Time-efficient implementation of the safe PI function–based algorithms for Tarski* and Tarski

Ascertain whether the O(log n)-query algorithm for Tarski*(n,3), and the induced O(log^2 n)-query algorithm for Tarski(n,4), can be implemented with efficient time complexity rather than relying on the currently non–time-efficient safe partial-information framework and brute-force steps.

Background

The paper achieves optimal query complexity for Tarski(n,4) and improved bounds for higher dimensions via a new framework based on safe partial-information (PI) functions. However, the resulting algorithms are not time-efficient for two reasons explicitly noted by the authors: (1) the safe PI function game framework is not currently time-efficient, and (2) selecting good query points uses brute-force counting arguments.

The authors explicitly elevate the question of time efficiency to an open problem, asking whether these query-efficient algorithms can be implemented to run efficiently in time.