Blog
The latest from Google Research
Balanced Partitioning and Hierarchical Clustering at Scale
Wednesday, March 14, 2018
Posted by Hossein Bateni, Research Scientist and Kevin Aydin, Software Engineer, NYC Algorithms and Optimization Research Team
Solving large-scale optimization problems often starts with
graph partitioning
, which means partitioning the vertices of the graph into clusters to be processed on different machines. The need to make sure that clusters are of near equal size gives rise to the
balanced graph partitioning problem
. In simple terms, we need to partition the vertices of a given graph into k almost equal clusters, while we minimize the number of edges that are cut by the partition. This
NP-hard
problem is notoriously difficult in practice because the
best approximation algorithms
for small instances rely on
semidefinite programming
which is impractical for larger instances.
This post presents the distributed algorithm
we developed
which is more applicable to large instances. We introduced this balanced graph-partitioning algorithm in our
WSDM 2016 paper
, and have applied this approach to several applications within Google. Our more recent
NIPS 2017 paper
provides more details of the algorithm via a theoretical and empirical study.
Balanced Partitioning via Linear Embedding
Our algorithm first embeds vertices of the graph onto a line, and then processes vertices in a distributed manner guided by the linear embedding order. We examine various ways to find the initial embedding, and apply four different techniques (such as local swaps and dynamic programming) to obtain the final partition. The best initial embedding is based on “
affinity clustering
”.
Affinity Hierarchical Clustering
Affinity clustering is an agglomerative hierarchical graph clustering based on
Borůvka’s classic Maximum-cost Spanning Tree algorithm
. As discussed above, this algorithm is a critical part of our balanced partitioning tool. The algorithm starts by placing each vertex in a cluster of its own: v0, v1, and so on. Then, in each iteration, the highest-cost edge out of each cluster is selected in order to induce larger merged clusters: A
0
, A
1
, A
2
, etc. in the first round and B
0
, B
1
, etc. in the second round and so on. The set of merges naturally produces a hierarchical clustering, and gives rise to a linear ordering of the leaf vertices (vertices with degree one). The image below demonstrates this, with the numbers at the bottom corresponding to the ordering of the vertices.
Our
NIPS’17 paper
explains how we run affinity clustering efficiently in the massively parallel computation (MPC) model, in particular using distributed hash tables (
DHTs
) to significantly reduce running time. This paper also presents a theoretical study of the algorithm. We report clustering results for graphs with tens of trillions of edges, and also observe that affinity clustering empirically beats other clustering algorithms such as k-means in terms of “quality of the clusters”.
This video
contains a summary of the result and explains how this parallel algorithm may produce higher-quality clusters even compared to a sequential single-linkage agglomerative algorithm.
Comparison to Previous Work
In comparing our algorithm to previous work in (distributed) balanced graph partitioning, we focus on
FENNEL
,
Spinner
,
METIS
, and a recent
label propagation-based algorithm
. We report results on several public social networks as well as a large private map graph. For a
Twitter followership graph
, we see a consistent improvement of 15–25% over previous results (
Ugander and Backstrom, 2013
), and for LiveJournal graph, our algorithm outperforms all the others for all cases except k = 2, where ours is slightly worse than FENNEL's.
The following table presents the fraction of cut edges in the Twitter graph obtained via different algorithms for various values of k, the number of clusters. The numbers given in parentheses denote the size imbalance factor: i.e., the relative difference of the sizes of largest and smallest clusters. Here “Vanilla Affinity Clustering” denotes the first stage of our algorithm where only the hierarchical clustering is built and no further processing is performed on the cuts. Notice that this is already as good as the best previous work (shown in the first two columns below), cutting a smaller fraction of edges while achieving a perfect (and thus better) balance (i.e., 0% imbalance). The last column in the table includes the final result of our algorithm with the post-processing.
k
UB13
(5%)
Spinner
(5%)
Vanilla Affinity
Clustering
(0%)
Final Algorithm
(0%)
20
37.0%
38.0%
35.71%
27.50%
40
43.0%
40.0%
40.83%
33.71%
60
46.0%
43.0%
43.03%
36.65%
80
47.5%
44.0%
43.27%
38.65%
100
49.0%
46.0%
45.05%
41.53%
Applications
We apply balanced graph partitioning to multiple applications including
Google Maps
driving directions, the serving backend for web search, and finding
treatment groups for experimental design
. For example, in Google Maps the World map graph is stored in several
shards
. The navigational queries spanning multiple shards are substantially more expensive than those handled within a shard. Using the methods described in our paper, we can reduce 21% of cross-shard queries by increasing the shard imbalance factor from 0% to 10%. As discussed in our paper, live experiments on real traffic show that the number of multi-shard queries from our cut-optimization techniques is 40% less compared to a baseline
Hilbert embedding
technique. This, in turn, results in less CPU usage in response to queries. In a future blog post, we will talk about application of this work in the web search serving backend, where balanced partitioning helped us design a cache-aware load balancing system that dramatically reduced our cache miss rate.
Acknowledgements
We especially thank Vahab Mirrokni whose guidance and technical contribution were instrumental in developing these algorithms and writing this post. We also thank our other co-authors and colleagues for their contributions: Raimondas Kiveris, Soheil Behnezhad, Mahsa Derakhshan, MohammadTaghi Hajiaghayi, Silvio Lattanzi, Aaron Archer and other members of
NYC Algorithms and Optimization research team
.
