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28
A New Technique For Distributed Symmetry Breaking
- In Symp. on Principles of Distributed Computing
, 2010
"... We introduce Multi-Trials, a new technique for symmetry breaking for distributed algorithms and apply it to various problems in general graphs. For instance, we present three randomized algorithms for distributed (vertex or edge) coloring improving on previous algorithms and showing a time/color tra ..."
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Cited by 34 (6 self)
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We introduce Multi-Trials, a new technique for symmetry breaking for distributed algorithms and apply it to various problems in general graphs. For instance, we present three randomized algorithms for distributed (vertex or edge) coloring improving on previous algorithms and showing a time/color trade-off. To get a ∆ + 1 coloring takes time O(log ∆ + √ log n). To obtain an O( ∆ + log 1+1 / log ∗ n n) coloring takes time O(log ∗ n). This is more than an exponential improvement in time for graphs of polylogarithmic degree. Our fastest algorithm works in constant time using O( ∆ log (c) n + log 1+1/c n) colors, where c denotes an arbitrary constant and log (c) n denotes the c times (recursively) applied logarithm to n. We also use the Multi-Trials technique to compute network decompositions and to compute maximal independent set (MIS), obtaining new results for several graph classes.
Deploying Wireless Networks with Beeps
"... We present the 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, do not have access to synchronized ..."
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Cited by 18 (2 self)
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We present the 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, do not 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. This model is interesting from a practical point of view, because it is possible to implement it (or emulate it) even in extremely restricted radio network environments. From a theory point of view, it shows that complex problems (such as vertex coloring) can be solved efficiently even without strong assumptions on properties of the communication model. We study the problem of 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 have disjoint resources. A k-interval coloring is one where every node gets at least a 1/k fraction of the 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 with probability 1 produces a O(∆)-interval coloring. This improves an O(log n) time algorithm with the same guarantees presented in [17], and accentuates the unrealistic assumptions of the continuous model. Under the more realistic discrete model, we present a Las Vegas algorithm that solves O(∆)-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.
The Locality of Distributed Symmetry Breaking
"... We present new bounds on the locality of several classical symmetry breaking tasks in distributed networks. A sampling of the results include 1) A randomized algorithm for computing a maximal matching (MM) in O(log ∆+(log log n) 4) rounds, where ∆ is the maximum degree. This improves a 25-year old ..."
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Cited by 16 (2 self)
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We present new bounds on the locality of several classical symmetry breaking tasks in distributed networks. A sampling of the results include 1) A randomized algorithm for computing a maximal matching (MM) in O(log ∆+(log log n) 4) rounds, where ∆ is the maximum degree. This improves a 25-year old randomized algorithm of Israeli and Itai that takes O(log n) rounds and is provably optimal for all log ∆ in the range [(log log n) 4, √ log n]. 2) A randomized maximal independent set (MIS) algorithm requiring O(log ∆ √ log n) rounds, for all ∆, and only 2 O(√log log n) rounds when ∆ = poly(log n). These improve on the 25-year old O(log n)-round randomized MIS algorithms of Luby and Alon, Babai, and Itai when log ∆ ≪ √ log n. 3) A randomized ( ∆ + 1)-coloring algorithm requiring O(log ∆ + 2 O(√log log n)) rounds, improving on an algorithm √ of Schneider and Wattenhofer that takes O(log ∆+ log n) rounds. This result implies that an O(∆)coloring can be computed in 2 O(√log log n) rounds for all ∆, improving on Kothapalli et al.’s O ( √ log n)-round algorithm. We also introduce a new technique for reducing symmetry breaking problems on low arboricity graphs to low degree graphs. Corollaries of this reduction include MM and MIS algorithms for low arboricity graphs (e.g., planar graphs and graphs that exclude any fixed minor) requiring O ( √ log n) and O(log 2/3 n) rounds w.h.p., respectively.
Fast Local Computation Algorithms
"... For input x, let F (x) denote the set of outputs that are the “legal ” answers for a computational problem F. Suppose x and members of F (x) are so large that there is not time to read them in their entirety. We propose a model of local computation algorithms which for a given input x, support queri ..."
