arXiv subjects

Fabian Kuhn

Publications and source records attributed to Fabian Kuhn.

83 records · Page 5Linked to original sources

arXiv2015

Near-Optimal Distributed Maximum Flow

We present a near-optimal distributed algorithm for $(1+o(1))$-approximation of single-commodity maximum flow in undirected weighted networks that runs in $(D+ \sqrt{n})\cdot n^{o(1)}$ communication rounds in the \Congest model. Here, $n$ and $D$ denote the number of nodes and the network diameter, respectively. This is the first improvement over the trivial bound of $O(n^2)$, and it nearly matches the $\tildeΩ(D+ \sqrt{n})$ round complexity lower bound. The development of the algorithm contains two results of independent interest: (i) A $(D+\sqrt{n})\cdot n^{o(1)}$-round distributed construction of a spanning tree of average stretch $n^{o(1)}$. (ii) A $(D+\sqrt{n})\cdot n^{o(1)}$-round distributed construction of an $n^{o(1)}$-congestion approximator consisting of the cuts induced by $O(\log n)$ virtual trees. The distributed representation of the cut approximator allows for evaluation in $(D+\sqrt{n})\cdot n^{o(1)}$ rounds. All our algorithms make use of randomization and succeed with high probability.

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arXiv2015

Tight Bounds for MIS in Multichannel Radio Networks

Daum et al. [PODC'13] presented an algorithm that computes a maximal independent set (MIS) within $O(\log^2 n/F+\log n \mathrm{polyloglog} n)$ rounds in an $n$-node multichannel radio network with $F$ communication channels. The paper uses a multichannel variant of the standard graph-based radio network model without collision detection and it assumes that the network graph is a polynomially bounded independence graph (BIG), a natural combinatorial generalization of well-known geographic families. The upper bound of that paper is known to be optimal up to a polyloglog factor. In this paper, we adapt algorithm and analysis to improve the result in two ways. Mainly, we get rid of the polyloglog factor in the runtime and we thus obtain an asymptotically optimal multichannel radio network MIS algorithm. In addition, our new analysis allows to generalize the class of graphs from those with polynomially bounded local independence to graphs where the local independence is bounded by an arbitrary function of the neighborhood radius.

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arXiv2013

Distributed Connectivity Decomposition

We present time-efficient distributed algorithms for decomposing graphs with large edge or vertex connectivity into multiple spanning or dominating trees, respectively. As their primary applications, these decompositions allow us to achieve information flow with size close to the connectivity by parallelizing it along the trees. More specifically, our distributed decomposition algorithms are as follows: (I) A decomposition of each undirected graph with vertex-connectivity $k$ into (fractionally) vertex-disjoint weighted dominating trees with total weight $Ω(\frac{k}{\log n})$, in $\widetilde{O}(D+\sqrt{n})$ rounds. (II) A decomposition of each undirected graph with edge-connectivity $λ$ into (fractionally) edge-disjoint weighted spanning trees with total weight $\lceil\frac{λ-1}{2}\rceil(1-\varepsilon)$, in $\widetilde{O}(D+\sqrt{nλ})$ rounds. We also show round complexity lower bounds of $\tildeΩ(D+\sqrt{\frac{n}{k}})$ and $\tildeΩ(D+\sqrt{\frac{n}λ})$ for the above two decompositions, using techniques of [Das Sarma et al., STOC'11]. Moreover, our vertex-connectivity decomposition extends to centralized algorithms and improves the time complexity of [Censor-Hillel et al., SODA'14] from $O(n^3)$ to near-optimal $\tilde{O}(m)$. As corollaries, we also get distributed oblivious routing broadcast with $O(1)$-competitive edge-congestion and $O(\log n)$-competitive vertex-congestion. Furthermore, the vertex connectivity decomposition leads to near-time-optimal $O(\log n)$-approximation of vertex connectivity: centralized $\widetilde{O}(m)$ and distributed $\tilde{O}(D+\sqrt{n})$. The former moves toward the 1974 conjecture of Aho, Hopcroft, and Ullman postulating an $O(m)$ centralized exact algorithm while the latter is the first distributed vertex connectivity approximation.

