Incremental computation

Incremental computing, also known as incremental computation, is a software feature which, whenever a piece of data changes, attempts to save time by only recomputing those outputs which depend on the changed data. When incremental computing is successful, it can be significantly faster than computing new outputs naively. For example, a spreadsheet software package might use incremental computation in its recalculation feature, to update only those cells containing formulas which depend (directly or indirectly) on the changed cells.

When incremental computing is implemented by a tool that can implement it for a variety of different pieces of code automatically, that tool is an example of a program analysis tool for optimization.

Incremental computing techniques can be broadly separated into two types of approaches:

Static approaches attempt to derive an incremental program from a conventional program P using, e.g., either manual design and refactoring, or automatic program transformations. These program transformations occur before any inputs or input changes are provided.

Dynamic approaches record information about executing program P on a particular input (I1) and use this information when the input changes (to I2) in order to update the output (from O1 to O2). The figure shows the relationship between program P, the change calculation function ΔP, which constitutes the core of the incremental program, and a pair of inputs and outputs, I1, O1 and I2, O2.

Some approaches to incremental computing are specialized, while others are general purpose. Specialized approaches require the programmer to explicitly specify the algorithms and data structures that will be used to preserve unchanged sub-calculations. General-purpose approaches, on the other hand, use language, compiler or algorithmic techniques to give incremental behavior to otherwise non-incremental programs.

Given a computation ${\displaystyle C=f(x_{1},x_{2},\dots x_{n})}$ and a potential change ${\displaystyle x_{j}:=\Delta _{x_{j}}}$, we can insert code before the change (the pre-derivative) and after the change (the post-derivative) to update the value of ${\displaystyle C}$ faster than rerunning ${\displaystyle f}$. Paige has written down a list of rules for formal differentiation of programs in SUBSETL.

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