Near-Optimal Distributed Approximation of Minimum-Weight Connected Dominating Set

This paper presents a near-optimal distributed approximation algorithm for the minimum-weight connected dominating set (MCDS) problem. The presented algorithm finds an $O(\log n)$ approximation in $\tilde{O}(D+\sqrt{n})$ rounds, where $D$ is the network diameter and $n$ is the number of nodes. MCDS is a classical NP-hard problem and the achieved approximation factor $O(\log n)$ is known to be optimal up to a constant factor, unless P=NP. Furthermore, the $\tilde{O}(D+\sqrt{n})$ round complexity is known to be optimal modulo logarithmic factors (for any approximation), following [Das Sarma et al.---STOC'11].Comment: An extended abstract version of this result appears in the proceedings of 41st International Colloquium on Automata, Languages, and Programming (ICALP 2014

Message and time efficient multi-broadcast schemes

We consider message and time efficient broadcasting and multi-broadcasting in wireless ad-hoc networks, where a subset of nodes, each with a unique rumor, wish to broadcast their rumors to all destinations while minimizing the total number of transmissions and total time until all rumors arrive to their destination. Under centralized settings, we introduce a novel approximation algorithm that provides almost optimal results with respect to the number of transmissions and total time, separately. Later on, we show how to efficiently implement this algorithm under distributed settings, where the nodes have only local information about their surroundings. In addition, we show multiple approximation techniques based on the network collision detection capabilities and explain how to calibrate the algorithms' parameters to produce optimal results for time and messages.Comment: In Proceedings FOMC 2013, arXiv:1310.459

Global Computation in a Poorly Connected World: Fast Rumor Spreading with No Dependence on Conductance

In this paper, we study the question of how efficiently a collection of interconnected nodes can perform a global computation in the widely studied GOSSIP model of communication. In this model, nodes do not know the global topology of the network, and they may only initiate contact with a single neighbor in each round. This model contrasts with the much less restrictive LOCAL model, where a node may simultaneously communicate with all of its neighbors in a single round. A basic question in this setting is how many rounds of communication are required for the information dissemination problem, in which each node has some piece of information and is required to collect all others. In this paper, we give an algorithm that solves the information dissemination problem in at most $O(D+\text{polylog}{(n)})$ rounds in a network of diameter $D$, withno dependence on the conductance. This is at most an additive polylogarithmic factor from the trivial lower bound of $D$, which applies even in the LOCAL model. In fact, we prove that something stronger is true: any algorithm that requires $T$ rounds in the LOCAL model can be simulated in $O(T +\mathrm{polylog}(n))$ rounds in the GOSSIP model. We thus prove that these two models of distributed computation are essentially equivalent

Distributed Symmetry Breaking in Hypergraphs

Fundamental local symmetry breaking problems such as Maximal Independent Set (MIS) and coloring have been recognized as important by the community, and studied extensively in (standard) graphs. In particular, fast (i.e., logarithmic run time) randomized algorithms are well-established for MIS and $\Delta +1$-coloring in both the LOCAL and CONGEST distributed computing models. On the other hand, comparatively much less is known on the complexity of distributed symmetry breaking in {\em hypergraphs}. In particular, a key question is whether a fast (randomized) algorithm for MIS exists for hypergraphs. In this paper, we study the distributed complexity of symmetry breaking in hypergraphs by presenting distributed randomized algorithms for a variety of fundamental problems under a natural distributed computing model for hypergraphs. We first show that MIS in hypergraphs (of arbitrary dimension) can be solved in $O(\log^2 n)$ rounds ($n$ is the number of nodes of the hypergraph) in the LOCAL model. We then present a key result of this paper --- an $O(\Delta^{\epsilon}\text{polylog}(n))$-round hypergraph MIS algorithm in the CONGEST model where $\Delta$ is the maximum node degree of the hypergraph and $\epsilon > 0$ is any arbitrarily small constant. To demonstrate the usefulness of hypergraph MIS, we present applications of our hypergraph algorithm to solving problems in (standard) graphs. In particular, the hypergraph MIS yields fast distributed algorithms for the {\em balanced minimal dominating set} problem (left open in Harris et al. [ICALP 2013]) and the {\em minimal connected dominating set problem}. We also present distributed algorithms for coloring, maximal matching, and maximal clique in hypergraphs.Comment: Changes from the previous version: More references adde

Distributed Strong Diameter Network Decomposition

For a pair of positive parameters $D,\chi$, a partition ${\cal P}$ of the vertex set $V$ of an $n$-vertex graph $G = (V,E)$ into disjoint clusters of diameter at most $D$ each is called a $(D,\chi)$ network decomposition, if the supergraph ${\cal G}({\cal P})$, obtained by contracting each of the clusters of ${\cal P}$, can be properly $\chi$-colored. The decomposition ${\cal P}$ is said to be strong (resp., weak) if each of the clusters has strong (resp., weak) diameter at most $D$, i.e., if for every cluster $C \in {\cal P}$ and every two vertices $u,v \in C$, the distance between them in the induced graph $G(C)$ of $C$ (resp., in $G$) is at most $D$. Network decomposition is a powerful construct, very useful in distributed computing and beyond. It was shown by Awerbuch \etal \cite{AGLP89} and Panconesi and Srinivasan \cite{PS92}, that strong $(2^{O(\sqrt{\log n})},2^{O(\sqrt{\log n})})$ network decompositions can be computed in $2^{O(\sqrt{\log n})}$ distributed time. Linial and Saks \cite{LS93} devised an ingenious randomized algorithm that constructs {\em weak} $(O(\log n),O(\log n))$ network decompositions in $O(\log^2 n)$ time. It was however open till now if {\em strong} network decompositions with both parameters $2^{o(\sqrt{\log n})}$ can be constructed in distributed $2^{o(\sqrt{\log n})}$ time. In this paper we answer this long-standing open question in the affirmative, and show that strong $(O(\log n),O(\log n))$ network decompositions can be computed in $O(\log^2 n)$ time. We also present a tradeoff between parameters of our network decomposition. Our work is inspired by and relies on the "shifted shortest path approach", due to Blelloch \etal \cite{BGKMPT11}, and Miller \etal \cite{MPX13}. These authors developed this approach for PRAM algorithms for padded partitions. We adapt their approach to network decompositions in the distributed model of computation

