While algebrisation constitutes a powerful technique in the design and analysis of centralised algorithms, to date there have been hardly any applications of algebraic techniques in the context of distributed graph algorithms. This work is a case stu
dy that demonstrates the potential of algebrisation in the distributed context. We will focus on distributed graph algorithms in the congested clique model; the graph problems that we will consider include, e.g., the triangle detection problem and the all-pairs shortest path problem (APSP). There is plenty of prior work on combinatorial algorithms in the congested clique model: for example, Dolev et al. (DISC 2012) gave an algorithm for triangle detection with a running time of $tilde O(n^{1/3})$, and Nanongkai (STOC 2014) gave an approximation algorithm for APSP with a running time of $tilde O(n^{1/2})$. In this work, we will use algebraic techniques -- in particular, algorithms based on fast matrix multiplication -- to solve both triangle detection and the unweighted APSP in time $O(n^{0.15715})$; for weighted APSP, we give a $(1+o(1))$-approximation with this running time, as well as an exact $tilde O(n^{1/3})$ solution.
We show that there is no deterministic local algorithm (constant-time distributed graph algorithm) that finds a $(7-epsilon)$-approximation of a minimum dominating set on planar graphs, for any positive constant $epsilon$. In prior work, the best low
er bound on the approximation ratio has been $5-epsilon$; there is also an upper bound of $52$.
We study the problem of clock synchronization in highly dynamic networks, where communication links can appear or disappear at any time. The nodes in the network are equipped with hardware clocks, but the rate of the hardware clocks can vary arbitrar
ily within specific bounds, and the estimates that nodes can obtain about the clock values of other nodes are inherently inaccurate. Our goal in this setting is to output a logical clock at each node such that the logical clocks of any two nodes are not too far apart, and nodes that remain close to each other in the network for a long time are better synchronized than distant nodes. This property is called gradient clock synchronization. Gradient clock synchronization has been widely studied in the static setting, where the network topology does not change. We show that the asymptotically optimal bounds obtained for the static case also apply to our highly dynamic setting: if two nodes remain at distance $d$ from each other for sufficiently long, it is possible to upper bound the difference between their clock values by $O(d log (D / d))$, where $D$ is the diameter of the network. This is known to be optimal even for static networks. Furthermore, we show that our algorithm has optimal stabilization time: when a path of length $d$ appears between two nodes, the time required until the clock skew between the two nodes is reduced to $O(d log (D / d))$ is $O(D)$, which we prove to be optimal. Finally, the techniques employed for the more intricate analysis of the algorithm for dynamic graphs provide additional insights that are also of interest for the static setting. In particular, we establish self-stabilization of the gradient property within $O(D)$ time.
