Given operators X and Y on a Hilbert space H, an interpolating operator is a bounded operator A such that AX = Y. In this article, we investigate compact interpolation problems for vectors in a tridiagonal algebra. Let L be a subspace lattice acting on a separable complex Hilbert space H and Alg L be a tridiagonal algebra. Let X = $(x_{ij});and;Y;=;(y_{ij})$ be operators acting on H. Then the following are equivalent: (1) There exists a compact operator A = $(x_{ij})$ in AlgL such that AX = Y. (2) There is a sequence {$alpha_n$} in $mathbb{C}$ such that {$alpha_n$} converges to zero and $$y_1;_j=alpha_1x_1;_j+alpha_2x_2;_j;y_{2k};_j=alpha_{4k-1}x_{2k;j};y_{2k+1;j}=alpha_{4k}x_{2k;j}+alpha_{4k+1}x_{2k+1;j}+alpha_{4k+2}x_{2k+2;j;for;all;k;epsilon;mathbb{N}$$.