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Cited by 9 (4 self)
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For input x, let F (x) denote the set of outputs that are the “legal ” answers for a computational problem F. Suppose x and members of F (x) are so large that there is not time to read them in their entirety. We propose a model of local computation algorithms which for a given input x, support queries by a user to values of specified locations yi in a legal output y ∈ F (x). When more than one legal output y exists for a given x, the local computation algorithm should output in a way that is consistent with at least one such y. Local computation algorithms are intended to distill the common features of several concepts that have appeared in various algorithmic subfields, including local distributed computation, local algorithms, locally decodable codes, and local reconstruction. We develop a technique, based on Beck’s analysis in his algorithmic approach to the Lovász Local Lemma, which under certain conditions can be applied to construct polylogarithmic time local computation algorithms. We apply this technique to maximal independent set computations, scheduling radio network broadcasts, hypergraph coloring and satisfying k-SAT formulas.
Locality and Checkability in Wait-free Computing
"... Abstract. This paper studies several notions of locality that are in-herent to the specification of distributed tasks and independent of the computing environment, and investigates the ability of a shared memory wait-free system to solve tasks satisfying various forms of locality. First, we define a ..."
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Cited by 8 (7 self)
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Abstract. This paper studies several notions of locality that are in-herent to the specification of distributed tasks and independent of the computing environment, and investigates the ability of a shared memory wait-free system to solve tasks satisfying various forms of locality. First, we define a task to be projection-closed if every partial output pi(t) for a full input s is also a valid output for the partial input pi(s) and prove that projection-closed tasks are precisely those tasks that are wait-free checkable. Our second main contribution is dealing with a stronger notion of lo-cality of topological nature. A task T = (I,O,∆) is said to be locality-preserving if and only if O is a covering complex of I, that is, each simplex s of I is mapped by ∆ to a set of simplexes of O each iso-morphic to s. This topological property yields obstacles for wait-free solvability different in nature from the classical agreement impossibility results. On the other hand, locality-preserving tasks are projection-closed and therefore always wait-free checkable. We provide a classification of locality-preserving tasks in term of their computational power, by estab-lishing a correspondence between locality-preserving tasks and subgroups of the edgepath group of a complex. Using this correspondence, we prove the existence of hierarchies of locality-preserving tasks, each one contain-ing a universal task (induced by the universal covering complex), and at the bottom the trivial identity task.
Combinatorial algorithms for distributed graph coloring
- In DISC
, 2011
"... Abstract. Numerous problems in Theoretical Computer Science can be solved very efficiently using powerful algebraic constructions. Computing shortest paths, constructing expanders, and proving the PCP Theorem, are just a few examples of this phenomenon. The quest for combinatorial algorithms that do ..."
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Cited by 7 (0 self)
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Abstract. Numerous problems in Theoretical Computer Science can be solved very efficiently using powerful algebraic constructions. Computing shortest paths, constructing expanders, and proving the PCP Theorem, are just a few examples of this phenomenon. The quest for combinatorial algorithms that do not use heavy algebraic machinery, but have the same (or better) efficiency has become a central field of study in this area. Com-binatorial algorithms are often simpler than their algebraic counterparts. Moreover, in many cases, combinatorial algorithms and proofs provide additional understanding of studied problems. In this paper we initiate the study of combinatorial algorithms for Distributed Graph Coloring problems. In a distributed setting a communication network is modeled by a graph G = (V,E) of maximum degree ∆. The vertices of G host the processors, and communication is performed over the edges of G. The goal of distributed vertex coloring is to color V with (∆+1) colors such that any two neighbors are colored with distinct colors. Currently, effi-
Distributed Coloring Depending on the Chromatic Number or the Neighborhood Growth
"... Abstract 1 We deterministically compute a ∆+1 coloring in time O(∆5c+2·(∆5) 2/c /(∆1) ɛ + (∆1) ɛ + log ∗ n) and O(∆5c+2 · (∆5) 1/c / ∆ ɛ + ∆ ɛ + (∆5) d log ∆5 log n) for arbitrary constants d, ɛ and arbitrary constant integer c, where ∆i is defined as the maximal number of nodes within distance i fo ..."
