Communication complexity theory is a powerful tool to show time complexity lower bounds of distributed algorithms for global problems such as minimum spanning tree (MST) and shortest path. While it often leads to nearly-tight lower bounds for many problems, polylogarithmic complexity gaps still lie between the currently best upper and lower bounds. In this paper, we propose a new approach for filling the gaps. Using this approach, we achieve tighter deterministic lower bounds for MST and shortest paths. Specifically, for those problems, we show the deterministic
-round lower bound for graphs of
) hop-count diameter, and the deterministic
lower bound for graphs of
) hop-count diameter. The main idea of our approach is to utilize the two-party communication complexity lower bound for a function we call
, which is newly introduced in this paper.