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arXiv2013

Distributed Minimum Cut Approximation

We study the problem of computing approximate minimum edge cuts by distributed algorithms. We use a standard synchronous message passing model where in each round, $O(\log n)$ bits can be transmitted over each edge (a.k.a. the CONGEST model). We present a distributed algorithm that, for any weighted graph and any $ε\in (0, 1)$, with high probability finds a cut of size at most $O(ε^{-1}λ)$ in $O(D) + \tilde{O}(n^{1/2 + ε})$ rounds, where $λ$ is the size of the minimum cut. This algorithm is based on a simple approach for analyzing random edge sampling, which we call the random layering technique. In addition, we also present another distributed algorithm, which is based on a centralized algorithm due to Matula [SODA '93], that with high probability computes a cut of size at most $(2+ε)λ$ in $\tilde{O}((D+\sqrt{n})/ε^5)$ rounds for any $ε>0$. The time complexities of both of these algorithms almost match the $\tildeΩ(D + \sqrt{n})$ lower bound of Das Sarma et al. [STOC '11], thus leading to an answer to an open question raised by Elkin [SIGACT-News '04] and Das Sarma et al. [STOC '11]. Furthermore, we also strengthen the lower bound of Das Sarma et al. by extending it to unweighted graphs. We show that the same lower bound also holds for unweighted multigraphs (or equivalently for weighted graphs in which $O(w\log n)$ bits can be transmitted in each round over an edge of weight $w$), even if the diameter is $D=O(\log n)$. For unweighted simple graphs, we show that even for networks of diameter $\tilde{O}(\frac{1}λ\cdot \sqrt{\frac{n}{αλ}})$, finding an $α$-approximate minimum cut in networks of edge connectivity $λ$ or computing an $α$-approximation of the edge connectivity requires $\tildeΩ(D + \sqrt{\frac{n}{αλ}})$ rounds.

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arXiv2013

A New Perspective on Vertex Connectivity

Edge connectivity and vertex connectivity are two fundamental concepts in graph theory. Although by now there is a good understanding of the structure of graphs based on their edge connectivity, our knowledge in the case of vertex connectivity is much more limited. An essential tool in capturing edge connectivity are edge-disjoint spanning trees. The famous results of Tutte and Nash-Williams show that a graph with edge connectivity $λ$ contains $\floor{λ/2}$ edge-disjoint spanning trees. We present connected dominating set (CDS) partition and packing as tools that are analogous to edge-disjoint spanning trees and that help us to better grasp the structure of graphs based on their vertex connectivity. The objective of the CDS partition problem is to partition the nodes of a graph into as many connected dominating sets as possible. The CDS packing problem is the corresponding fractional relaxation, where CDSs are allowed to overlap as long as this is compensated by assigning appropriate weights. CDS partition and CDS packing can be viewed as the counterparts of the well-studied edge-disjoint spanning trees, focusing on vertex disjointedness rather than edge disjointness. We constructively show that every $k$-vertex-connected graph with $n$ nodes has a CDS packing of size $Ω(k/\log n)$ and a CDS partition of size $Ω(k/\log^5 n)$. We prove that the $Ω(k/\log n)$ CDS packing bound is existentially optimal. Using CDS packing, we show that if vertices of a $k$-vertex-connected graph are independently sampled with probability $p$, then the graph induced by the sampled vertices has vertex connectivity $\tildeΩ(kp^2)$. Moreover, using our $Ω(k/\log n)$ CDS packing, we get a store-and-forward broadcast algorithm with optimal throughput in the networking model where in each round, each node can send one bounded-size message to all its neighbors.

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arXiv2012

Lower Bounds on Information Dissemination in Dynamic Networks

We study lower bounds on information dissemination in adversarial dynamic networks. Initially, k pieces of information (henceforth called tokens) are distributed among n nodes. The tokens need to be broadcast to all nodes through a synchronous network in which the topology can change arbitrarily from round to round provided that some connectivity requirements are satisfied. If the network is guaranteed to be connected in every round and each node can broadcast a single token per round to its neighbors, there is a simple token dissemination algorithm that manages to deliver all k tokens to all the nodes in O(nk) rounds. Interestingly, in a recent paper, Dutta et al. proved an almost matching Omega(n + nk/log n) lower bound for deterministic token-forwarding algorithms that are not allowed to combine, split, or change tokens in any way. In the present paper, we extend this bound in different ways. If nodes are allowed to forward b < k tokens instead of only one token in every round, a straight-forward extension of the O(nk) algorithm disseminates all k tokens in time O(nk/b). We show that for any randomized token-forwarding algorithm, Omega(n + nk/(b^2 log n log log n)) rounds are necessary. If nodes can only send a single token per round, but we are guaranteed that the network graph is c-vertex connected in every round, we show a lower bound of Omega(nk/(c log^{3/2} n)), which almost matches the currently best O(nk/c) upper bound. Further, if the network is T-interval connected, a notion that captures connection stability over time, we prove that Omega(n + nk/(T^2 log n)) rounds are needed. The best known upper bound in this case manages to solve the problem in O(n + nk/T) rounds. Finally, we show that even if each node only needs to obtain a delta-fraction of all the tokens for some delta in [0,1], Omega(nk delta^3 log n) are still required.