Labels
accessibility
ACL
ACM
Acoustic Modeling
Adaptive Data Analysis
ads
adsense
adwords
Africa
AI
AI for Social Good
Algorithms
Android
Android Wear
API
App Engine
App Inventor
April Fools
Art
Audio
Augmented Reality
Australia
Automatic Speech Recognition
AutoML
Awards
BigQuery
Cantonese
Chemistry
China
Chrome
Cloud Computing
Collaboration
Compression
Computational Imaging
Computational Photography
Computer Science
Computer Vision
conference
conferences
Conservation
correlate
Course Builder
crowd-sourcing
CVPR
Data Center
Data Discovery
data science
datasets
Deep Learning
DeepDream
DeepMind
distributed systems
Diversity
Earth Engine
economics
Education
Electronic Commerce and Algorithms
electronics
EMEA
EMNLP
Encryption
entities
Entity Salience
Environment
Europe
Exacycle
Expander
Faculty Institute
Faculty Summit
Flu Trends
Fusion Tables
gamification
Gboard
Gmail
Google Accelerated Science
Google Books
Google Brain
Google Cloud Platform
Google Docs
Google Drive
Google Genomics
Google Maps
Google Photos
Google Play Apps
Google Science Fair
Google Sheets
Google Translate
Google Trips
Google Voice Search
Google+
Government
grants
Graph
Graph Mining
Hardware
HCI
Health
High Dynamic Range Imaging
ICCV
ICLR
ICML
ICSE
Image Annotation
Image Classification
Image Processing
Inbox
India
Information Retrieval
internationalization
Internet of Things
Interspeech
IPython
Journalism
jsm
jsm2011
K-12
Kaggle
KDD
Keyboard Input
Klingon
Korean
Labs
Linear Optimization
localization
Low-Light Photography
Machine Hearing
Machine Intelligence
Machine Learning
Machine Perception
Machine Translation
Magenta
MapReduce
market algorithms
Market Research
materials science
Mixed Reality
ML
ML Fairness
MOOC
Moore's Law
Multimodal Learning
NAACL
Natural Language Processing
Natural Language Understanding
Network Management
Networks
Neural Networks
NeurIPS
Nexus
Ngram
NIPS
NLP
On-device Learning
open source
operating systems
Optical Character Recognition
optimization
osdi
osdi10
patents
Peer Review
ph.d. fellowship
PhD Fellowship
PhotoScan
Physics
PiLab
Pixel
Policy
Professional Development
Proposals
Public Data Explorer
publication
Publications
Quantum AI
Quantum Computing
Recommender Systems
Reinforcement Learning
renewable energy
Research
Research Awards
resource optimization
Responsible AI
Robotics
schema.org
Search
search ads
Security and Privacy
Self-Supervised Learning
Semantic Models
Semi-supervised Learning
SIGCOMM
SIGMOD
Site Reliability Engineering
Social Networks
Software
Sound Search
Speech
Speech Recognition
statistics
Structured Data
Style Transfer
Supervised Learning
Systems
TensorBoard
TensorFlow
TPU
Translate
trends
TTS
TV
UI
University Relations
UNIX
Unsupervised Learning
User Experience
video
Video Analysis
Virtual Reality
Vision Research
Visiting Faculty
Visualization
VLDB
Voice Search
Wiki
wikipedia
WWW
Year in Review
YouTube
Archive
2022
Aug
Jul
Jun
May
Apr
Mar
Feb
Jan
2021
Dec
Nov
Oct
Sep
Aug
Jul
Jun
May
Apr
Mar
Feb
Jan
2020
Dec
Nov
Oct
Sep
Aug
Jul
Jun
May
Apr
Mar
Feb
Jan
2019
Dec
Nov
Oct
Sep
Aug
Jul
Jun
May
Apr
Mar
Feb
Jan
2018
Dec
Nov
Oct
Sep
Aug
Jul
Jun
May
Apr
Mar
Feb
Jan
2017
Dec
Nov
Oct
Sep
Aug
Jul
Jun
May
Apr
Mar
Feb
Jan
2016
Dec
Nov
Oct
Sep
Aug
Jul
Jun
May
Apr
Mar
Feb
Jan
2015
Dec
Nov
Oct
Sep
Aug
Jul
Jun
May
Apr
Mar
Feb
Jan
2014
Dec
Nov
Oct
Sep
Aug
Jul
Jun
May
Apr
Mar
Feb
Jan
2013
Dec
Nov
Oct
Sep
Aug
Jul
Jun
May
Apr
Mar
Feb
Jan
2012
Dec
Oct
Sep
Aug
Jul
Jun
May
Apr
Mar
Feb
Jan
2011
Dec
Nov
Sep
Aug
Jul
Jun
May
Apr
Mar
Feb
Jan
2010
Dec
Nov
Oct
Sep
Aug
Jul
Jun
May
Apr
Mar
Feb
Jan
2009
Dec
Nov
Aug
Jul
Jun
May
Apr
Mar
Feb
Jan
2008
Dec
Nov
Oct
Sep
Jul
May
Apr
Mar
Feb
2007
Oct
Sep
Aug
Jul
Jun
Feb
2006
Dec
Nov
Sep
Aug
Jul
Jun
Apr
Mar
Feb
Feed
Follow @googleai
Give us feedback in our
Product Forums
.
