Encyclopedia of Algorithms

Living Edition
| Editors: Ming-Yang Kao

Adaptive Partitions

  • Ping Deng
  • Weili Wu
  • Eugene Shragowitz
  • Ding-Zhu Du
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27848-8_2-2

Years and Authors of Summarized Original Work

1986; Du, Pan, Shing

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 Note

The adaptive partition was first introduced to the design of an approximation algorithm by Du et al. [4] with a guillotine cut while they studied the minimum edge-length rectangular partition (MELRP)...

Keywords

Steiner Tree Steiner Minimum Tree VLSI Technology Rectilinear Polygon Rectangular Partition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Recommended Reading

  1. 1.
    Arora S (1996) Polynomial-time approximation schemes for Euclidean TSP and other geometric problems. In: Proceedings of the 37th IEEE symposium on foundations of computer science, pp 2–12Google Scholar
  2. 2.
    Arora S (1997) Nearly linear time approximation schemes for Euclidean TSP and other geometric problems. In: Proceedings of the 38th IEEE symposium on foundations of computer science, pp 554–563Google Scholar
  3. 3.
    Arora S (1998) Polynomial-time approximation schemes for Euclidean TSP and other geometric problems. J ACM 45:753–782CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Du D-Z, Pan, L-Q, Shing, M-T (1986) Minimum edge length guillotine rectangular partition. Technical report 0241886, Math. Sci. Res. Inst., Univ. California, BerkeleyGoogle Scholar
  5. 5.
    Du D-Z, Hsu DF, Xu K-J (1987) Bounds on guillotine ratio. Congressus Numerantium 58: 313–318MathSciNetGoogle Scholar
  6. 6.
    Du DZ, Hwang FK, Shing MT, Witbold T (1988) Optimal routing trees. IEEE Trans Circuits 35:1335–1337CrossRefGoogle Scholar
  7. 7.
    Gonzalez T, Zheng SQ (1985) Bounds for partitioning rectilinear polygons. In: Proceedings of the 1st symposium on computational geometryGoogle Scholar
  8. 8.
    Gonzalez T, Zheng SQ (1989) Improved bounds for rectangular and guillotine partitions. J Symb Comput 7:591–610CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Lingas A (1983) Heuristics for minimum edge length rectangular partitions of rectilinear figures. In: Proceedings of the 6th GI-conference, Dortmund. SpringerGoogle Scholar
  10. 10.
    Lingas A, Pinter RY, Rivest RL, Shamir A (1982) Minimum edge length partitioning of rectilinear polygons. In: Proceedings of the 20th Allerton conference on communication, control, and computing, IllinoisGoogle Scholar
  11. 11.
    Min M, Huang SC-H, Liu J, Shragowitz E, Wu W, Zhao Y, Zhao Y (2003) An approximation scheme for the rectilinear Steiner minimum tree in presence of obstructions. Fields Inst Commun 37:155–164MathSciNetGoogle Scholar
  12. 12.
    Mitchell JSB (1996) Guillotine subdivisions approximate polygonal subdivisions: a simple new method for the geometric k-MST problem. In: Proceedings of the 7th ACM-SIAM symposium on discrete algorithms, pp 402–408Google Scholar
  13. 13.
    Mitchell JSB (1997) Guillotine subdivisions approximate polygonal subdivisions: part III – faster polynomial-time approximation scheme for geometric network optimization, manuscript, State University of New York, Stony BrookGoogle Scholar
  14. 14.
    Mitchell JSB (1999) Guillotine subdivisions approximate polygonal subdivisions: part II – a simple polynomial-time approximation scheme for geometric k-MST, TSP, and related problem. SIAM J Comput 29(2):515–544CrossRefMathSciNetGoogle Scholar
  15. 15.
    Mitchell JSB, Blum A, Chalasani P, Vempala S (1999) A constant-factor approximation algorithm for the geometric k-MST problem in the plane. SIAM J Comput 28(3):771–781CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Ping Deng
    • 1
  • Weili Wu
    • 1
  • Eugene Shragowitz
    • 2
  • Ding-Zhu Du
    • 3
  1. 1.Department of Computer ScienceUniversity of Texas at DallasRichardsonUSA
  2. 2.Department of Computer Science and EngineeringUniversity of MinnesotaMinneapolisUSA
  3. 3.Computer ScienceUniversity of MinnesotaMinneapolisUSA