Sparse arrays increase aperture and resolution capability for direction-of-arrival estimation. Spatial smoothing step in coarray MUSIC algorithm prevents us from using the full coarray; so, there is a limitation for aperture N_α and degrees of freedom that are determined by minimum redundancy arrays. Interpolation methods can reduce this limitation and make it possible to use the full coarray of partially augmentable arrays. In interpolation methods based on matrix completion techniques, we should complete an N_α × N_α Toeplitz matrix. For this purpose, a semidefinite programming (SDP) with O(N_α⁶) complexity should be solved. In this letter, we introduce a set that contains arrays that have the equal aperture with original array and have filled coarray. By selecting the array that have minimum number of sensors in this set, we formulate a new structured matrix completion problem. Numerical examples indicate that the proposed structured matrix completion not only decreases the complexity of SDP problem but also, in many instances, increases the estimation accuracy and probability of resolution.