Deterministic Fully Dynamic SSSP and More

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 and 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.

我们介绍了第一个非平凡的完全动态算法，该算法在未加权图中维持精确的单源距离。这解决了Sankowski [Cocoon 2005]和Van den Brand和Nanongkai [Focs 2019]提出的一个空旷的问题。以前的完全动态的单源距离数据结构都是近似的，但是到目前为止，确切设置的非平凡动态算法只能排除在多项式加权图的情况下（Abboud和Vassilevska Williams，[FOCS 2014]）。确切的未加权案例仍然是主要情况，该案例既不知道，次级动态算法和二次下限都不知道。我们的动态算法在有向图上起作用，并且是确定性的，并且可以在亚次级时间中报告单源最短路径树。因此，我们还获得了第一个确定性的完全动态数据结构，以进行次级更新和查询时间的可及性（传递闭合）。这回答了Van Den Brand，Nanongkai和Saranurak的空缺问题[FOCS 2019]。最后，使用相同的框架，我们获得了第一个完全动态的数据结构，以维护all对$（1+ \ epsilon）$ - 非平凡sub- $ n^{\ omega} $差的距最佳时间近似最短路径报告同时报告。这种数据结构也是确定性的，因此意味着第一个已知的非平凡的确定性最差案例结合，用于重新计算Digraph的传递闭合。