Fill Array Zeros with Positive Integers to Satisfy Adjacent |diff| ≤ 1

Do not assume a unique formulation from this title. First clarify whether the condition is absolute difference exactly one, whether zero means an unknown placeholder, and whether the goal is any valid fill, a count, or a minimum-cost fill. With fixed positive anchors a at index l and b at r, feasibility depends on distance d: a path of d unit moves can reach b only when d is at least |a-b| and d minus |a-b| is even, while values must remain positive. For a constructive any-solution version, build steps of plus or minus one between anchors and use harmless up-down pairs to consume extra distance without dropping below one. Prefixes or suffixes with no fixed anchor can be filled by alternating 1,2. If constraints add costs or choices, define DP[position][value] with transitions only from value minus one and value plus one. Validate every boundary. Construction is O(n); bounded-value DP is O(nV) time and O(V) rolling space.