Theory of linear and integer programming
Theory of linear and integer programming
The design and analysis of spatial data structures
The design and analysis of spatial data structures
Building a scaleable geo-spatial DBMS: technology, implementation, and evaluation
SIGMOD '97 Proceedings of the 1997 ACM SIGMOD international conference on Management of data
The DEDALE system for complex spatial queries
SIGMOD '98 Proceedings of the 1998 ACM SIGMOD international conference on Management of data
Incremental distance join algorithms for spatial databases
SIGMOD '98 Proceedings of the 1998 ACM SIGMOD international conference on Management of data
Data mining: concepts and techniques
Data mining: concepts and techniques
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
R-trees: a dynamic index structure for spatial searching
SIGMOD '84 Proceedings of the 1984 ACM SIGMOD international conference on Management of data
Towards a Formal Model for Multi-Resolution Spatial Maps
SSD '95 Proceedings of the 4th International Symposium on Advances in Spatial Databases
Multiscale Visualization Using Data Cubes
IEEE Transactions on Visualization and Computer Graphics
Constant information density in zoomable interfaces
AVI '98 Proceedings of the working conference on Advanced visual interfaces
IEEE Transactions on Visualization and Computer Graphics
A survey of top-k query processing techniques in relational database systems
ACM Computing Surveys (CSUR)
by chance enhancing interaction with large data sets through statistical sampling
Proceedings of the Working Conference on Advanced Visual Interfaces
Optimizing active ranges for consistent dynamic map labeling
Computational Geometry: Theory and Applications
A Multi-Threading Architecture to Support Interactive Visual Exploration
IEEE Transactions on Visualization and Computer Graphics
Google fusion tables: web-centered data management and collaboration
Proceedings of the 2010 ACM SIGMOD International Conference on Management of data
Trust me, i'm partially right: incremental visualization lets analysts explore large datasets faster
Proceedings of the SIGCHI Conference on Human Factors in Computing Systems
Efficient spatial sampling of large geographical tables
SIGMOD '12 Proceedings of the 2012 ACM SIGMOD International Conference on Management of Data
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Large-scale map visualization systems play an increasingly important role in presenting geographic datasets to end-users. Since these datasets can be extremely large, a map rendering system often needs to select a small fraction of the data to visualize them in a limited space. This article addresses the fundamental challenge of thinning: determining appropriate samples of data to be shown on specific geographical regions and zoom levels. Other than the sheer scale of the data, the thinning problem is challenging because of a number of other reasons: (1) data can consist of complex geographical shapes, (2) rendering of data needs to satisfy certain constraints, such as data being preserved across zoom levels and adjacent regions, and (3) after satisfying the constraints, an optimal solution needs to be chosen based on objectives such as maximality, fairness, and importance of data. This article formally defines and presents a complete solution to the thinning problem. First, we express the problem as an integer programming formulation that efficiently solves thinning for desired objectives. Second, we present more efficient solutions for maximality, based on DFS traversal of a spatial tree. Third, we consider the common special case of point datasets, and present an even more efficient randomized algorithm. Fourth, we show that contiguous regions are tractable for a general version of maximality for which arbitrary regions are intractable. Fifth, we examine the structure of our integer programming formulation and show that for point datasets, our program is integral. Finally, we have implemented all techniques from this article in Google Maps [Google 2005] visualizations of fusion tables [Gonzalez et al. 2010], and we describe a set of experiments that demonstrate the trade-offs among the algorithms.