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**1 - 2**of**2**### Ami Paz Technion

"... 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 multipli-cation implementations to the congested clique, obtaining an O(n1−2/ω) round matrix multiplication algorithm, where ω & ..."

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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 multipli-cation implementations to the congested clique, obtaining an O(n1−2/ω) round matrix multiplication algorithm, where ω < 2.3728639 is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorith-mics, 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(n0.158) rounds, im-proving upon the O(n1/3) triangle counting algorithm of Dolev et al. [DISC 2012], – a (1 + o(1))-approximation of all-pairs shortest paths in O(n0.158) rounds, improving upon the Õ(n1/2)-round (2+o(1))-approximation algorithm of Nanongkai [STOC 2014], and – computing the girth in O(n0.158) 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.

### Fast Partial Distance Estimation and Applications

"... We study approximate distributed solutions to the weighted all-pairs-shortest-paths (APSP) problem in the congest model. We obtain the following results. • A deterministic (1 + ε)-approximation to APSP with running time O(ε−2n logn) rounds. The best previously known algorithm was randomized and slow ..."

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We study approximate distributed solutions to the weighted all-pairs-shortest-paths (APSP) problem in the congest model. We obtain the following results. • A deterministic (1 + ε)-approximation to APSP with running time O(ε−2n logn) rounds. The best previously known algorithm was randomized and slower by a Θ(logn) factor. In many cases, routing schemes involve relabeling, i.e., assigning new names to nodes and that are used in distance and routing queries. It is known that relabeling is necessary to achieve running times of o(n / logn). In the relabeling model, we obtain the following results. • A randomized O(k)-approximation to APSP, for any in-teger k> 1, running in Õ(n1/2+1/k +D) rounds, where D is the hop diameter of the network. This algorithm simplifies the best previously known result and reduces its approxima-tion ratio from O(k log k) to O(k). Also, the new algorithm uses O(logn)-bit labels, which is asymptotically optimal. • A randomized O(k)-approximation to APSP, for any integer k> 1, running in time Õ((nD)1/2 · n1/k + D) and producing compact routing tables of size Õ(n1/k). The node labels consist of O(k logn) bits. This improves on the ap-proximation ratio of Θ(k2) for tables of that size achieved by the best previously known algorithm, which terminates faster, in Õ(n1/2+1/k +D) rounds. In addition, we improve on the time complexity of the best known deterministic algorithm for distributed approximate Steiner forest.