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Determine the deterministic round complexity of replacement paths

Determine the optimal deterministic round complexity for computing all edge-failure replacement path distances for a given s–t path in the CONGEST model, and ascertain whether an O~(n^{2/3}+D) deterministic algorithm exists or whether higher deterministic lower bounds are necessary.

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Background

The paper’s O~(n{2/3}+D) algorithms are randomized, with long-detour handling inherently randomized.

The authors explicitly raise the question of the optimal deterministic complexity, suggesting possible use of derandomization techniques from deterministic APSP.

References

In our \widetilde{O}(n{2/3} + D)-round algorithms, the way we compute long-detour replacement paths is inherently randomized. 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: Conclusions and Open Problems (Deterministic algorithms)