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arXiv2012

Beeping a Maximal Independent Set

We consider the problem of computing a maximal independent set (MIS) in an extremely harsh broadcast model that relies only on carrier sensing. The model consists of an anonymous broadcast network in which nodes have no knowledge about the topology of the network or even an upper bound on its size. Furthermore, it is assumed that an adversary chooses at which time slot each node wakes up. At each time slot a node can either beep, that is, emit a signal, or be silent. At a particular time slot, beeping nodes receive no feedback, while silent nodes can only differentiate between none of its neighbors beeping, or at least one of its neighbors beeping. We start by proving a lower bound that shows that in this model, it is not possible to locally converge to an MIS in sub-polynomial time. We then study four different relaxations of the model which allow us to circumvent the lower bound and find an MIS in polylogarithmic time. First, we show that if a polynomial upper bound on the network size is known, it is possible to find an MIS in O(log^3 n) time. Second, if we assume sleeping nodes are awoken by neighboring beeps, then we can also find an MIS in O(log^3 n) time. Third, if in addition to this wakeup assumption we allow sender-side collision detection, that is, beeping nodes can distinguish whether at least one neighboring node is beeping concurrently or not, we can find an MIS in O(log^2 n) time. Finally, if instead we endow nodes with synchronous clocks, it is also possible to find an MIS in O(log^2 n) time.

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arXiv2011

Computing a Maximal Independent Set Using Beeps

We consider the problem of finding a maximal independent set (MIS) in the discrete beeping model. At each time, a node in the network can either beep (i.e., emit a signal) or be silent. Silent nodes can only differentiate between no neighbor beeping, or at least one neighbor beeping. This basic communication model relies only on carrier-sensing. Furthermore, we assume nothing about the underlying communication graph and allow nodes to wake up (and crash) arbitrarily. We show that if a polynomial upper bound on the size of the network n is known, then with high probability every node becomes stable in O(\log^3 n) time after it is woken up. To contrast this, we establish a polynomial lower bound when no a priori upper bound on the network size is known. This holds even in the much stronger model of local message broadcast with collision detection. Finally, if we assume nodes have access to synchronized clocks or we consider a somewhat restricted wake up, we can solve the MIS problem in O(\log^2 n) time without requiring an upper bound on the size of the network, thereby achieving the same bit complexity as Luby's MIS algorithm.

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arXiv2010

Deploying Wireless Networks with Beeps

We present the \emph{discrete beeping} communication model, which assumes nodes have minimal knowledge about their environment and severely limited communication capabilities. Specifically, nodes have no information regarding the local or global structure of the network, don't have access to synchronized clocks and are woken up by an adversary. Moreover, instead on communicating through messages they rely solely on carrier sensing to exchange information. We study the problem of \emph{interval coloring}, a variant of vertex coloring specially suited for the studied beeping model. Given a set of resources, the goal of interval coloring is to assign every node a large contiguous fraction of the resources, such that neighboring nodes share no resources. To highlight the importance of the discreteness of the model, we contrast it against a continuous variant described in [17]. We present an O(1$ time algorithm that terminates with probability 1 and assigns an interval of size $Ω(T/Δ)$ that repeats every $T$ time units to every node of the network. This improves an $O(\log n)$ time algorithm with the same guarantees presented in \cite{infocom09}, and accentuates the unrealistic assumptions of the continuous model. Under the more realistic discrete model, we present a Las Vegas algorithm that solves $Ω(T/Δ)$-interval coloring in $O(\log n)$ time with high probability and describe how to adapt the algorithm for dynamic networks where nodes may join or leave. For constant degree graphs we prove a lower bound of $Ω(\log n)$ on the time required to solve interval coloring for this model against randomized algorithms. This lower bound implies that our algorithm is asymptotically optimal for constant degree graphs.

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arXiv2009

Gradient Clock Synchronization using Reference Broadcasts

In this paper we suggest a method by which reference broadcast synchronization (RBS), and other methods of estimating clock values, can be incorporated in standard clock synchronization algorithms to improve synchronization quality. We advocate a logical separation of the task of estimating the clock values of other nodes in the network from the task of using these estimates to output a logical clock value. The separation is achieved by means of a virtual estimate graph, overlaid on top of the real network graph, which represents the information various nodes can obtain about each other. RBS estimates are represented in the estimate graph as edges between nodes at distance 2 from each other in the original network graph. A clock synchronization algorithm then operates on the estimate graph as though it were the original network. To illustrate the merits of this approach, we modify a recent optimal gradient clock synchronization algorithm to work in this setting. The modified algorithm transparently takes advantage of RBS estimates and any other means by which nodes can estimate each others' clock values.

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arXiv2009

Local Multicoloring Algorithms: Computing a Nearly-Optimal TDMA Schedule in Constant Time

The described multicoloring problem has direct applications in the context of wireless ad hoc and sensor networks. In order to coordinate the access to the shared wireless medium, the nodes of such a network need to employ some medium access control (MAC) protocol. Typical MAC protocols control the access to the shared channel by time (TDMA), frequency (FDMA), or code division multiple access (CDMA) schemes. Many channel access schemes assign a fixed set of time slots, frequencies, or (orthogonal) codes to the nodes of a network such that nodes that interfere with each other receive disjoint sets of time slots, frequencies, or code sets. Finding a valid assignment of time slots, frequencies, or codes hence directly corresponds to computing a multicoloring of a graph $G$. The scarcity of bandwidth, energy, and computing resources in ad hoc and sensor networks, as well as the often highly dynamic nature of these networks require that the multicoloring can be computed based on as little and as local information as possible.

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