We present almost optimal distributed algorithms for finding $2$-ruling sets in the classical LOCAL model. Our main contribution is a randomized algorithm that w.h.p. computes a $2$-ruling set on any $n$-node graph with bounded arboricity in $O(\log \log n)$ rounds.

In this work, we make further advancements in the study of directed LDDs. We reveal a natural and intuitive (in hindsight) connection to Expander Decompositions, and leveraging this connection along with additional techniques, we establish the existence of an LDD with an edge-cutting probability of $O(\frac{1}{D} \cdot \log n \log\log n)$.

We give the first parallel algorithm with optimal $\tilde{O}(m)$ work for the classical problem of computing Single-Source Shortest Paths in general graphs with negative-weight edges.

In this work, we study the vertex-coloring problem in sparse graphs parameterized by their arboricity $\alpha$, a standard measure for sparsity. We give deterministic algorithms that in constant, or almost constant, time give $\text{poly} \alpha$ and $O(\alpha)$-colorings, where $\alpha$ can be arbitrarily smaller than $\Delta$.

We show the first conditionally optimal deterministic algorithm for $3$-coloring forests in the low-space massively parallel computation (MPC) model. Our algorithm runs in $O(\log \log n)$ rounds and uses optimal global space.

We study the problem of finding connected components in the Adaptive Massively Parallel Computation (AMPC) model. We show that when we require the total space to be linear in the size of the input graph the problem can be solved in $O(\log^* {n})$ rounds in forests (with high probability) and $2^{O(\log^*{n})}$ expected rounds in general graphs.

We present a deterministic algorithm for solving a wide range of dynamic programming problems in trees in $O(\log D)$ rounds in the massively parallel computation model (MPC), with $O(n^\delta)$ words of local memory per machine, for any given constant $0 < \delta < 1$. Our algorithm can solve many classical graph optimization problems such as maximum weight independent set, maximum weight matching, minimum weight dominating set, and minimum weight vertex cover.

We give an algorithm that identifies connected components in $O(\log D)$ deterministic rounds. The total memory required is $O(n+m)$ words, where $m$ is the number of edges in the input graph, which is optimal as it is only enough to store the input graph. We complement our upper bound results by showing that $\Omega(\log D)$ time is necessary even for component-unstable algorithms, conditioned on the widely believed 1 vs. 2 cycles conjecture.

In this work, we initiate the study of LCL problems in the low-space Massively Parallel Computation (MPC) model. In particular, on forests, we provide a method that, given the complexity of an LCL problem in the LOCAL model, automatically provides an exponentially faster algorithm for the low-space MPC setting that uses optimal global memory, that is, truly linear.

We establish scalable Massively Parallel Computation (MPC) algorithms for a family of fundamental graph problems on trees. We give a general method that, for a wide range of LCL problems, turns their message passing counterparts into exponentially faster algorithms in the sublinear MPC model.