instanceoptimality

Instance optimality is a concept in algorithm analysis that measures how well an algorithm performs on each input instance relative to the best possible algorithm for that same instance. An algorithm A is instance-optimal with respect to a class of algorithms C and a cost function cost(.,.) if there exists a constant c ≥ 1 such that for every input instance x, cost(A, x) ≤ c · min_{B ∈ C} cost(B, x). In words, A never does worse than the optimal algorithm for the same input by more than a fixed multiplicative factor c.

In many formulations the constant c is required to be independent of the instance and the problem size, while other versions allow additive or instance-specific constants. Instance optimality thus provides a per-instance benchmark that contrasts with worst-case or average-case optimality, focusing on how closely an algorithm can match the best possible choice for each particular input.

The notion is often discussed in contexts where algorithms can adapt to input structure or distribution, such as adaptive data structures, selection and sorting problems, or geometric and graph problems. It emphasizes the potential for an algorithm to tailor its effort to the intrinsic difficulty of an individual instance rather than apply a single global bound.

Existence and usefulness of instance-optimal algorithms depend on the problem class and the chosen comparison set. Some problem families admit nontrivial instance-optimal algorithms, while others do not. The concept is related to competitive analysis and adaptive approaches in algorithms and data structures. See also: adaptive algorithms, competitive analysis, instance-based learning.