontiers'' across which every optimal path must pass. The framework yields near-optimal traceback bounds for asymmetric and banded sequence alignment, one-dimensional recurrences, and dynamic-programming formulations on graphs of bounded pathwidth. We also show that an space term (in bits) is unavoidable in forward single-pass models and discuss conjectured -type barriers in streaming settings, supporting the view that space-efficient traceback is a structural property of width-bounded DP DAGs rather than a peculiarity of grid-based algorithms.