Between the deterministic and non-deterministic query complexity
Dániel Gerbner
TL;DR
The paper introduces a spectrum of query complexities $D_k(P,n)$ that interpolate between non-deterministic and deterministic settings by allowing an adversary to change the input up to $k$ times during querying. It analyzes canonical problems—search theory/group testing, sorting, max/min, and graph connectivity—providing exact or tight bounds and illustrating how input dynamism affects certificate-based strategies. Through adversarial constructions and graph-theoretic tools (e.g., Turán graphs), the work reveals when and how robustness to input changes reshapes query complexity. The findings offer insights into the resilience of algorithms under dynamic inputs and lay groundwork for further exploration of dynamic/adversarial models and potential extensions with lies or additional constraints.
Abstract
We consider problems that can be solved by asking certain queries. The deterministic query complexity $D(P,n)$ of a problem $P$ is the smallest number of queries needed to ask in order to find the solution with an input of size $n$ (in the worst case), while the non-deterministic query complexity $D_0(P,n)$ is the smallest number of queries needed to ask, in case we know the solution, to prove that it is indeed the solution (in the worst case). Equivalently, $D(P,n)$ is the largest number of queries needed to find the solution in case an Adversary is answering the queries, while $D_0(P)$ is the largest number of queries needed to find the solution in case an Adversary chooses the input. We define a series of quantities between these two values, $D_k(P,n)$ is the largest number of queries needed to find the solution in case an Adversary chooses the input, and answers the queries, but he can change the input at most $k$ times. We give bounds on $D_k(P,n)$ for various problems $P$.