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There are three useful distinctions between dynamic programming and a conventional divide-and-conquer algorithm.

These are paradigm-level distinctions, not mutually exclusive definitions: a recursive algorithm can use memoization, and dynamic programming need not be implemented iteratively. For optimization, the dynamic programming state and optimal substructure justify combining stored optimum values; storing repeated work alone does not prove the state is sufficient.

Different move orders can lead to the same position, and the two occupancy lists also retain irrelevant insertion order. Thus this search tree repeatedly solves the same remaining game. For example, swapping the order of two O moves while leaving the intervening X moves legal can reach the same occupied sets.

Canonicalize each state by sorted occupancy sets, or by two bitboards, and apply memoization or a visited-state set. Expand each canonical position once, stopping at terminal states, and collect terminal positions in a set. There are at most $3^9$ cell assignments, many of which are illegal; this is much smaller than a tree of move permutations. Store positions, not histories, when the requested output is distinct final positions. Board rotations and reflections can reduce work further if handled as an additional equivalence: if the output must retain every oriented board, reconstruct those distinct orientations rather than silently returning one representative per symmetry class.

Evaluate numeric formula dependencies using depth-first search and memoization, or a topological ordering. Marking active evaluations detects cycles; repeated numerical sweeps need not terminate or determine a unique result.

A spreadsheet dependency graph turns a page of spreadsheet formulas into a plan: compute each prerequisite before the spreadsheet cell that consumes it. This section uses a deliberately small model with explicit cell references, fixed spreadsheet formulas, and deterministic computation from numerical inputs. There are no external data changes or formulas that change their references while running. Within these assumptions, a topological order gives a reproducible answer without repeatedly sweeping the visible grid.

Take two input spreadsheet cells and five spreadsheet formulas:

Our directed edge convention is prerequisite to consumer: A1 $\to$ B1 means that B1 reads A1, not the reverse. The seven directed edges are A1 $\to$ B1, A2 $\to$ B1, A1 $\to$ B2, B1 $\to$ C1, B2 $\to$ C1, C1 $\to$ D1, and A2 $\to$ E1. This spreadsheet dependency graph is a Directed acyclic graph. One topological order is A1,A2,B1,B2,C1,D1,E1; E1 could instead be computed immediately after A2. A topological order constrains prerequisites, not the physical positions of spreadsheet cells.

In that order, the initial arithmetic operations give $B1=3+4=7$, $B2=2\cdot 3=6$, $C1=7+6=13$, $D1=13/2$, and $E1=4^2=16$. The spreadsheet formulas define one value per spreadsheet cell: an outline by induction along any topological order shows that every prerequisite already has its uniquely determined value when its consumer is computed. Different permitted orders therefore agree in this model. The useful idea is that information flows through the Directed acyclic graph even when the visible grid does not display that flow.

A concrete implementation history illustrates this separation. Microsoft's documentation distinguishes the per-worksheet calculation chains used in Excel 2002 and earlier from calculation using multiple software threads introduced in Excel 2007 for independent chain sections. This is a documented implementation example, not a definition of spreadsheet dependency evaluation.^[ref]

A two-player game on nine positions arranged in a three-by-three grid. Players alternate marking empty positions and stop when one has a full row, column or diagonal, or when the board is full. A game search must stop at the first win. Distinct move histories can reach the same position; canonical occupancy sets or bitboards make that equality available to memoization. Quotienting by board symmetry changes the requested output unless distinct oriented positions are subsequently restored.