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Register a new userSuperlinear Lower Bounds for Distributed Subgraph Detection
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Orr Fischer
Publication date
2017
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Informatics Engineering
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English
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In the distributed subgraph-freeness problem, we are given a graph $H$, and asked to determine whether the network graph contains $H$ as a subgraph or not. Subgraph-freeness is an extremely local problem: if the network had no bandwidth constraints, we could detect any subgraph $H$ in $|H|$ rounds, by having each node of the network learn its entire $|H|$-neighborhood. However, when bandwidth is limited, the problem becomes harder. Upper and lower bounds in the presence of congestion have been established for several classes of subgraphs, including cycles, trees, and more complicated subgraphs. All bounds shown so far have been linear or sublinear. We show that the subgraph-freeness problem is not, in general, solvable in linear time: for any $k geq 2$, there exists a subgraph $H_k$ such that $H_k$-freeness requires $Omega( n^{2-1/k} / (Bk) )$ rounds to solve. Here $B$ is the bandwidth of each communication link. The lower bound holds even for diameter-3 subgraphs and diameter-3 network graphs. In particular, taking $k = Theta(log n)$, we obtain a lower bound of $Omega(n^2 / (B log n))$.
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The minimum-weight $2$-edge-connected spanning subgraph (2-ECSS) problem is a natural generalization of the well-studied minimum-weight spanning tree (MST) problem, and it has received considerable attention in the area of network design. The latter problem asks for a minimum-weight subgraph with an edge connectivity of $1$ between each pair of vertices while the former strengthens this edge-connectivity requirement to $2$. Despite this resemblance, the 2-ECSS problem is considerably more complex than MST. While MST admits a linear-time centralized exact algorithm, 2-ECSS is NP-hard and the best known centralized approximation algorithm for it (that runs in polynomial time) gives a $2$-approximation. In this paper, we give a deterministic distributed algorithm with round complexity of $widetilde{O}(D+sqrt{n})$ that computes a $(5+epsilon)$-approximation of 2-ECSS, for any constant $epsilon>0$. Up to logarithmic factors, this complexity matches the $widetilde{Omega}(D+sqrt{n})$ lower bound that can be derived from Das Sarma et al. [STOC11], as shown by Censor-Hillel and Dory [OPODIS17]. Our result is the first distributed constant approximation for 2-ECSS in the nearly optimal time and it improves on a recent randomized algorithm of Dory [PODC18], which achieved an $O(log n)$-approximation in $widetilde{O}(D+sqrt{n})$ rounds. We also present an alternative algorithm for $O(log n)$-approximation, whose round complexity is linear in the low-congestion shortcut parameter of the network, following a framework introduced by Ghaffari and Haeupler [SODA16]. This algorithm has round complexity $widetilde{O}(D+sqrt{n})$ in worst-case networks but it provably runs much faster in many well-behaved graph families of interest. For instance, it runs in $widetilde{O}(D)$ time in planar networks and those with bounded genus, bounded path-width or bounded tree-width.
Subgraph counting is a fundamental problem in analyzing massive graphs, often studied in the context of social and complex networks. There is a rich literature on designing efficient, accurate, and scalable algorithms for this problem. In this work, we tackle this challenge and design several new algorithms for subgraph counting in the Massively Parallel Computation (MPC) model: Given a graph $G$ over $n$ vertices, $m$ edges and $T$ triangles, our first main result is an algorithm that, with high probability, outputs a $(1+varepsilon)$-approximation to $T$, with optimal round and space complexity provided any $S geq max{(sqrt m, n^2/m)}$ space per machine, assuming $T=Omega(sqrt{m/n})$. Our second main result is an $tilde{O}_{delta}(log log n)$-rounds algorithm for exactly counting the number of triangles, parametrized by the arboricity $alpha$ of the input graph. The space per machine is $O(n^{delta})$ for any constant $delta$, and the total space is $O(malpha)$, which matches the time complexity of (combinatorial) triangle counting in the sequential model. We also prove that this result can be extended to exactly counting $k$-cliques for any constant $k$, with the same round complexity and total space $O(malpha^{k-2})$. Alternatively, allowing $O(alpha^2)$ space per machine, the total space requirement reduces to $O(nalpha^2)$. Finally, we prove that a recent result of Bera, Pashanasangi and Seshadhri (ITCS 2020) for exactly counting all subgraphs of size at most $5$, can be implemented in the MPC model in $tilde{O}_{delta}(sqrt{log n})$ rounds, $O(n^{delta})$ space per machine and $O(malpha^3)$ total space. Therefore, this result also exhibits the phenomenon that a time bound in the sequential model translates to a space bound in the MPC model.
