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Partially dynamic SSSP with O(mn) total time for m updates and n^2 queries

Determine whether there exists a partially dynamic Single-Source Shortest Paths (SSSP) algorithm on directed graphs with nonnegative edge weights that, over m edge updates and n^2 distance queries (from a fixed source to arbitrary targets), achieves O(mn) total time for all updates and queries combined.

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Background

Roditty and Zwick considered a query model of partially dynamic SSSP where, instead of reporting all distances after each update, the data structure must answer distance queries from a fixed source to any target node. They posed whether one can achieve total time proportional to O(mn) when there are m edge updates and n2 queries.

An affirmative answer would imply an O(mn)-time algorithm for the classic partially dynamic SSSP setting (which has at most m queries), thereby substantially improving upon the naive recomputation approach and resolving a central barrier in dynamic shortest paths.

References

Roditty and Zwick considered partially dynamic where instead of outputting the distances from $s$ to all nodes in $V$ after each update, the data structure only needs to support distance queries between $s$ and any queried node $v$. They explicitly stated the following open question:

Is partially dynamic $$ with $m$ updates and $n2$ queries in $O(mn)$ time?

— Fine-Grained Optimality of Partially Dynamic Shortest Paths and More (2407.09651 - Saha et al., 12 Jul 2024) in Open Question (label open:RZ), Introduction