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Optimal deterministic complexity for the replacement paths problem

Determine the optimal deterministic round complexity in the CONGEST model for computing, for every edge e on a given s–t shortest path P in a graph G, the replacement-path distances |st ⧫ e| (the Replacement Paths problem).

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Background

All upper bounds in the paper are randomized, with the long-detour component relying on random sampling of landmark vertices.

The authors highlight that the deterministic complexity of Replacement Paths remains unsettled and suggest that derandomization techniques from deterministic APSP might be applicable.

References

It remains an open question to determine the optimal deterministic complexity for the replacement path problem.

— Optimal Distributed Replacement Paths (2502.15378 - Chang et al., 21 Feb 2025) in Section 10 (Conclusions and Open Problems), bullet “Deterministic algorithms”