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 O(n 1−2/ω ) round matrix multiplication algorithm, where ω < 2.3728639 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: -triangle and 4-cycle counting in O(n 0.158 ) rounds, improving upon the O(n 1/3 ) algorithm of Dolev et al. [DISC 2012], -a (1 + o(1))-approximation of all-pairs shortest paths in O(n 0.158 ) rounds, improving upon theÕ(n 1/2 )-round (2 + o(1))-approximation algorithm of Nanongkai [STOC 2014], and computing the girth in O(n 0.158 ) 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.

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We present a simple deterministic single-pass (2 + )-approximation algorithm for the maximum weight matching problem in the semi-streaming model. This improves upon the currently best known approximation ratio of (3.5 + ).Our algorithm uses O(n log 2 n) space for constant values of . It relies on a variation of the local-ratio theorem, which may be of independent interest in the semi-streaming model.

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