481 research outputs found
Sublinear-Time Distributed Algorithms for Detecting Small Cliques and Even Cycles
In this paper we give sublinear-time distributed algorithms in the CONGEST model for subgraph detection for two classes of graphs: cliques and even-length cycles. We show for the first time that all copies of 4-cliques and 5-cliques in the network graph can be listed in sublinear time, O(n^{5/6+o(1)}) rounds and O(n^{21/22+o(1)}) rounds, respectively. Prior to our work, it was not known whether it was possible to even check if the network contains a 4-clique or a 5-clique in sublinear time.
For even-length cycles, C_{2k}, we give an improved sublinear-time algorithm, which exploits a new connection to extremal combinatorics. For example, for 6-cycles we improve the running time from O~(n^{5/6}) to O~(n^{3/4}) rounds. We also show two obstacles on proving lower bounds for C_{2k}-freeness: First, we use the new connection to extremal combinatorics to show that the current lower bound of Omega~(sqrt{n}) rounds for 6-cycle freeness cannot be improved using partition-based reductions from 2-party communication complexity, the technique by which all known lower bounds on subgraph detection have been proven to date. Second, we show that there is some fixed constant delta in (0,1/2) such that for any k, a Omega(n^{1/2+delta}) lower bound on C_{2k}-freeness implies new lower bounds in circuit complexity.
For general subgraphs, it was shown in [Orr Fischer et al., 2018] that for any fixed k, there exists a subgraph H of size k such that H-freeness requires Omega~(n^{2-Theta(1/k)}) rounds. It was left as an open problem whether this is tight, or whether some constant-sized subgraph requires truly quadratic time to detect. We show that in fact, for any subgraph H of constant size k, the H-freeness problem can be solved in O(n^{2 - Theta(1/k)}) rounds, nearly matching the lower bound of [Orr Fischer et al., 2018]
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 round matrix multiplication algorithm, where
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 rounds, improving upon the
triangle detection algorithm of Dolev et al. [DISC 2012],
-- a -approximation of all-pairs shortest paths in
rounds, improving upon the -round -approximation algorithm of Nanongkai [STOC 2014], and
-- computing the girth in 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
Simpler Reductions from Exact Triangle
In this paper, we provide simpler reductions from Exact Triangle to two
important problems in fine-grained complexity: Exact Triangle with Few
Zero-Weight -Cycles and All-Edges Sparse Triangle.
Exact Triangle instances with few zero-weight -cycles was considered by
Jin and Xu [STOC 2023], who used it as an intermediate problem to show SUM
hardness of All-Edges Sparse Triangle with few -cycles (independently
obtained by Abboud, Bringmann and Fischer [STOC 2023]), which is further used
to show SUM hardness of a variety of problems, including -Cycle
Enumeration, Offline Approximate Distance Oracle, Dynamic Approximate Shortest
Paths and All-Nodes Shortest Cycles. We provide a simple reduction from Exact
Triangle to Exact Triangle with few zero-weight -cycles. Our new reduction
not only simplifies Jin and Xu's previous reduction, but also strengthens the
conditional lower bounds from being under the SUM hypothesis to the even
more believable Exact Triangle hypothesis. As a result, all conditional lower
bounds shown by Jin and Xu [STOC 2023] and by Abboud, Bringmann and Fischer
[STOC 2023] using All-Edges Sparse Triangle with few -cycles as an
intermediate problem now also hold under the Exact Triangle hypothesis.
We also provide two alternative proofs of the conditional lower bound of the
All-Edges Sparse Triangle problem under the Exact Triangle hypothesis, which
was originally proved by Vassilevska Williams and Xu [FOCS 2020]. Both of our
new reductions are simpler, and one of them is also deterministic -- all
previous reductions from Exact Triangle or 3SUM to All-Edges Sparse Triangle
(including P\u{a}tra\c{s}cu's seminal work [STOC 2010]) were randomized.Comment: To appear in SOSA 202
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