Finding any given 2-factor in sparse pseudorandom graphs efficiently

Given an (Formula presented.) -vertex pseudorandom graph (Formula presented.) and an (Formula presented.) -vertex graph (Formula presented.) with maximum degree at most two, we wish to find a copy of (Formula presented.) in (Formula presented.), that is, an embedding (Formula presented.) so that (Formula presented.) for all (Formula presented.). Particular instances of this problem include finding a triangle-factor and finding a Hamilton cycle in (Formula presented.). Here, we provide a deterministic polynomial time algorithm that finds a given (Formula presented.) in any suitably pseudorandom graph (Formula presented.). The pseudorandom graphs we consider are (Formula presented.) -bijumbled graphs of minimum degree which is a constant proportion of the average degree, that is, (Formula presented.). A (Formula presented.) -bijumbled graph is characterised through the discrepancy property: (Formula presented.) for any two sets of vertices (Formula presented.) and (Formula presented.). Our condition (Formula presented.) on bijumbledness is within a log factor from being tight and provides a positive answer to a recent question of Nenadov. We combine novel variants of the absorption-reservoir method, a powerful tool from extremal graph theory and random graphs. Our approach builds on our previous work, incorporating the work of Nenadov, together with additional ideas and simplifications.