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Register a new userA Conditional Lower Bound on Graph Connectivity in MapReduce
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Benjamin Moseley
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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MapReduce (and its open source implementation Hadoop) has become the de facto platform for processing large data sets. MapReduce offers a streamlined computational framework by interleaving sequential and parallel computation while hiding underlying system issues from the programmer. Due to the popularity of MapReduce, there have been attempts in the theoretical computer science community to understand the power and limitations of the MapReduce framework. In the most widely studied MapReduce models each machine has memory sub-linear in the input size to the problem, hence cannot see the entire input. This restriction places many limitations on algorithms that can be developed for the model; however, the current understanding of these restrictions is still limited. In this paper, our goal is to work towards understanding problems which do not admit efficient algorithms in the MapReduce model. We study the basic question of determining if a graph is connected or not. We concentrate on instances of this problem where an algorithm is to determine if a graph consists of a single cycle or two disconnected cycles. In this problem, locally every part of the graph is similar and the goal is to determine the global structure of the graph. We consider a natural class of algorithms that can store/process/transfer the information only in the form of paths and show that no randomized algorithm cannot answer the decision question in a sub-logarithmic number of rounds. Currently, there are no absolute super constant lower bounds on the number of rounds known for any problem in MapReduce. We introduce some of the first lower bounds for a natural graph problem, albeit for a restricted class of algorithms. We believe our result makes progress towards understanding the limitations of MapReduce.
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Let $G = (V,w)$ be a weighted undirected graph with $m$ edges. The cut dimension of $G$ is the dimension of the span of the characteristic vectors of the minimum cuts of $G$, viewed as vectors in ${0,1}^m$. For every $n ge 2$ we show that the cut dimension of an $n$-vertex graph is at most $2n-3$, and construct graphs realizing this bound. The cut dimension was recently defined by Graur et al. cite{GPRW20}, who show that the maximum cut dimension of an $n$-vertex graph is a lower bound on the number of cut queries needed by a deterministic algorithm to solve the minimum cut problem on $n$-vertex graphs. For every $nge 2$, Graur et al. exhibit a graph on $n$ vertices with cut dimension at least $3n/2 -2$, giving the first lower bound larger than $n$ on the deterministic cut query complexity of computing mincut. We observe that the cut dimension is even a lower bound on the number of emph{linear} queries needed by a deterministic algorithm to solve mincut, where a linear query can ask any vector $x in mathbb{R}^{binom{n}{2}}$ and receives the answer $w^T x$. Our results thus show a lower bound of $2n-3$ on the number of linear queries needed by a deterministic algorithm to solve minimum cut on $n$-vertex graphs, and imply that one cannot show a lower bound larger than this via the cut dimension. We further introduce a generalization of the cut dimension which we call the $ell_1$-approximate cut dimension. The $ell_1$-approximate cut dimension is also a lower bound on the number of linear queries needed by a deterministic algorithm to compute minimum cut. It is always at least as large as the cut dimension, and we construct an infinite family of graphs on $n=3k+1$ vertices with $ell_1$-approximate cut dimension $2n-2$, showing that it can be strictly larger than the cut dimension.
Given two strings $S$ and $P$, the Episode Matching problem is to compute the length of the shortest substring of $S$ that contains $P$ as a subsequence. The best known upper bound for this problem is $tilde O(nm)$ by Das et al. (1997), where $n,m$ are the lengths of $S$ and $P$, respectively. Although the problem is well studied and has many applications in data mining, this bound has never been improved. In this paper we show why this is the case by proving that an $O((nm)^{1-epsilon})$ algorithm (even for binary strings) would refute the popular Strong Exponential Time Hypothesis (SETH). The proof is based on a simple reduction from Orthogonal Vectors.
M.Aleknovich et al. have recently proposed a model of algorithms, called BT model, which generalizes both the priority model of Borodin, Nielson and Rackoff, as well as a simple dynamic programming model by Woeginger. BT model can be further divided into three kinds of fixed, adaptive and fully adaptive ones. They have proved exponential time lower bounds of exact and approximation algorithms under adaptive BT model for Knapsack problem. Their exact lower bound is $Omega(2^{0.5n}/sqrt{n})$, in this paper, we slightly improve the exact lower bound to about $Omega(2^{0.69n}/sqrt{n})$, by the same technique, with related parameters optimized.
The view of a node in a port-labeled network is an infinite tree encoding all walks in the network originating from this node. We prove that for any integers $ngeq Dgeq 1$, there exists a port-labeled network with at most $n$ nodes and diameter at most $D$ which contains a pair of nodes whose (infinite) views are different, but whose views truncated to depth $Omega(Dlog (n/D))$ are identical.
We prove that with high probability over the choice of a random graph $G$ from the ErdH{o}s-Renyi distribution $G(n,1/2)$, a natural $n^{O(varepsilon^2 log n)}$-time, degree $O(varepsilon^2 log n)$ sum-of-squares semidefinite program cannot refute the existence of a valid $k$-coloring of $G$ for $k = n^{1/2 +varepsilon}$. Our result implies that the refutation guarantee of the basic semidefinite program (a close variant of the Lovasz theta function) cannot be appreciably improved by a natural $o(log n)$-degree sum-of-squares strengthening, and this is tight up to a $n^{o(1)}$ slack in $k$. To the best of our knowledge, this is the first lower bound for coloring $G(n,1/2)$ for even a single round strengthening of the basic SDP in any SDP hierarchy. Our proof relies on a new variant of instance-preserving non-pointwise complete reduction within SoS from coloring a graph to finding large independent sets in it. Our proof is (perhaps surprisingly) short, simple and does not require complicated spectral norm bounds on random matrices with dependent entries that have been otherwise necessary in the proofs of many similar results [BHK+16, HKP+17, KB19, GJJ+20, MRX20]. Our result formally holds for a constraint system where vertices are allowed to belong to multiple color classes; we leave the extension to the formally stronger formulation of coloring, where vertices must belong to unique colors classes, as an outstanding open problem.
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