Linear cost is sometimes quadratic

In analysis of programs and many other kinds of computation, algorithms are commonly written to have an input P and an output Q, where both P and Q are large and complicated objects. For example, P might be a routing problem and Q might be a solution to P. Although documented and discussed in this exhaustive style, algorithms are sometimes intended for use in contexts with two departures from the one-time analysis of an entire new input. First, the current value of P is the result of a small change to a previous value of P. We want to update the results of the previous analysis without redoing all of the work. Second, accurate information is only wanted in some designated portion of the large output Q. Possibly inaccurate information may appear elsewhere in Q. We want the analysis to be demand-driven: accurate where accuracy is demanded, but not burdened by the cost of providing much more than is demanded. This paper studies demand-driven algorithms capable of updating without extensive reanalysls. Such algorithms are called incremental, in contrast with the one-time analysis of an entire new input contemplated by exhaustive algorithms. In some cases, it is shown that an exhaustive algorithm can be easily recast in an efficient incremental style. Other algorithms for the same problem may be much less amenable to incremental use. It is shown that traditional exhaustive computational complexity bounds are poor predictors of incremental performance. The main examples are taken from global data flow analysis.