We present the first non-trivial fully dynamic algorithm maintaining exact single-source distances in unweighted graphs. This resolves an open problem stated by Sankowski [COCOON 2005] and van den Brand and Nanongkai [FOCS 2019]. Previous fully dynamic single-source distances data structures were all approximate, but so far, non-trivial dynamic algorithms for the exact setting could only be ruled out for polynomially weighted graphs (Abboud and Vassilevska Williams, [FOCS 2014]). The exact unweighted case remained the main case for which neither a subquadratic dynamic algorithm nor a quadratic lower bound was known. Our dynamic algorithm works on directed graphs, is deterministic, and can report a single-source shortest paths tree in subquadratic time as well. Thus we also obtain the first deterministic fully dynamic data structure for reachability (transitive closure) with subquadratic update and query time. This answers an open problem of van den Brand, Nanongkai, and Saranurak [FOCS 2019]. Finally, using the same framework we obtain the first fully dynamic data structure maintaining all-pairs $(1+\epsilon)$-approximate distances within non-trivial sub-$n^\omega$ worst-case update time while supporting optimal-time approximate shortest path reporting at the same time. This data structure is also deterministic and therefore implies the first known non-trivial deterministic worst-case bound for recomputing the transitive closure of a digraph.

翻译：我们提出了首个非平凡的全动态算法，用于在无权重图中维护精确的单源距离。这解决了Sankowski [COCOON 2005] 以及 van den Brand 和 Nanongkai [FOCS 2019] 提出的开放问题。此前，全动态单源距离数据结构均为近似算法，而对于精确设置，非平凡动态算法仅能在多项式加权图中被排除（Abboud 和 Vassilevska Williams，[FOCS 2014]）。精确的无权重情况仍是主要情形，此前既无次二次动态算法，也无二次下界已知。我们的动态算法适用于有向图，具有确定性，并且能够在次二次时间内报告单源最短路径树。因此，我们还首次获得了用于可达性（传递闭包）的确定性全动态数据结构，并具有次二次更新和查询时间。这回答了 van den Brand、Nanongkai 和 Saranurak [FOCS 2019] 的开放问题。最后，利用同一框架，我们首次获得了在非平凡次$n^\omega$最坏情况更新时间内维护所有点对$(1+\epsilon)$近似距离的全动态数据结构，同时支持最优时间的近似最短路径报告。该数据结构也是确定性的，因此隐含着已知首个非平凡确定性最坏情况界，用于重新计算有向图的传递闭包。