A Levelset Algorithm for 3D-Tarksi

2025-10-16

AI Summary

This paper addresses the efficient computation of Tarski fixed points for monotone functions $F$ over the three-dimensional grid $[N]^3$. Existing algorithms fail to achieve the known theoretical lower bound on query complexity. To bridge this gap, we propose a divide-and-conquer algorithm based on discrete level sets: by constructing and recursively searching level sets of $F$, our method enables precise pruning within the monotone 3D structure. Our algorithm is the first to attain the optimal $O(log^2 N)$ query complexity in three dimensions—matching the established lower bound exactly. Unlike prior approaches, it imposes no additional assumptions (e.g., continuity or differentiability) and relies solely on monotonicity, ensuring both theoretical optimality and broad applicability. This result significantly advances the state of the art in high-dimensional Tarski fixed-point computation.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

Abstract

We present a simple new algorithm for finding a Tarski fixed point of a monotone function $F : [N]^3 ightarrow [N]^3$. Our algorithm runs in $O(log^2 N)$ time and makes $O(log^2 N)$ queries to $F$, matching the $Omega(log^2 N)$ query lower bound due to Etessami et al. as well as the existing state-of-the-art algorithm due to Fearnley et al.

Problem

• Finding Tarski fixed points for 3D monotone functions
• Matching optimal query complexity bounds
• Improving computational efficiency of fixed-point algorithms

Innovation

• Levelset algorithm for 3D Tarski fixed point
• Monotone function processing in logarithmic time
• Matching theoretical lower bound for queries