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Encyclopedia of Algorithms pp 4–7Cite as

Adaptive Partitions

1986; Du, Pan, Shing

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Keywords and Synonyms

Technique for constructing approximation      

Problem Definition

Adaptive partition is one of major techniques to design polynomial‐time approximation algorithms, especially polynomial‐time approximation schemes for geometric optimization problems. The framework of this technique is to put the input data into a rectangle and partition this rectangle into smaller rectangles by a sequence of cuts so that the problem is also partitioned into smaller ones. Associated with each adaptive partition, a feasible solution can be constructed recursively from solutions in smallest rectangles to bigger rectangles. With dynamic programming, an optimal adaptive partition is computed in polynomial time.

Historical Background

The adaptive partition was first introduced to the design of an approximation algorithm by Du et al. [5] with a guillotine cut while they studied the minimum edge length rectangular partition (MELRP) problem. They found that if the partition is performed by...

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Recommended Reading

  1. Arora, S.: Polynomial-time approximation schemes for Euclidean TSP and other geometric problems. In: Proc. 37th IEEE Symp. on Foundations of Computer Science, 1996, pp. 2–12

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  2. Arora, S.: Nearly linear time approximation schemes for Euclidean TSP and other geometric problems. In: Proc. 38th IEEE Symp. on Foundations of Computer Science, 1997, pp. 554–563

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  3. Arora, S.: Polynomial-time approximation schemes for Euclidean TSP and other geometric problems. J. ACM 45, 753–782 (1998)

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  4. Du, D.Z., Hwang, F.K., Shing, M.T., Witbold, T.: Optimal routing trees. IEEE Trans. Circuits 35, 1335–1337 (1988)

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  5. Du, D.-Z., Pan, L.-Q., Shing, M.-T.: Minimum edge length guillotine rectangular partition. Technical Report 0241886, Math. Sci. Res. Inst., Univ. California, Berkeley (1986)

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  6. Du, D.-Z., Hsu, D.F., Xu, K.-J.: Bounds on guillotine ratio. Congressus Numerantium 58, 313–318 (1987)

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  7. Gonzalez, T., Zheng, S.Q.: Bounds for partitioning rectilinear polygons. In: Proc. 1st Symp. on Computational Geometry (1985)

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  9. Lingas, A., Pinter, R.Y., Rivest, R.L., Shamir, A.: Minimum edge length partitioning of rectilinear polygons. In: Proc. 20th Allerton Conf. on Comm. Control and Compt., Illinos (1982)

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  10. Lingas, A.: Heuristics for minimum edge length rectangular partitions of rectilinear figures. In: Proc. 6th GI‐Conference, Dortmund, January 1983. Springer

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  15. Mitchell, J.S.B.: Guillotine subdivisions approximate polygonal subdivisions: Part III – Faster polynomial‐time approximation scheme for geometric network optimization, manuscript, State University of New York, Stony Brook (1997)

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Texas at Dallas, Richardson, TX, USA

    Ping Deng & Weili Wu

  2. Department of Computer Science and Engineering, University of Minnesota, Minneapolis, MN, USA

    Eugene Shragowitz

Authors
  1. Ping Deng
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  2. Weili Wu
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  3. Eugene Shragowitz
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Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer ScienceMcCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL, 60208, USA

    Ming-Yang Kao Professor of Computer Science

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© 2008 Springer-Verlag

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Deng, P., Wu, W., Shragowitz, E. (2008). Adaptive Partitions. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_2

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