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 $O(n^{1-2/\omega})$ round matrix multiplication algorithm, where $\omega < 2.3728639$ 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 $O(n^{1/3})$ triangle detection 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 the $\tilde{O} (n^{1/2})$-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.Comment: This is work is a merger of arxiv:1412.2109 and arxiv:1412.266

Counting thin subgraphs via packings faster than meet-in-the-middle time

Vassilevska and Williams (STOC'09) showed how to count simple paths on k vertices and matchings on k/2 edges in ann-vertex graph in time nk/2+O(1). In the same year, two different algorithms with the same runtime were given by Koutis and Williams (ICALP'09), and Bjorklund et al. (ESA'09), via nst/2+O(1)-time algorithms for counting t-tuples of pairwise disjoint sets drawn from a given family of s-sized subsets of an n-element universe. Shortly afterwards, Alon and Gutner (TALG'10) showed that these problems have O(n st/2) and O(nk/2) lower bounds when counting by color coding. Here, we show that one can do better-we show that the "meet-in-the-middle" exponent st/2 can be beaten and give an algorithm that counts in time n0.45470382st+O(1) for t a multiple of three. This implies algorithms for counting occurrences of a fixed subgraph on k vertices and pathwidth pk in an n-vertex graph in n0.45470382k+2p+O(1) time, improving on the three mentioned algorithms for paths and matchings, and circumventing the color-coding lower bound. We also give improved bounds for counting t-tuples of disjoint s-sets for s = 2, 3, 4. Our algorithms use fast matrix multiplication. We show an argument that this is necessary to go below the meet-in-the-middle barrier

Counting Thin Subgraphs via Packings Faster Than Meet-in-the-Middle Time

Vassilevska and Williams (STOC 2009) showed how to count simple paths on k vertices and matchings on k/2 edges in an n-vertex graph in time n^{k/2+O(1)}. In the same year, two different algorithms with the same runtime were given by Koutis and Williams (ICALP 2009), and Björklund et al. (ESA 2009), via nst/2+O(1)-time algorithms for counting t-tuples of pairwise disjoint sets drawn from a given family of s-sized subsets of an n-element universe. Shortly afterwards, Alon and Gutner (TALG 2010) showed that these problems have Ω(n^{⌊st/2⌋}) and Ω(n^{⌊k/2⌋}) lower bounds when counting by color coding. Here we show that one can do better, namely, we show that the “meet-in-the-middle” exponent st/2 can be beaten and give an algorithm that counts in time n^{0.4547st+O(1)} for t a multiple of three. This implies algorithms for counting occurrences of a fixed subgraph on k vertices and pathwidth p ≪ k in an n-vertex graph in n^{0.4547k+2p+O(1)} time, improving on the three mentioned algorithms for paths and matchings, and circumventing the color-coding lower bound