We present deterministic constant-round protocols for the graph connectivity problem in the model where each of the $n$ nodes of a graph receives a row of the adjacency matrix, and broadcasts a single sublinear size message to all other nodes. Communication rounds are synchronous. This model is sometimes called the broadcast congested clique. Specifically, we exhibit a deterministic protocol that computes the connected components of the input graph in $\lceil 1/ε\rceil$ rounds, each player communicating $\mathcal{O}(n^ε \cdot \log n)$ bits per round, with $0 < ε\leq 1$.
We also provide a deterministic one-round protocol for connectivity, in the model when each node receives as input the graph induced by the nodes at distance at most $r>0$, and communicates $\mathcal{O}(n^{1/r} \cdot \log n)$ bits. This result is based on a $d$-pruning protocol, which consists in successively removing nodes of degree at most $d$ until obtaining a graph with minimum degree larger than $d$. Our technical novelty is the introduction of deterministic sparse linear sketches: a linear compression function that permits to recover sparse Boolean vectors deterministically.
We give a simple deterministic constant-round algorithm in the congested clique model for reducing the number of edges in a graph to $n^{1+\varepsilon}$ while preserving the minimum spanning forest, where $\varepsilon > 0$ is any constant. This implies that in the congested clique model, it is sufficient to improve MST and other connectivity algorithms on graphs with slightly superlinear number of edges to obtain a general improvement. As a byproduct, we also obtain a simple alternative proof showing that MST can be computed deterministically in $O(\log \log n)$ rounds.
The main results of this paper are (I) a simulation algorithm which, under quite general constraints, transforms algorithms running on the Congested Clique into algorithms running in the MapReduce model, and (II) a distributed $O(Δ)$-coloring algorithm running on the Congested Clique which has an expected running time of (i) $O(1)$ rounds, if $Δ\geq Θ(\log^4 n)$; and (ii) $O(\log \log n)$ rounds otherwise. Applying the simulation theorem to the Congested-Clique $O(Δ)$-coloring algorithm yields an $O(1)$-round $O(Δ)$-coloring algorithm in the MapReduce model. Our simulation algorithm illustrate…
The congested clique is a synchronous, message-passing model of distributed computing in which each computational unit (node) in each round can send message of O(log n) bits to each other node of the network, where n is the number of nodes. This model has been considered under two extreme scanarios: unicast or broadcast. In the unicast model, a node can send (possibly) different message to each other node of the network. In contrast, in the broadcast model each node sends a single (the same) message to all other nodes. We study the congested clique model parametrized by the range r, the maximu…
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.…
Censor-Hillel et al. [PODC'15] recently showed how to efficiently implement centralized algebraic algorithms for matrix multiplication in the congested clique model, a model of distributed computing that has received increasing attention in the past few years. This paper develops further algebraic techniques for designing algorithms in this model. We present deterministic and randomized algorithms, in the congested clique model, for efficiently computing multiple independent instances of matrix products, computing the determinant, the rank and the inverse of a matrix, and solving systems of li…
We give a maximal independent set (MIS) algorithm that runs in $O(\log \log Δ)$ rounds in the congested clique model, where $Δ$ is the maximum degree of the input graph. This improves upon the $O(\frac{\log(Δ) \cdot \log \log Δ}{\sqrt{\log n}} + \log \log Δ)$ rounds algorithm of [Ghaffari, PODC '17], where $n$ is the number of vertices of the input graph. In the first stage of our algorithm, we simulate the first $O(\frac{n}{\text{poly} \log n})$ iterations of the sequential random order Greedy algorithm for MIS in the congested clique model in $O(\log \log Δ)$ rounds. This thins out the input…
In this paper, we show that the Minimum Spanning Tree problem can be solved \emph{deterministically}, in $\mathcal{O}(1)$ rounds of the $\mathsf{Congested}$ $\mathsf{Clique}$ model. In the $\mathsf{Congested}$ $\mathsf{Clique}$ model, there are $n$ players that perform computation in synchronous rounds. Each round consist of a phase of local computation and a phase of communication, in which each pair of players is allowed to exchange $\mathcal{O}(\log n)$ bit messages. The studies of this model began with the MST problem: in the paper by Lotker et al.[SPAA'03, SICOMP'05] that defines the $\ma…
We give the first Congested Clique algorithm that computes a sparse hopset with polylogarithmic hopbound in polylogarithmic time. Given a graph $G=(V,E)$, a $(β,ε)$-hopset $H$ with "hopbound" $β$, is a set of edges added to $G$ such that for any pair of nodes $u$ and $v$ in $G$ there is a path with at most $β$ hops in $G \cup H$ with length within $(1+ε)$ of the shortest path between $u$ and $v$ in $G$. Our hopsets are significantly sparser than the recent construction of Censor-Hillel et al. [6], that constructs a hopset of size $\tilde{O}(n^{3/2})$, but with a smaller polylogarithmic hopboun…
