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Register a new userLower Bounds for Structuring Unreliable Radio Networks
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Calvin Newport
Publication date
2014
fields
Informatics Engineering
and research's language is
English
Authors
Calvin Newport
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In this paper, we study lower bounds for randomized solutions to the maximal independent set (MIS) and connected dominating set (CDS) problems in the dual graph model of radio networks---a generalization of the standard graph-based model that now includes unreliable links controlled by an adversary. We begin by proving that a natural geographic constraint on the network topology is required to solve these problems efficiently (i.e., in time polylogarthmic in the network size). We then prove the importance of the assumption that nodes are provided advance knowledge of their reliable neighbors (i.e, neighbors connected by reliable links). Combined, these results answer an open question by proving that the efficient MIS and CDS algorithms from [Censor-Hillel, PODC 2011] are optimal with respect to their dual graph model assumptions. They also provide insight into what properties of an unreliable network enable efficient local computation.
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Theoreticians have studied distributed algorithms in the radio network model for close to three decades. A significant fraction of this work focuses on lower bounds for basic communication problems such as wake-up (symmetry breaking among an unknown set of nodes) and broadcast (message dissemination through an unknown network topology). In this paper, we introduce a new technique for proving this type of bound, based on reduction from a probabilistic hitting game, that simplifies and strengthens much of this existing work. In more detail, in this single paper we prove new expected time and high probability lower bounds for wake-up and global broadcast in single and multichann
In this paper, we study local and global broadcast in the dual graph model, which describes communication in a radio network with both reliable and unreliable links. Existing work proved that efficient solutions to these problems are impossible in the dual graph model under standard assumptions. In real networks, however, simple back-off strategies tend to perform well for solving these basic communication tasks. We address this apparent paradox by introducing a new set of constraints to the dual graph model that better generalize the slow/fast fading behavior common in real networks. We prove that in the context of these new constraints, simple back-off strategies now provide efficient solutions to local and global broadcast in the dual graph model. We also precisely characterize how this efficiency degrades as the new constraints are reduced down to non-existent, and prove new lower bounds that establish this degradation as near optimal for a large class of natural algorithms. We conclude with a preliminary investigation of the performance of these strategies when we include additional generality to the model. These results provide theoretical foundations for the practical observation that simple back-off algorithms tend to work well even amid the complicated link dynamics of real radio networks.
Given a graph $G = (V,E)$, an $(alpha, beta)$-ruling set is a subset $S subseteq V$ such that the distance between any two vertices in $S$ is at least $alpha$, and the distance between any vertex in $V$ and the closest vertex in $S$ is at most $beta$. We present lower bounds for distributedly computing ruling sets. More precisely, for the problem of computing a $(2, beta)$-ruling set in the LOCAL model, we show the following, where $n$ denotes the number of vertices, $Delta$ the maximum degree, and $c$ is some universal constant independent of $n$ and $Delta$. $bullet$ Any deterministic algorithm requires $Omegaleft(min left{ frac{log Delta}{beta log log Delta} , log_Delta n right} right)$ rounds, for all $beta le c cdot minleft{ sqrt{frac{log Delta}{log log Delta}} , log_Delta n right}$. By optimizing $Delta$, this implies a deterministic lower bound of $Omegaleft(sqrt{frac{log n}{beta log log n}}right)$ for all $beta le c sqrt[3]{frac{log n}{log log n}}$. $bullet$ Any randomized algorithm requires $Omegaleft(min left{ frac{log Delta}{beta log log Delta} , log_Delta log n right} right)$ rounds, for all $beta le c cdot minleft{ sqrt{frac{log Delta}{log log Delta}} , log_Delta log n right}$. By optimizing $Delta$, this implies a randomized lower bound of $Omegaleft(sqrt{frac{log log n}{beta log log log n}}right)$ for all $beta le c sqrt[3]{frac{log log n}{log log log n}}$. For $beta > 1$, this improves on the previously best lower bound of $Omega(log^* n)$ rounds that follows from the 30-year-old bounds of Linial [FOCS87] and Naor [J.Disc.Math.91]. For $beta = 1$, i.e., for the problem of computing a maximal independent set, our results improve on the previously best lower bound of $Omega(log^* n)$ on trees, as our bounds already hold on trees.
We develop lower bounds on communication in the memory hierarchy or between processors for nested bilinear algorithms, such as Strassens algorithm for matrix multiplication. We build on a previous framework that establishes communication lower bounds by use of the rank expansion, or the minimum rank of any fixed size subset of columns of a matrix, for each of the three matrices encoding the bilinear algorithm. This framework provides lower bounds for any way of computing a bilinear algorithm, which encompasses a larger space of algorithms than by fixing a particular dependency graph. Nested bilinear algorithms include fast recursive algorithms for convolution, matrix multiplication, and contraction of tensors with symmetry. Two bilinear algorithms can be nested by taking Kronecker products between their encoding matrices. Our main result is a lower bound on the rank expansion of a matrix constructed by a Kronecker product derived from lower bounds on the rank expansion of the Kronecker products operands. To prove this bound, we map a subset of columns from a submatrix to a 2D grid, collapse them into a dense grid, expand the grid, and use the size of the expanded grid to bound the number of linearly independent columns of the submatrix. We apply the rank expansion lower bounds to obtain novel communication lower bounds for nested Toom-Cook convolution, Strassens algorithm, and fast algorithms for partially symmetric contractions.
This paper proves strong lower bounds for distributed computing in the CONGEST model, by presenting the bit-gadget: a new technique for constructing graphs with small cuts. The contribution of bit-gadgets is twofold. First, developing careful sparse graph constructions with small cuts extends known techniques to show a near-linear lower bound for computing the diameter, a result previously known only for dense graphs. Moreover, the sparseness of the construction plays a crucial role in applying it to approximations of various distance computation problems, drastically improving over what can be obtained when using dense graphs. Second, small cuts are essential for proving super-linear lower bounds, none of which were known prior to this work. In fact, they allow us to show near-quadratic lower bounds for several problems, such as exact minimum vertex cover or maximum independent set, as well as for coloring a graph with its chromatic number. Such strong lower bounds are not limited to NP-hard problems, as given by two simple graph problems in P which are shown to require a quadratic and near-quadratic number of rounds. All of the above are optimal up to logarithmic factors. In addition, in this context, the complexity of the all-pairs-shortest-paths problem is discussed. Finally, it is shown that graph constructions for CONGEST lower bounds translate to lower bounds for the semi-streaming model, despite being very different in its nature.
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