Partition-Merge: Distributed Inference and Modularity Optimization

Vincent Blondel; Kyomin Jung; Pushmeet Kohli; Devavrat Shah

Home/Archive/arXiv

arXiv · 1309.6129

Partition-Merge: Distributed Inference and Modularity Optimization

Vincent Blondel·Kyomin Jung·Pushmeet Kohli·Devavrat Shah

View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper presents a novel meta algorithm, Partition-Merge (PM), which takes existing centralized algorithms for graph computation and makes them distributed and faster. In a nutshell, PM divides the graph into small subgraphs using our novel randomized partitioning scheme, runs the centralized algorithm on each partition separately, and then stitches the resulting solutions to produce a global solution. We demonstrate the efficiency of the PM algorithm on two popular problems: computation of Maximum A Posteriori (MAP) assignment in an arbitrary pairwise Markov Random Field (MRF), and modularity optimization for community detection. We show that the resulting distributed algorithms for these problems essentially run in time linear in the number of nodes in the graph, and perform as well -- or even better -- than the original centralized algorithm as long as the graph has geometric structures. Here we say a graph has geometric structures, or polynomial growth property, when the number of nodes within distance r of any given node grows no faster than a polynomial function of r. More precisely, if the centralized algorithm is a C-factor approximation with constant C \ge 1, the resulting distributed algorithm is a (C+δ)-factor approximation for any small δ>0; but if the centralized algorithm is a non-constant (e.g. logarithmic) factor approximation, then the resulting distributed algorithm becomes a constant factor approximation. For general graphs, we compute explicit bounds on the loss of performance of the resulting distributed algorithm with respect to the centralized algorithm.

Explore related subjects

cs.DS cs.AI cs.SI

Keep this discovery

BibTeX RIS

Vincent Blondel, Kyomin Jung, Pushmeet Kohli, Devavrat Shah. 2013-09-24. Partition-Merge: Distributed Inference and Modularity Optimization. https://arxiv.org/abs/1309.6129

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗

arXiv2026

A deterministic $(1+\varepsilon)^n$ approximation for the permanent of a nonnegative matrix

For every fixed $0<\varepsilon\le1$, we give a deterministic strongly polynomial algorithm that, given a nonnegative matrix $A\in\mathbb{R}_{\ge0}^{n\times n}$, returns $Q$ satisfying $\operatorname{per} A\le Q\le(1+\varepsilon)^n\operatorname{per} A$.

cs.DS↗

arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