اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدMassively Parallel Correlation Clustering in Bounded Arboricity Graphs
87
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Identifying clusters of similar elements in a set is a common task in data analysis. With the immense growth of data and physical limitations on single processor speed, it is necessary to find efficient parallel algorithms for clustering tasks. In this paper, we study the problem of correlation clustering in bounded arboricity graphs with respect to the Massively Parallel Computation (MPC) model. More specifically, we are given a complete graph where the edges are either positive or negative, indicating whether pairs of vertices are similar or dissimilar. The task is to partition the vertices into clusters with as few disagreements as possible. That is, we want to minimize the number of positive inter-cluster edges and negative intra-cluster edges. Consider an input graph $G$ on $n$ vertices such that the positive edges induce a $lambda$-arboric graph. Our main result is a 3-approximation ($textit{in expectation}$) algorithm to correlation clustering that runs in $mathcal{O}(log lambda cdot textrm{poly}(log log n))$ MPC rounds in the $textit{strongly sublinear memory regime}$. This is obtained by combining structural properties of correlation clustering on bounded arboricity graphs with the insights of Fischer and Noever (SODA 18) on randomized greedy MIS and the $texttt{PIVOT}$ algorithm of Ailon, Charikar, and Newman (STOC 05). Combined with known graph matching algorithms, our structural property also implies an exact algorithm and algorithms with $textit{worst case}$ $(1+epsilon)$-approximation guarantees in the special case of forests, where $lambda=1$.
قيم البحث
اقرأ أيضاً
Given a similarity graph between items, correlation clustering (CC) groups similar items together and dissimilar ones apart. One of the most popular CC algorithms is KwikCluster: an algorithm that serially clusters neighborhoods of vertices, and obta
ins a 3-approximation ratio. Unfortunately, KwikCluster in practice requires a large number of clustering rounds, a potential bottleneck for large graphs. We present C4 and ClusterWild!, two algorithms for parallel correlation clustering that run in a polylogarithmic number of rounds and achieve nearly linear speedups, provably. C4 uses concurrency control to enforce serializability of a parallel clustering process, and guarantees a 3-approximation ratio. ClusterWild! is a coordination free algorithm that abandons consistency for the benefit of better scaling; this leads to a provably small loss in the 3-approximation ratio. We provide extensive experimental results for both algorithms, where we outperform the state of the art, both in terms of clustering accuracy and running time. We show that our algorithms can cluster billion-edge graphs in under 5 seconds on 32 cores, while achieving a 15x speedup.
We introduce the Adaptive Massively Parallel Computation (AMPC) model, which is an extension of the Massively Parallel Computation (MPC) model. At a high level, the AMPC model strengthens the MPC model by storing all messages sent within a round in a
distributed data store. In the following round, all machines are provided with random read access to the data store, subject to the same constraints on the total amount of communication as in the MPC model. Our model is inspired by the previous empirical studies of distributed graph algorithms using MapReduce and a distributed hash table service. This extension allows us to give new graph algorithms with much lower round complexities compared to the best known solutions in the MPC model. In particular, in the AMPC model we show how to solve maximal independent set in $O(1)$ rounds and connectivity/minimum spanning tree in $O(loglog_{m/n} n)$ rounds both using $O(n^delta)$ space per machine for constant $delta < 1$. In the same memory regime for MPC, the best known algorithms for these problems require polylog $n$ rounds. Our results imply that the 2-Cycle conjecture, which is widely believed to hold in the MPC model, does not hold in the AMPC model.
The Massively Parallel Computation (MPC) model serves as a common abstraction of many modern large-scale data processing frameworks, and has been receiving increasingly more attention over the past few years, especially in the context of classical gr
aph problems. So far, the only way to argue lower bounds for this model is to condition on conjectures about the hardness of some specific problems, such as graph connectivity on promise graphs that are either one cycle or two cycles, usually called the one cycle vs. two cycles problem. This is unlike the traditional arguments based on conjectures about complexity classes (e.g., $textsf{P} eq textsf{NP}$), which are often more robust in the sense that refuting them would lead to groundbreaking algorithms for a whole bunch of problems. In this paper we present connections between problems and classes of problems that allow the latter type of arguments. These connections concern the class of problems solvable in a sublogarithmic amount of rounds in the MPC model, denoted by $textsf{MPC}(o(log N))$, and some standard classes concerning space complexity, namely $textsf{L}$ and $textsf{NL}$, and suggest conjectures that are robust in the sense that refuting them would lead to many surprisingly fast new algorithms in the MPC model. We also obtain new conditional lower bounds, and prove new reductions and equivalences between problems in the MPC model.
Analyzing massive complex networks yields promising insights about our everyday lives. Building scalable algorithms to do so is a challenging task that requires a careful analysis and an extensive evaluation. However, engineering such algorithms is o
ften hindered by the scarcity of publicly~available~datasets. Network generators serve as a tool to alleviate this problem by providing synthetic instances with controllable parameters. However, many network generators fail to provide instances on a massive scale due to their sequential nature or resource constraints. Additionally, truly scalable network generators are few and often limited in their realism. In this work, we present novel generators for a variety of network models that are frequently used as benchmarks. By making use of pseudorandomization and divide-and-conquer schemes, our generators follow a communication-free paradigm. The resulting generators are thus embarrassingly parallel and have a near optimal scaling behavior. This allows us to generate instances of up to $2^{43}$ vertices and $2^{47}$ edges in less than 22 minutes on 32768 cores. Therefore, our generators allow new graph families to be used on an unprecedented scale.
The Minimum Dominating Set (MDS) problem is not only one of the most fundamental problems in distributed computing, it is also one of the most challenging ones. While it is well-known that minimum dominating sets cannot be approximated locally on gen
eral graphs, over the last years, several breakthroughs have been made on computing local approximations on sparse graphs. This paper presents a deterministic and local constant factor approximation for minimum dominating sets on bounded genus graphs, a very large family of sparse graphs. Our main technical contribution is a new analysis of a slightly modified, first-order definable variant of an existing algorithm by Lenzen et al. Interestingly, unlike existing proofs for planar graphs, our analysis does not rely on any topological arguments. We believe that our techniques can be useful for the study of local problems on sparse graphs beyond the scope of this paper.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
