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Memoization retains a completed function result under a memoization key and returns that result when an equivalent request recurs. For a terminating pure function, repeated evaluation can therefore become lookup rather than repeated work. The benefit is especially striking with recursion and overlapping subproblems: a large tree of calls can refer to a small Directed acyclic graph of distinct answers. The memoization cache contract determines whether reuse is correct; the memoization cache lifetime determines whether an answer is still present when wanted.

Memoization records computed answers, whereas a memory cache retains data from computer memory. A CPU cache hit does not by itself prevent a program from evaluating the same function again. Spreadsheet dependency evaluation adds an explicit mechanism for invalidating affected cached cell values after inputs change; a general memoization table does not discover such changes automatically.

A documented early motivation was to let programs improve their efficiency during execution through rote learning: the abstract of a 1968 paper explicitly proposes such a programming language facility.^[ref]

For a terminating pure function, a reusable entry must record its correct completed result, and equal memoization keys must imply equal results throughout the entry's memoization cache lifetime. Reusing that result must also preserve the behavior callers require. These conditions separate correctness from speed: a complete key can be correct even if it rarely repeats.

A procedure using deterministic computation can still have a required side effect. If a procedure returns $n^2$ and appends an audit record on every call, serving the second call from a table preserves the numerical answer but skips the second record. Likewise, if each call promises a fresh mutable list, returning the same stored mutable list changes observable sharing and allows a caller to damage later answers. Plain result reuse is unsuitable for those contracts.^[ref]

Completion matters for recursion. Suppose computing key $A$ needs key $B$, and computing $B$ needs $A$. A table containing only completed answers has no answer to return for the second $A$; remembering completed values does not make this directed cycle terminate. Marking a key ACTIVE before descending detects re-entry, as in spreadsheet dependency evaluation, but detection is not a solution of the circular equations. If computation fails, clear its active mark and do not publish an unfinished value. The example using Fibonacci numbers avoids this issue through a strictly decreasing argument.

The call counts above assume sequential execution and no eviction. Concurrent requests for an absent key can both begin evaluation before either publishes its result; keeping a table coherent alone does not ensure one body execution per key.^[ref]

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]