Fast matrix determinant calculation with specific structure

Question

I have a k*k squared matrix with diagonal elements x>0 and all other elements y>0. The values of k, x, y are all subject to change.

Now I need the determinant of this matrix. I know there won't be a closed-form formula for it, but is there a way to calculate it faster than the commonly used LU-decomposition which takes O(K^3) time complexity (considering its special structure)?

(I am using R as my coding language, and the built-in det() function in R uses the LU-decomposition.)