207 research outputs found
Connectivity Lower Bounds in Broadcast Congested Clique
We prove three new lower bounds for graph connectivity in the 1-bit broadcast congested clique model, BCC(1). First, in the KT-0 version of BCC(1), in which nodes are aware of neighbors only through port numbers, we show an ?(log n) round lower bound for Connectivity even for constant-error randomized Monte Carlo algorithms. The deterministic version of this result can be obtained via the well-known "edge-crossing" argument, but, the randomized version of this result requires establishing new combinatorial results regarding the indistinguishability graph induced by inputs. In our second result, we show that the ?(log n) lower bound result extends to the KT-1 version of the BCC(1) model, in which nodes are aware of IDs of all neighbors, though our proof works only for deterministic algorithms. This result substantially improves upon the existing ?(log^* n) deterministic lower bound (Jurdzi?ski et el., SIROCCO 2018) for this problem. Since nodes know IDs of their neighbors in the KT-1 model, it is no longer possible to play "edge-crossing" tricks; instead we present a reduction from the 2-party communication complexity problem Partition in which Alice and Bob are given two set partitions on [n] and are required to determine if the join of these two set partitions equals the trivial one-part set partition. While our KT-1 Connectivity lower bound holds only for deterministic algorithms, in our third result we extend this ?(log n) KT-1 lower bound to constant-error Monte Carlo algorithms for the closely related ConnectedComponents problem. We use information-theoretic techniques to obtain this result. All our results hold for the seemingly easy special case of Connectivity in which an algorithm has to distinguish an instance with one cycle from an instance with multiple cycles. Our results showcase three rather different lower bound techniques and lay the groundwork for further improvements in lower bounds for Connectivity in the BCC(1) model
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
Survey of Distributed Decision
We survey the recent distributed computing literature on checking whether a
given distributed system configuration satisfies a given boolean predicate,
i.e., whether the configuration is legal or illegal w.r.t. that predicate. We
consider classical distributed computing environments, including mostly
synchronous fault-free network computing (LOCAL and CONGEST models), but also
asynchronous crash-prone shared-memory computing (WAIT-FREE model), and mobile
computing (FSYNC model)
Towards a complexity theory for the congested clique
The congested clique model of distributed computing has been receiving
attention as a model for densely connected distributed systems. While there has
been significant progress on the side of upper bounds, we have very little in
terms of lower bounds for the congested clique; indeed, it is now know that
proving explicit congested clique lower bounds is as difficult as proving
circuit lower bounds.
In this work, we use various more traditional complexity-theoretic tools to
build a clearer picture of the complexity landscape of the congested clique:
-- Nondeterminism and beyond: We introduce the nondeterministic congested
clique model (analogous to NP) and show that there is a natural canonical
problem family that captures all problems solvable in constant time with
nondeterministic algorithms. We further generalise these notions by introducing
the constant-round decision hierarchy (analogous to the polynomial hierarchy).
-- Non-constructive lower bounds: We lift the prior non-uniform counting
arguments to a general technique for proving non-constructive uniform lower
bounds for the congested clique. In particular, we prove a time hierarchy
theorem for the congested clique, showing that there are decision problems of
essentially all complexities, both in the deterministic and nondeterministic
settings.
-- Fine-grained complexity: We map out relationships between various natural
problems in the congested clique model, arguing that a reduction-based
complexity theory currently gives us a fairly good picture of the complexity
landscape of the congested clique
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