The Congested Clique model proposed by Lotker et al.[SICOMP'05] was introduced in order to provide a simple abstraction for overlay networks. Congested Clique is a model of distributed (or parallel) computing, in which there are $n$ players with unique identifiers from set [n], which perform computations in synchronous rounds. Each round consists of the phase of unlimited local computation and the communication phase. While communicating, each pair of players is allowed to exchange a single message of size $O(\log n)$ bits. Since, in a single round, each player can communicate with even $Θ(n)$…
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…
While algebrisation constitutes a powerful technique in the design and analysis of centralised algorithms, to date there have been hardly any applications of algebraic techniques in the context of distributed graph algorithms. This work is a case study that demonstrates the potential of algebrisation in the distributed context. We will focus on distributed graph algorithms in the congested clique model; the graph problems that we will consider include, e.g., the triangle detection problem and the all-pairs shortest path problem (APSP). There is plenty of prior work on combinatorial algorithms …
We design fast deterministic algorithms for distance computation in the congested clique model. Our key contributions include: -- A $(2+ε)$-approximation for all-pairs shortest paths in $O(\log^2{n} / ε)$ rounds on unweighted undirected graphs. With a small additional additive factor, this also applies for weighted graphs. This is the first sub-polynomial constant-factor approximation for APSP in this model. -- A $(1+ε)$-approximation for multi-source shortest paths from $O(\sqrt{n})$ sources in $O(\log^2{n} / ε)$ rounds on weighted undirected graphs. This is the first sub-polynomial algorithm…
We present a distributed randomized algorithm finding Minimum Spanning Tree (MST) of a given graph in O(1) rounds, with high probability, in the Congested Clique model. The input graph in the Congested Clique model is a graph of n nodes, where each node initially knows only its incident edges. The communication graph is a clique with limited edge bandwidth: each two nodes (not necessarily neighbours in the input graph) can exchange $O(\log n)$ bits. As in previous works, the key part of the MST algorithm is an efficient Connected Components (CC) algorithm. However, unlike the former approaches…
The {Congested Clique} is a distributed-computing model for single-hop networks with restricted bandwidth that has been very intensively studied recently. It models a network by an $n$-vertex graph in which any pair of vertices can communicate one with another by transmitting $O(\log n )$ bits in each round. Various problems have been studied in this setting, but for some of them the best-known results are those for general networks. In this paper we devise significantly improved algorithms for various symmetry-breaking problems, such as forests-decompositions, vertex-colorings, and maximal in…
This paper presents constant-time and near-constant-time distributed algorithms for a variety of problems in the congested clique model. We show how to compute a 3-ruling set in expected $O(\log \log \log n)$ rounds and using this, we obtain a constant-approximation to metric facility location, also in expected $O(\log \log \log n)$ rounds. In addition, assuming an input metric space of constant doubling dimension, we obtain constant-round algorithms to compute constant-factor approximations to the minimum spanning tree and the metric facility location problems. These results significantly imp…
We present improved deterministic algorithms for approximating shortest paths in the Congested Clique model of distributed computing. We obtain $poly(\log\log n)$-round algorithms for the following problems in unweighted undirected $n$-vertex graphs: -- $(1+ε)$-approximation of multi-source shortest paths (MSSP) from $O(\sqrt{n})$ sources. -- $(2+ε)$-approximation of all pairs shortest paths (APSP). -- $(1+ε,β)$-approximation of APSP where $β=O(\frac{\log\log n}ε)^{\log\log n}$. These bounds improve exponentially over the state-of-the-art poly-logarithmic bounds due to [Censor-Hillel et al., P…
Consider a clique of n nodes, where in each synchronous round each pair of nodes can exchange O(log n) bits. We provide deterministic constant-time solutions for two problems in this model. The first is a routing problem where each node is source and destination of n messages of size O(log n). The second is a sorting problem where each node i is given n keys of size O(log n) and needs to receive the ith batch of n keys according to the global order of the keys. The latter result also implies deterministic constant-round solutions for related problems such as selection or determining modes.
