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A three-pair network consists of a directed acyclic graph with two sets $S={s_1,s_2,s_3}$ and $T={t_1,t_2,t_3}$  such that each $s_{i}in S$ generates a message $X_{i}$ and each $t_{i}in T$ wants to get the message $X_{i}$.  This paper presents an $O(|E|^2)$ time algorithm to diagnose the solvability or unsolvability of three-pair networks with common bottleneck links, where $|E|$ is the number of links. The algorithm is a region decomposition method, which improves upon the previous polynomial time of $O(|E|^3)$ due to cite{CH}.