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Cited by 5 (4 self)
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Abstract 1 We deterministically compute a ∆+1 coloring in time O(∆5c+2·(∆5) 2/c /(∆1) ɛ + (∆1) ɛ + log ∗ n) and O(∆5c+2 · (∆5) 1/c / ∆ ɛ + ∆ ɛ + (∆5) d log ∆5 log n) for arbitrary constants d, ɛ and arbitrary constant integer c, where ∆i is defined as the maximal number of nodes within distance i for a node and ∆: = ∆1. Our greedy algorithm improves the state-of-the-art ∆+1 coloring algorithms for a large class of graphs, e.g. graphs of moderate neighborhood growth. We also state and analyze a randomized coloring algorithm in terms of the chromatic number, the run time and the used colors. If ∆ ∈ Ω(log 1+1 / log ∗ n n) and χ ∈ O(∆ / log 1+1 / log ∗ n n) then our algorithm executes in time O(log χ + log ∗ n) with high probability. For graphs of polylogarithmic chromatic number the analysis reveals an exponential gap compared to the fastest ∆ + 1 coloring algorithm running in time O(log ∆ + √ log n). The algorithm works without knowledge of χ and uses less than ∆ colors, i.e., (1 − 1/O(χ)) ∆ with high probability. To the best of our knowledge this is the first distributed algorithm for (such) general graphs taking the chromatic number χ into account. 1
Distributed Channel Allocation Protocols for Wireless Sensor Networks
"... Interference between concurrent transmissions can cause severe performance degradation in wireless sensor networks (WSNs). While multiple channels available in WSN technology such as IEEE 802.15.4 can be exploited to mitigate interference, channel allocation can have a significant impact on the per ..."
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Cited by 3 (0 self)
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Interference between concurrent transmissions can cause severe performance degradation in wireless sensor networks (WSNs). While multiple channels available in WSN technology such as IEEE 802.15.4 can be exploited to mitigate interference, channel allocation can have a significant impact on the performance of multi-channel communication. This paper proposes a set of distributed protocols for channel allocation in WSNs with theoretical bounds. We first consider the problem of minimizing the number of channels needed to remove interference in a WSN, and propose both receiver-based and linkbased distributed channel allocation protocols. Then, for WSNs with an insufficient number of channels, we formulate a fair channel allocation problem whose objective is to minimize the maximum interference (MinMax) experienced by any transmission link in the network. We prove that MinMax channel allocation is NP-hard, and propose a distributed link-based MinMax channel allocation protocol. Finally, we propose a distributed protocol for link scheduling based on MinMax channel allocation that creates a conflict-free schedule for transmissions. The proposed decentralized protocols are efficient, scalable, and adaptive to channel condition and network dynamics. Simulations based on the topologies and data traces collected from a WSN testbed of 74 TelosB motes have shown that our channel allocation protocols significantly outperform a state-of-the-art channel allocation protocol.
Node-Disjoint Multipath Spanners and their Relationship with Fault-Tolerant Spanners
, 2011
"... Motivated by multipath routing, we introduce a multi-connected variant of spanners. For that purpose we introduce the p-multipath cost between two nodes u and v as the minimum weight of a collection of p internally vertex-disjoint paths between u and v. Given a weighted graph G, a subgraph H is a p- ..."
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Cited by 2 (1 self)
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Motivated by multipath routing, we introduce a multi-connected variant of spanners. For that purpose we introduce the p-multipath cost between two nodes u and v as the minimum weight of a collection of p internally vertex-disjoint paths between u and v. Given a weighted graph G, a subgraph H is a p-multipath s-spanner if for all u, v, the p-multipath cost between u and v in H is at most s times the p-multipath cost in G. The s factor is called the stretch. Building upon recent results on fault-tolerant spanners, we show how to build p-multipath spanners of constant stretch and of Õ(n1+1/k) edges 1, for fixed parameters p and k, n being the number of nodes of the graph. Such spanners can be constructed by a distributed algorithm running in O(k) rounds. Additionally, we give an improved construction for the case p = k = 2. Our spanner H has O(n 3/2) edges and the p-multipath cost in H between any two node is at most twice the corresponding one in G plus O(W), W being the maximum edge weight.
