Achieving linear complexity is crucial for demonstrating optimal convergence rates in adaptive refinement. It has been shown that the existing linear complexity local refinement algorithm for ${\rm T}$-splines generally produces more degrees of freedom than the existing greedy refinement, which lacks linear complexity. This paper introduces a novel greedy local refinement algorithm for analysis-suitable ${\rm T}$-splines, which achieves linear complexity and requires fewer control points than existing algorithms with linear complexity. Our approach is based on the observation that confining refinements around each ${\rm T}$-junction to a preestablished feasible region ensures the algorithm’s linear complexity. Building on this constraint, we propose a greedy optimization local refinement algorithm that upholds linear complexity while significantly reducing the degrees of freedom relative to previous linear complexity local refinement methods.