1. Prepared for
International Seminar on Electoral Boundaries Delimitation
Instituto Federal Electoral
Mexico City, November 2012
Redistricting Algorithms
Micah Altman <micah_altman@alumni.brown.edu>
Director of Research -- MIT Libraries,
Massachusetts Institute of Technology
Non-Resident Senior Fellow,
Brookings Institution

2. Collaborators*
 Michael P. McDonald
Associate Professor
Department of Public and International Affairs
George Mason University
Web: http://elections.gmu.edu
 Karin Mac Donald
Director, Statewide Database & Election Administration
Research Center
U.C. Berkeley
Web: http://swdb.berkeley.edu/
 Research Support
Thanks to the Sloan Foundation, the Joyce Foundation, National
Science Foundation
* And co-conspirators
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3. Related Work
Reprints available from:
http://micahaltman.com
 M. Altman, (1997). "Is Automation the Answer? The Computational Complexity
of Automated Redistricting", Rutgers Computer and Technology Law Journal 23 (1).
 M. Altman, (1998). "Modeling the Effect of Mandatory District Compactness on
Partisan Gerrymanders", Political Geography 17 (8): 989-1012.
 M. Altman, (2002). "A Bayesian Approach to Detecting Electoral Manipulation"
Political Geography 22(1).
 M. Altman, K. Mac Donald, and M. P. McDonald, (2005). "From Crayons to
Computers: The Evolution of Computer Use in Redistricting" Social Science
Computer Review 23(3).
 M. Altman, K. Mac Donald, and M. P. McDonald, (2005). "Pushbutton
Gerrymanders", in Party Lines: Competition, Partisanship, and Congressional
Redistricting Thomas E. Mann and Bruce E. Cain (eds), Brookings Press.
 M. Altman & M.P. McDonald. (2010) “The Promises and Perils of Computer Use
in Redistricting”, Duke Constitutional Law and Policy Journal, 5(69).
 M. Altman & M.P. McDonald. (2011). "BARD: Better Automated Redistricting."
Journal of Statistical Software 42(4).
 M. Altman, & M. P. McDonald, (2012). ”Technology for Public Participation in
Rdistricting", in Redistricting and Reapportionment in the West, G. Moncrief (ed.),
Lexington Press.
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4. How are computers used in redistricting?
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5. Evolution of Computing for Redistricting
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6. Automated Redistricting Is Easy
-- If you ignore solution quality
 Choose a starting point
 Examine local trades
– choose one that yield most improvement
 Repeat until no further improvement is possible
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7. The Quality Problem – Local Optimum
 Automated redistricting algorithms yield a solution
 Practical algorithms not guaranteed to yield best solution
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8. Automated Redistricting is Fundamentally Difficult
 Why not look at all possible solutions?
1  r! 
r
S ( n,r) = ∑ ( −1) ( r − i)
r n
=
r! i= 0  ( r − i) !i!
(TOO MANY)
 Redistricting using common criteria is NP-complete
[Altman 1997; Puppe & Tasnadi 2008, 2009]
 Not mathematically possible to find optimal solutions
to general redistricting criteria!
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9. State of the Art -- Exact solutions
 Enumeration – [30-50 geographical units]
 Explicit enumeration intractable even for small #’s of units
 Early work with implicit enumeration (branch and cut) yielded
solutions for 30-50 units [E.g. Mehohtra, et. al 1998]
 Integer Programming – [100’s of units]
 School districting problem solved for < 500 units.
[Caro et. al 2004]
 Integer programming applied to < 400 units, but used early
termination, rendering solution non-exact. [Shirabe 2009]
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10. State of the Art – Non-Exact Algorithms
 “Redistricter” [Olson 2008]
 Specialized for compactness and population
 uses kmeans with ad-hoc refinements (including annealing) to solve
 Using 500000 census blocks can find solutions within 1% of population
 General Metaheuristics [Altman & McDonald 2010]
 Framework for multiple metaheuristics & criteria
 iRedistrict [Guo 2011]
 General criteria
 Tabu search, agglomeration, enhanced by connected-components trading
 Successful for 1000’s of units
 IFE System [Trelles 2007]
 Complete GIS interface for redistricting – not just an optimization algorithm
 Successfully used for automated redistricting of 1000’s of units in Mexico
 Other notable algorithms
 Q State Pott’s Model [Chou and Li 2007]
 Shortest Split-line [Kai et al 2007]
 Ad Hoc Greedy Heuristics [Sakguchi and Wado 2008]
 Genetic Algorithm w/TSP Encoding [Forman and Yu 2003]
 Annealing [Andrade & Garcia 2009]
 Tabu Seach [Bozkaya et. al 2003]
 Weighted Voronoi Diagrams [Ricca, et. al 2008]
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11. Conclusions
Algorithms are an advance in redistricting – and they are a part of the solution
Solutions depend on starting values
Solutions depend on good data
Some algorithms assume particular criteria
Some criteria are more tractable to optimization
And it is difficult to answer the question how good is this solution?