Energy Constrained Dominating Set for Clustering in Wireless Sensor Networks

Using partitioning in wireless sensor networks to create clusters for routing, data management, and other protocols has been proven as a way to ensure scalability and to deal with sensor network shortcomings such as limited communication ranges and energy. Choosing a cluster head within each cluster is important because cluster heads use additional energy for their responsibilities and that burden needs to be carefully passed around. Many existing protocols either choose cluster heads randomly or use nodes with the highest remaining energy. We introduce the energy constrained minimum dominating set (ECDS) to model the problem of optimally choosing cluster heads with energy constraints. We propose a distributed algorithm for the constrained dominating set which runs in O(log n log Δ) rounds with high probability. We experimentally show that the distributed algorithm performs well in terms of energy usage, node lifetime, and clustering time and, thus, is very suitable for wireless sensor networks

Energy efficient clustering and secure data aggregation in wireless sensor networks

Communication consumes the majority of a wireless sensor network\u27s limited energy. There are several ways to reduce the communication cost. Two approaches used in this work are clustering and in-network aggregation. The choice of a cluster head within each cluster is important because cluster heads use additional energy for their responsibilities and that burden needs to be carefully distributed. We introduce the energy constrained minimum dominating set (ECDS) to model the problem of optimally choosing cluster heads in the presence of energy constraints. We show its applicability to sensor networks and give an approximation algorithm of O(log n) for solving the ECDS problem. We propose a distributed algorithm for the constrained dominating set which runs in O(log n log [triangle]) rounds with high probability. We show experimentally that the distributed algorithm performs well in terms of energy usage, node lifetime, and clustering time and thus is very suitable for wireless sensor networks. Using aggregation in wireless sensor networks is another way to reduce the overall communication cost. However, changes in security are necessary when in- network aggregation is applied. Traditional end-to-end security is not suitable for use with in-network aggregation. A corrupted sensor has access to the intermediate data and can falsify results. Additively homomorphic encryption allows for aggregation of encrypted values, with the result being the same as the result as if unencrypted data were aggregated. Using public key cryptography, digital signatures can be used to achieve integrity. We propose a new algorithm using homomorphic encryption and additive digital signatures to achieve confidentiality, integrity and availability for in- network aggregation in wireless sensor networks. We prove that our digital signature algorithm which is based on Elliptic Curve Digital Signature Algorithm (ECDSA) is at least as secure as ECDSA. Even without in-network aggregation, security is a challenge in wireless sensor networks. In wireless sensor networks, not all messages need to be secured with the same level of encryption. We propose a new algorithm which provides adequate levels of security while providing much higher availablility [sic] than other security protocols. Our approach uses similar amounts of energy as a network without security --Abstract, page iv

Constant-time distributed dominating set approximation

Abstract.: Finding a small dominating set is one of the most fundamental problems of classical graph theory. In this paper, we present a new fully distributed approximation algorithm based on LP relaxation techniques. For an arbitrary, possibly constant parameter k and maximum node degree $\Delta$ , our algorithm computes a dominating set of expected size ${\rm O}(k\Delta^{2/k}{\rm log}(\Delta)\vert DS_{\rm {OPT}}\vert)$ in ${\rm O}{(k^2)}$ rounds. Each node has to send ${\rm O}{(k^2\Delta)}$ messages of size ${\rm O}({\rm log}\Delta)$ . This is the first algorithm which achieves a non-trivial approximation ratio in a constant number of round

On the Complexity of Local Distributed Graph Problems

This paper is centered on the complexity of graph problems in the well-studied LOCAL model of distributed computing, introduced by Linial [FOCS '87]. It is widely known that for many of the classic distributed graph problems (including maximal independent set (MIS) and $(\Delta+1)$-vertex coloring), the randomized complexity is at most polylogarithmic in the size $n$ of the network, while the best deterministic complexity is typically $2^{O(\sqrt{\log n})}$. Understanding and narrowing down this exponential gap is considered to be one of the central long-standing open questions in the area of distributed graph algorithms. We investigate the problem by introducing a complexity-theoretic framework that allows us to shed some light on the role of randomness in the LOCAL model. We define the SLOCAL model as a sequential version of the LOCAL model. Our framework allows us to prove completeness results with respect to the class of problems which can be solved efficiently in the SLOCAL model, implying that if any of the complete problems can be solved deterministically in $\log^{O(1)} n$ rounds in the LOCAL model, we can deterministically solve all efficient SLOCAL-problems (including MIS and $(\Delta+1)$-coloring) in $\log^{O(1)} n$ rounds in the LOCAL model. We show that a rather rudimentary looking graph coloring problem is complete in the above sense: Color the nodes of a graph with colors red and blue such that each node of sufficiently large polylogarithmic degree has at least one neighbor of each color. The problem admits a trivial zero-round randomized solution. The result can be viewed as showing that the only obstacle to getting efficient determinstic algorithms in the LOCAL model is an efficient algorithm to approximately round fractional values into integer values