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Register a new userDeterministic subgraph detection in broadcast CONGEST
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Added by
Janne H. Korhonen
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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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 $d$-degenerate graphs can be improved to optimal complexity $O(d/log n)$ and $O(d^2/log n)$, respectively, in the supported CONGEST model, which can be seen as an intermediate model between CONGEST and the congested clique.
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We study the maximum cardinality matching problem in a standard distributed setting, where the nodes $V$ of a given $n$-node network graph $G=(V,E)$ communicate over the edges $E$ in synchronous rounds. More specifically, we consider the distributed CONGEST model, where in each round, each node of $G$ can send an $O(log n)$-bit message to each of its neighbors. We show that for every graph $G$ and a matching $M$ of $G$, there is a randomized CONGEST algorithm to verify $M$ being a maximum matching of $G$ in time $O(|M|)$ and disprove it in time $O(D + ell)$, where $D$ is the diameter of $G$ and $ell$ is the length of a shortest augmenting path. We hope that our algorithm constitutes a significant step towards developing a CONGEST algorithm to compute a maximum matching in time $tilde{O}(s^*)$, where $s^*$ is the size of a maximum matching.
We give efficient randomized and deterministic distributed algorithms for computing a distance-$2$ vertex coloring of a graph $G$ in the CONGEST model. In particular, if $Delta$ is the maximum degree of $G$, we show that there is a randomized CONGEST model algorithm to compute a distance-$2$ coloring of $G$ with $Delta^2+1$ colors in $O(logDeltacdotlog n)$ rounds. Further if the number of colors is slightly increased to $(1+epsilon)Delta^2$ for some $epsilon>1/{rm polylog}(n)$, we show that it is even possible to compute a distance-$2$ coloring deterministically in polylog$(n)$ time in the CONGEST model. Finally, we give a $O(Delta^2 + log^* n)$-round deterministic CONGEST algorithm to compute distance-$2$ coloring with $Delta^2+1$ colors.
Energy is often the most constrained resource in networks of battery-powered devices, and as devices become smaller, they spend a larger fraction of their energy on communication (transceiver usage) not computation. As an imperfect proxy for true energy usage, we define energy complexity to be the number of time slots a device transmits/listens; idle time and computation are free. In this paper we investigate the energy complexity of fundamental communication primitives such as broadcast in multi-hop radio networks. We consider models with collision detection (CD) and without (No-CD), as well as both randomized and deterministic algorithms. Some take-away messages from this work include: 1. The energy complexity of broadcast in a multi-hop network is intimately connected to the time complexity of leader election in a single-hop (clique) network. Many existing lower bounds on time complexity immediately transfer to energy complexity. For example, in the CD and No-CD models, we need $Omega(log n)$ and $Omega(log^2 n)$ energy, respectively. 2. The energy lower bounds above can almost be achieved, given sufficient ($Omega(n)$) time. In the CD and No-CD models we can solve broadcast using $O(frac{log nloglog n}{logloglog n})$ energy and $O(log^3 n)$ energy, respectively. 3. The complexity measures of Energy and Time are in conflict, and it is an open problem whether both can be minimized simultaneously. We give a tradeoff showing it is possible to be nearly optimal in both measures simultaneously. For any constant $epsilon>0$, broadcast can be solved in $O(D^{1+epsilon}log^{O(1/epsilon)} n)$ time with $O(log^{O(1/epsilon)} n)$ energy, where $D$ is the diameter of the network.
Broadcast is one of the fundamental network communication primitives. One node of a network, called the $mathit{source}$, has a message that has to be learned by all other nodes. We consider the feasibility of deterministic broadcast in radio networks. If nodes of the network do not have any labels, deterministic broadcast is impossible even in the four-cycle. On the other hand, if all nodes have distinct labels, then broadcast can be carried out, e.g., in a round-robin fashion, and hence $O(log n)$-bit labels are sufficient for this task in $n$-node networks. In fact, $O(log Delta)$-bit labels, where $Delta$ is the maximum degree, are enough to broadcast successfully. Hence, it is natural to ask if very short labels are sufficient for broadcast. Our main result is a positive answer to this question. We show that every radio network can be labeled using 2 bits in such a way that broadcast can be accomplished by some universal deterministic algorithm that does not know the network topology nor any bound on its size. Moreover, at the expense of an extra bit in the labels, we get the additional strong property that there exists a common round in which all nodes know that broadcast has been completed. Finally, we show that 3-bit labels are also sufficient to solve bo
In this paper, we consider the Byzantine reliable broadcast problem on authenticated and partially connected networks. The state-of-the-art method to solve this problem consists in combining two algorithms from the literature. Handling asynchrony and faulty senders is typically done thanks to Gabriel Brachas authenticated double-echo broadcast protocol, which assumes an asynchronous fully connected network. Danny Dolevs algorithm can then be used to provide reliable communications between processes in the global fault model, where up to f processes among N can be faulty in a communication network that is at least 2f+1-connected. Following recent works that showed that Dolevs protocol can be made more practical thanks to several optimizations, we show that the state-of-the-art methods to solve our problem can be optimized thanks to layer-specific and cross-layer optimizations. Our simulations with the Omnet++ network simulator show that these optimizations can be efficiently combined to decrease the total amount of information transmitted or the protocols latency (e.g., respectively, -25% and -50% with a 16B payload, N=31 and f=4) compared to the state-of-the-art combination of Brachas and Dolevs protocols.
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