Optimal bounds on finding fixed points of contraction mappings

Abstract

The Banach fixed-point theorem states that a contraction mapping on a complete metric space has a unique fixed point. Given an oracle access to a finite metric space ( M , d ) and a contraction mapping f : M → M on it, we show that the fixed point of f can be found with an expected O ( | M | ) oracle queries. We also show that every randomized algorithm for finding a fixed point must make an expected Ω ( | M | ) oracle queries to ( M , d ) and f for some finite metric space ( M , d ) and some contraction mapping f : M → M on it. As a generalization of the Banach fixed-point theorem, the Caristi–Kirk fixed-point theorem gives weaker conditions on ( M , d ) and f guaranteeing the existence of a fixed point of f . We show that every randomized algorithm that finds a fixed point must make the expected Ω ( | M | ) oracle queries to ( M , d ) and f for some finite metric space ( M , d ) and some function f : M → M satisfying the conditions of the Caristi–Kirk fixed-point theorem.