Some implications
Algorithms matter
-- Same criteria + same data + different algorithm = different result
Code Matters
-- Difficult to externally verify implementation of a complex algorithm
Transparency and public participation matters
 Open documentation allows for external replication of algorithms
 Open source allows external verification of implementation of algorithms
 Public input provides local community data for use in algorithmic redistricting
 Publicly submitted plans can provide good starting points for algorithmic refinement
 Public review of algorithmically created plans can help verify the quality of the solution
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12. Additional References
 J. Aerts, C.J.H,. Erwin Eisinger,Gerard B.M. Heuvelink and Theodor J. Stewart, 2003. “Using Linear Integer Programming for Multi-Site Land-Use
Allocation”, Geographical Analysis 35(2) 148-69.
 M. Andrade and E. Garcia 2009, “Redistricting by Square Cells”, A. Hernández Aguirre et al. (Eds.): MICAI 2009, LNAI 5845, pp. 669–679, 2009.
 J. Barabas & J. Jerit, 2004. "Redistricting Principles and Racial Representation," State and Politics Quarterly¸4 (4): 415-435.
 B. Bozkaya, E. Erkut and G. Laporte 2003, A Tabu Search Heuristic and Adaptive Memory Procedure for Political Districting. European Journal of
Operational Re- search 144(1) 12-26.
 F. Caro et al . , School redistricting: embedding GIS tools with integer programming Journal of the Operational Research Society (2004) 55, 836–849
 PG di Cortona, Manzi C, Pennisi A, Ricca F, Simeone B (1999). Evaluation and Optimization of Electoral Systems. SIAM Pres, Philadelphia.
 J.C. Duque, 2007. "Supervised Regionalization Methods: A Survey" International Regional Science Review, Vol. 30, No. 3, 195-220
 S Forman & Y. Yue 2003, Congressional Districting Using a TSP-Based Genetic Algorithm
 Guo D. and H. Jin (2011). "iRedistrict: Geovisual Analytics for Redistricting Optimization", Journal of Visual Languages and Computing,
doi:10.1016/j.jvlc.2011.03.001
 P. Kai, Tan Yue, Jiang Sheng, 2007, “The study of a new gerrymandering methodology”, Manuscript. http://arxiv.org/abs/0708.2266
 J. Kalcsics, S. Nickel, M. Schroeder, 2009. A Geometric Approach to Territory Design and Districting, Fraunhofer Insititut techno und
Wirtshaftsmethematik. Dissertation.
 A. Mehrotra, E.L. Johnson, G.L. Nemhauser (1998), An optimization based heuristic for political districting, Management Science 44, 1100-1114.
 B. Olson, 2008 Redistricter. Software Package. URL: http://code.google.com/p/redistricter/
 C. Puppe,, Attlia Tasnadi, 2009. "Optimal redistricting under geographical constraints: Why “pack and crack” does not work", Economics Letter
105:93-96
 C. Puppe,, Attlia Tasnadi, 2008. "A computational approach to unbiased districting", Mathematical and Computer Modeling 48(9-10), November 2008,
Pages 1455-1460
 F. Ricca, A. Scozzari and B. Simeone, Weighted Voronoi Region Algorithms for Political Districting. Mathematical Computer Modelling forthcoming
(2008).
 F. Ricci, C, Bruno Simeone, 2008, "Local search algorithms for political districting", European Journal of Operational Research189, Issue 3, 16
September 2008, Pages 1409-1426
 T. Shirabe, 2009. District modeling with exact contiguity constraints, Environment and Planning B (35) 1-14
 Trelles, A. 2007. Electoral Boundaries, The Contribution of Mexico´s Redistricting Model to California. Mexico DF: ITAM.
 S. ,Toshihiro and Junichiro Wado. 2008, "Automating the Districting Process: An Experiment Using a Japanese Case Study" in Lisa Handley and
Bernard Grofman (ed.) Redistricting in Comparative Perspective, Oxford University Press
 D.H. Wolpert, Macready, W.G. (1997), "No Free Lunch Theorems for Optimization," IEEE Transactions on Evolutionary Computation 1, 67
 N. Xiao, 2003. Geographical Optimization using Evolutionay Alogroithms, University of Iowa. Dissertation
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13. Questions & Contact
http://micahaltman.com
micah_altman@alumni.brown. edu
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