In the subgraph-freeness problem, we are given a constant-size graph $H$, and wish to determine whether the network contains $H$ as a subgraph or not. The emph{property-testing} relaxation of the problem only requires us to distinguish graphs that are $H$-free from graphs that are $epsilon$-far from $H$-free, meaning an $epsilon$-fraction of their edges must be removed to obtain an $H$-free graph. Recently, Censor-Hillel et. al. and Fraigniaud et al. showed that in the property-testing regime it is possible to test $H$-freeness for any graph $H$ of size 4 in constant time, $O(1/epsilon^2)$ rounds, regardless of the network size. However, Fraigniaud et. al. also showed that their techniques for graphs $H$ of size 4 cannot test $5$-cycle-freeness in constant time. In this paper we revisit the subgraph-freeness problem and show that $5$-cycle-freeness, and indeed $H$-freeness for many other graphs $H$ comprising more than 4 vertices, can be tested in constant time. We show that $C_k$-freeness can be tested in $O(1/epsilon)$ rounds for any cycle $C_k$, improving on the running time of $O(1/epsilon^2)$ of the previous algorithms for triangle-freeness and $C_4$-freeness. In the special case of triangles, we show that triangle-freeness can be solved in $O(1)$ rounds independently of $epsilon$, when $epsilon$ is not too small with respect to the number of nodes and edges. We also show that $T$-freeness for any constant-size tree $T$ can be tested in $O(1)$ rounds, even without the property-testing relaxation. Building on these results, we define a general class of graphs for which we can test subgraph-freeness in $O(1/epsilon)$ rounds. This class includes all graphs over 5 vertices except the 5-clique, $K_5$. For cliques $K_s$ over $s geq 3$ nodes, we show that $K_s$-freeness can be tested in $O(m^{1/2-1/(s-2)}/epsilon^{1/2+1/(s-2)})$ rounds, where $m$ is the number of edges.
We present improved distributed algorithms for triangle detection and its variants in the CONGEST model. We show that Triangle Detection, Counting, and Enumeration can be solved in $tilde{O}(n^{1/2})$ rounds. In contrast, the previous state-of-the-art bounds for Triangle Detection and Enumeration were $tilde{O}(n^{2/3})$ and $tilde{O}(n^{3/4})$, respectively, due to Izumi and LeGall (PODC 2017). The main technical novelty in this work is a distributed graph partitioning algorithm. We show that in $tilde{O}(n^{1-delta})$ rounds we can partition the edge set of the network $G=(V,E)$ into three parts $E=E_mcup E_scup E_r$ such that (a) Each connected component induced by $E_m$ has minimum degree $Omega(n^delta)$ and conductance $Omega(1/text{poly} log(n))$. As a consequence the mixing time of a random walk within the component is $O(text{poly} log(n))$. (b) The subgraph induced by $E_s$ has arboricity at most $n^{delta}$. (c) $|E_r| leq |E|/6$. All of our algorithms are based on the following generic framework, which we believe is of interest beyond this work. Roughly, we deal with the set $E_s$ by an algorithm that is efficient for low-arboricity graphs, and deal with the set $E_r$ using recursive calls. For each connected component induced by $E_m$, we are able to simulate congested clique algorithms with small overhead by applying a routing algorithm due to Ghaffari, Kuhn, and Su (PODC 2017) for high conductance graphs.
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.
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