We settle the complexity of the $(Δ+1)$-coloring and $(Δ+1)$-list coloring problems in the CONGESTED CLIQUE model by presenting a simple deterministic algorithm for both problems running in a constant number of rounds. This matches the complexity of the recent breakthrough randomized constant-round $(Δ+1)$-list coloring algorithm due to Chang et al. (PODC'19), and significantly improves upon the state-of-the-art $O(\log Δ)$-round deterministic $(Δ+1)$-coloring bound of Parter (ICALP'18). A remarkable property of our algorithm is its simplicity. Whereas the state-of-the-art randomized algorithm…
In this paper, we study the graph realization problem in the Congested Clique model of distributed computing under crash faults. We consider {\em degree-sequence realization}, in which each node $v$ is associated with a degree value $d(v)$, and the resulting degree sequence is realizable if it is possible to construct an overlay network with the given degrees. Our main result is a $O(f)$-round deterministic algorithm for the degree-sequence realization problem in a $n$-node Congested Clique, of which $f$ nodes could be faulty ($f<n$). The algorithm uses $O(n^2)$ messages. We complement the res…
This paper gives simple distributed algorithms for the fundamental problem of computing graph distances in the Congested Clique model. One of the main components of our algorithms is fast matrix multiplication, for which we show an $O(n^{1/3})$-round algorithm when the multiplication needs to be performed over a semi-ring, and an $O(n^{0.157})$-round algorithm when the computation can be performed over a field. We propose to denote by $κ$ the exponent of matrix multiplication in this model, which gives $κ< 0.157$. We show how to compute all-pairs-shortest-paths (APSP) in $O(n^{1/3}\log{n})$ ro…
We develop techniques to prove lower bounds for the BCAST(log n) Broadcast Congested Clique model (a distributed message passing model where in each round, each processor can broadcast an O(log n)-sized message to all other processors). Our techniques are built to prove bounds for natural input distributions. So far, all lower bounds for problems in the model relied on constructing specifically tailored graph families for the specific problem at hand, resulting in lower bounds for artificially constructed inputs, instead of natural input distributions. One of our results is a lower bound for t…
Graph problems are troublesome when it comes to MapReduce. Typically, to be able to design algorithms that make use of the advantages of MapReduce, assumptions beyond what the model imposes, such as the density of the input graph, are required. In a recent shift, a simple and robust model of MapReduce for graph problems, where the space per machine is set to be O(|V|), has attracted considerable attention. We term this model semi-MapReduce, or in short, semiMPC, and focus on its computational power. We show through a set of simulation methods that semiMPC is, perhaps surprisingly, equivalent t…
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 $Ω(\log n)$ 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 se…
We multiply two $n \times n$ matrices $S,T$ over semirings in the Congested Clique model, where $n$ fully connected nodes communicate synchronously using $O(\log n)$-bit messages, within $O(nz(S)^{1/3} nz(T)^{1/3}/n + 1)$ rounds of communication, where $nz(A)$ denotes the number of non-zero elements in a matrix $A$. By leveraging the sparsity of the input matrices, our algorithm greatly reduces communication compared with general algorithms [Censor-Hillel et al., PODC 2015], improving upon the state-of-the-art for matrices with $o(n^2)$ non-zero elements. Our algorithm exhibits the additional …
In a sequence of recent results (PODC 2015 and PODC 2016), the running time of the fastest algorithm for the \emph{minimum spanning tree (MST)} problem in the \emph{Congested Clique} model was first improved to $O(\log \log \log n)$ from $O(\log \log n)$ (Hegeman et al., PODC 2015) and then to $O(\log^* n)$ (Ghaffari and Parter, PODC 2016). All of these algorithms use $Θ(n^2)$ messages independent of the number of edges in the input graph. This paper positively answers a question raised in Hegeman et al., and presents the first "super-fast" MST algorithm with $o(m)$ message complexity for inpu…
