Complexityfrom

Complexityfrom is a proposed concept in theoretical computer science that references the resource cost required to produce a solution to a problem starting from a given initial configuration. It is not a standard measure in mainstream complexity theory, but appears in informal discussions and exploratory frameworks concerned with the role of prior state, precomputation, and data reuse in computation.

Definition: For a problem P, a computational model M, and an initial configuration I, complexityfrom of P with respect to I and M, denoted CF_M(P|I), is the minimal amount of a resource R required by any valid computation under M that produces a correct solution to P from I. When R is time, CF_M(P|I) is the minimal time from I; when R is space, CF_M(P|I) is the minimal peak memory usage. Variants may fix I or allow variability in I.

Relationships: CF is not a single complexity class but a family of measures parameterized by problem, model, and resource. It generalizes conventional time and space complexity by incorporating the influence of initial data and preexisting state. In some settings CF(P|I) may be tightly linked to amortized or online complexity, or to reusability of computed information.

Examples: In a sorting problem where I already contains a partially sorted sequence, CF_time may be lower than the worst-case time of standard sorting, reflecting warm-start advantages. In cryptographic protocols, CF_energy could model the energy cost of finishing a protocol fragment with a given key material available at start.

Status and criticism: Because complexityfrom is not standardized, definitions vary across authors and models. It is primarily used as a conceptual tool to reason about how starting conditions impact resource usage, rather than a formal framework with universal theorems.

See also: Computational complexity, Amortized analysis, Online algorithms, Parameterized complexity, Information complexity.