Fully dynamic all-pairs shortest paths with worst-case update-time revisited

We revisit the classic problem of dynamically maintaining shortest paths between all pairs of nodes of a directed weighted graph. The allowed updates are insertions and deletions of nodes and their incident edges. We give worst-case guarantees on the time needed to process a single update (in contrast to related results, the update time is not amortized over a sequence of updates). Our main result is a simple randomized algorithm that for any parameter $c>1$ has a worst-case update time of $O(cn^{2+2/3} \log^{4/3}{n})$ and answers distance queries correctly with probability $1-1/n^c$, against an adaptive online adversary if the graph contains no negative cycle. The best deterministic algorithm is by Thorup [STOC 2005] with a worst-case update time of $\tilde O(n^{2+3/4})$ and assumes non-negative weights. This is the first improvement for this problem for more than a decade. Conceptually, our algorithm shows that randomization along with a more direct approach can provide better bounds.Comment: To be presented at the Symposium on Discrete Algorithms (SODA) 201

Fully Dynamic All-pairs Shortest Paths with Worst-case Update-time revisited

We revisit the classic problem of dynamically maintaining shortest paths between all pairs of nodes of a directed weighted graph. The allowed updates are insertions and deletions of nodes and their incident edges. We give worst-case guarantees on the time needed to process a single update (in contrast to related results, the update time is not amortized over a sequence of updates). Our main result is a simple randomized algorithm that for any parameter $c>1$ has a worst-case update time of $O(cn^{2+2/3} \log^{4/3}{n})$ and answers distance queries correctly with probability $1-1/n^c$, against an adaptive online adversary if the graph contains no negative cycle. The best deterministic algorithm is by Thorup [STOC 2005] with a worst-case update time of $\tilde O(n^{2+3/4})$ and assumes non-negative weights. This is the first improvement for this problem for more than a decade. Conceptually, our algorithm shows that randomization along with a more direct approach can provide better bounds

Dynamic Algorithms for the Massively Parallel Computation Model

The Massive Parallel Computing (MPC) model gained popularity during the last decade and it is now seen as the standard model for processing large scale data. One significant shortcoming of the model is that it assumes to work on static datasets while, in practice, real-world datasets evolve continuously. To overcome this issue, in this paper we initiate the study of dynamic algorithms in the MPC model. We first discuss the main requirements for a dynamic parallel model and we show how to adapt the classic MPC model to capture them. Then we analyze the connection between classic dynamic algorithms and dynamic algorithms in the MPC model. Finally, we provide new efficient dynamic MPC algorithms for a variety of fundamental graph problems, including connectivity, minimum spanning tree and matching.Comment: Accepted to the 31st ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 2019

On the robustness of the metric dimension of grid graphs to adding a single edge

The metric dimension (MD) of a graph is a combinatorial notion capturing the minimum number of landmark nodes needed to distinguish every pair of nodes in the graph based on graph distance. We study how much the MD can increase if we add a single edge to the graph. The extra edge can either be selected adversarially, in which case we are interested in the largest possible value that the MD can take, or uniformly at random, in which case we are interested in the distribution of the MD. The adversarial setting has already been studied by [Eroh et. al., 2015] for general graphs, who found an example where the MD doubles on adding a single edge. By constructing a different example, we show that this increase can be as large as exponential. However, we believe that such a large increase can occur only in specially constructed graphs, and that in most interesting graph families, the MD at most doubles on adding a single edge. We prove this for $d$-dimensional grid graphs, by showing that $2d$ appropriately chosen corners and the endpoints of the extra edge can distinguish every pair of nodes, no matter where the edge is added. For the special case of $d=2$, we show that it suffices to choose the four corners as landmarks. Finally, when the extra edge is sampled uniformly at random, we conjecture that the MD of 2-dimensional grids converges in probability to $3+\mathrm{Ber}(8/27)$, and we give an almost complete proof