The \emph{Steiner tree} problem is one of the fundamental and classical problems in combinatorial optimization. In this paper, we study this problem in the $\mathcal{CONGESTED}$ $\mathcal{CLIQUE}$ model of distributed computing and present two deterministic distributed approximation algorithms for the same. The first algorithm computes a Steiner tree in $\tilde{O}(n^{1/3})$ rounds and $\tilde{O}(n^{7/3})$ messages for a given connected undirected weighted graph of $n$ nodes. Note here that $\tilde{O}(\cdot)$ notation hides polylogarithmic factors in $n$. The second one computes a Steiner tree …
We study the broadcast version of the CONGEST CLIQUE model of distributed computing. In this model, in each round, any node in a network of size $n$ can send the same message (i.e. broadcast a message) of limited size to every other node in the network. Nanongkai presented in [STOC'14] a randomized $(2+o(1))$-approximation algorithm to compute all pairs shortest paths (APSP) in time $\tilde{O}(\sqrt{n})$ on weighted graphs, where we use the convention that $\tildeΩ(f(n))$ is essentially $Ω(f(n)/$polylog$f(n))$ and $\tilde{O}(f(n))$ is essentially $O(f(n) $polylog$f(n))$. We complement this res…
This paper considers the \textit{minimum spanning tree (MST)} problem in the Congested Clique model and presents an algorithm that runs in $O(\log \log \log n)$ rounds, with high probability. Prior to this, the fastest MST algorithm in this model was a deterministic algorithm due to Lotker et al.~(SIAM J on Comp, 2005) from about a decade ago. A key step along the way to designing this MST algorithm is a \textit{connectivity verification} algorithm that not only runs in $O(\log \log \log n)$ rounds with high probability, but also has low message complexity. This allows the fast computation of …
This paper provides three nearly-optimal algorithms for scheduling $t$ jobs in the $\mathsf{CLIQUE}$ model. First, we present a deterministic scheduling algorithm that runs in $O(\mathsf{GlobalCongestion} + \mathsf{dilation})$ rounds for jobs that are sufficiently efficient in terms of their memory. The $\mathsf{dilation}$ is the maximum round complexity of any of the given jobs, and the $\mathsf{GlobalCongestion}$ is the total number of messages in all jobs divided by the per-round bandwidth of $n^2$ of the $\mathsf{CLIQUE}$ model. Both are inherent lower bounds for any scheduling algorithm. …
The congested clique model is a message-passing model of distributed computation where the underlying communication network is the complete graph of $n$ nodes. In this paper we consider the situation where the joint input to the nodes is an $n$-node labeled graph $G$, i.e., the local input received by each node is the indicator function of its neighborhood in $G$. Nodes execute an algorithm, communicating with each other in synchronous rounds and their goal is to compute some function that depends on $G$. In every round, each of the $n$ nodes may send up to $n-1$ different $b$-bit messages thr…
We present simple deterministic algorithms for subgraph finding and enumeration in the broadcast CONGEST model of distributed computation: -- For any constant $k$, detecting $k$-paths and trees on $k$ nodes can be done in $O(1)$ rounds. -- For any constant $k$, detecting $k$-cycles and pseudotrees on $k$ nodes can be done in $O(n)$ rounds. -- On $d$-degenerate graphs, cliques and $4$-cycles can be enumerated in $O(d + \log n)$ rounds, and $5$-cycles in $O(d^2 + \log n)$ rounds. In many cases, these bounds are tight up to logarithmic factors. Moreover, we show that the algorithms for $d$-degene…
We show that the $(degree+1)$-list coloring problem can be solved deterministically in $O(D \cdot \log n \cdot\log^2Δ)$ rounds in the \CONGEST model, where $D$ is the diameter of the graph, $n$ the number of nodes, and $Δ$ the maximum degree. Using the recent polylogarithmic-time deterministic network decomposition algorithm by Rozhoň and Ghaffari [STOC 2020], this implies the first efficient (i.e., $\poly\log n$-time) deterministic \CONGEST algorithm for the $(Δ+1)$-coloring and the $(\mathit{degree}+1)$-list coloring problem. Previously the best known algorithm required $2^{O(\sqrt{\log n})}…
In this paper we give fast distributed graph algorithms for detecting and listing small subgraphs, and for computing or approximating the girth. Our algorithms improve upon the state of the art by polynomial factors, and for girth, we obtain an constant-time algorithm for additive +1 approximation in the Congested Clique, and the first parametrized algorithm for exact computation in CONGEST. In the Congested Clique, we develop a technique for learning small neighborhoods, and apply it to obtain an $O(1)$-round algorithm that computes the girth with only an additive +1 error. Next, we introduce…
