Remove the O(h_st) additive term for weighted undirected replacement paths

Eliminate the O(h_st) additive dependence in the best-known randomized round complexity for computing replacement paths in weighted undirected graphs in the CONGEST model, achieving a bound that matches the single-source shortest paths round complexity without additive dependence on the path length h_st.

Background

For weighted undirected graphs, the prior best upper bound for replacement paths is O(T_SSSP + h_st), where T_SSSP denotes the round complexity of weighted undirected SSSP. Lower bounds of Ω~(√n+D) are known for SSSP.

The authors ask whether the extra O(h_st) additive term can be removed, especially in small-diameter graphs where existing bounds are nearly matched except for this term.