Algebraic methods in the congested clique

In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an (Formula presented.) round matrix multiplication algorithm, where (Formula presented.) is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorithmics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include:1.triangle and 4-cycle counting in (Formula presented.) rounds, improving upon the (Formula presented.) algorithm of Dolev et al. [DISC 2012],2.a (Formula presented.)-approximation of all-pairs shortest paths in (Formula presented.) rounds, improving upon the (Formula presented.)-round (Formula presented.)-approximation algorithm given by Nanongkai [STOC 2014], and3.computing the girth in (Formula presented.) rounds, which is the first non-trivial solution in this model.In addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles.