  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 study 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.