Over the past decade, there has been increasing interest in distributed/parallel algorithms for processing large-scale graphs. By now, we have quite fast algorithms -- usually sublogarithmic-time and often $poly(\log\log n)$-time, or even faster -- for a number of fundamental graph problems in the massively parallel computation (MPC) model. This model is a widely-adopted theoretical abstraction of MapReduce style settings, where a number of machines communicate in an all-to-all manner to process large-scale data. Contributing to this line of work on MPC graph algorithms, we present $poly(\log …
We study the effect of limiting the number of different messages a node can transmit simultaneously on the verification complexity of proof-labeling schemes (PLS). In a PLS, each node is given a label, and the goal is to verify, by exchanging messages over each link in each direction, that a certain global predicate is satisfied by the system configuration. We consider a single parameter r that bounds the number of distinct messages that can be sent concurrently by any node: in the case r=1, each node may only send the same message to all its neighbors (the broadcast model), in the case r is a…
In this paper we present a deterministic $O(\log\log n)$-round algorithm for the 2-ruling set problem in the Massively Parallel Computation model with $\tilde{O}(n)$ memory; this algorithm also runs in $O(\log\log n)$ rounds in the Congested Clique model. This is exponentially faster than the fastest known deterministic 2-ruling set algorithm for these models, which is simply the $O(\log Δ)$-round deterministic Maximal Independent Set algorithm due to Czumaj, Davies, and Parter (SPAA 2020). Our result is obtained by derandomizing the 2-ruling set algorithm of Kothapalli and Pemmaraju (FSTTCS 2…
In this work we address the question of efficiency of distributed computing in anonymous, congested and highly dynamic and not-always-connected networks/systems. More precisely, the system consists of an unknown number of anonymous nodes with congestion on links and local computation. Links can change arbitrarily from round to round, with only limitation that the union of any T consecutive networks must form a temporarily connected (multi-)graph on all nodes (knowledge of T is the only information the nodes require, otherwise the communication would not be feasible). Nodes do not have any IDs,…
In this paper we present a deterministic parallel algorithm solving the multiple selection problem in congested clique model. In this problem for given set of elements S and a set of ranks $K = \{k_1 , k_2 , ..., k_r \}$ we are asking for the $k_i$-th smallest element of $S$ for $1 \leq i \leq r$. The presented algorithm is deterministic, time optimal , and needs $O(\log^*_{r+1} (n))$ communication rounds, where $n$ is the size of the input set, and $r$ is the size of the rank set. This algorithm may be of theoretical interest, as for $r = 1$ (classic selection problem) it gives an improvement…
We give a randomized $Δ$-coloring algorithm in the LOCAL model that runs in $\text{poly} \log \log n$ rounds, where $n$ is the number of nodes of the input graph and $Δ$ is its maximum degree. This means that randomized $Δ$-coloring is a rare distributed coloring problem with an upper and lower bound in the same ballpark, $\text{poly}\log\log n$, given the known $Ω(\log_Δ\log n)$ lower bound [Brandt et al., STOC '16]. Our main technical contribution is a constant time reduction to a constant number of $(\text{deg}+1)$-list coloring instances, for $Δ= ω(\log^4 n)$, resulting in a $\text{poly} \…
This paper addresses the cornerstone family of \emph{local problems} in distributed computing, and investigates the curious gap between randomized and deterministic solutions under bandwidth restrictions. Our main contribution is in providing tools for derandomizing solutions to local problems, when the $n$ nodes can only send $O(\log n)$-bit messages in each round of communication. We combine bounded independence, which we show to be sufficient for some algorithms, with the method of conditional expectations and with additional machinery, to obtain the following results. Our techniques give a…
