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Faster Optimal Algorithms for Segment Minimization with Small Maximal Value

  • Therese Biedl
  • Stephane Durocher
  • Céline Engelbeen
  • Samuel Fiorini
  • Maxwell Young
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6844)

Abstract

The segment minimization problem consists of finding the smallest set of integer matrices (segments) that sum to a given intensity matrix, such that each summand has only one non-zero value (the segment-value), and the non-zeroes in each row are consecutive. This has direct applications in intensity-modulated radiation therapy, an effective form of cancer treatment.

We study here the special case when the largest value H in the intensity matrix is small. We show that for an intensity matrix with one row, this problem is fixed-parameter tractable (FPT) in H; our algorithm obtains a significant asymptotic speedup over the previous best FPT algorithm. We also show how to solve the full-matrix problem faster than all previously known algorithms. Finally, we address a closely related problem that deals with minimizing the number of segments subject to a minimum beam-on-time, defined as the sum of the segment-values. Here, we obtain an almost-quadratic speedup.

Keywords

Intensity Matrix Segmentation Problem Multileaf Collimator Information Processing Letter Discrete Apply Mathematic 
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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References

  1. 1.
    Baatar, D., Boland, N., Brand, S., Stuckey, P.J.: Minimum cardinality matrix decomposition into consecutive-ones matrices: CP and IP approaches. In: Van Hentenryck, P., Wolsey, L.A. (eds.) CPAIOR 2007. LNCS, vol. 4510, pp. 1–15. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  2. 2.
    Baatar, D., Hamacher, H.W.: New LP model for multileaf collimators in radiation therapy. In: Contribution to the Conference ORP3. Universität Kaiserslautern (2003)Google Scholar
  3. 3.
    Baatar, D., Hamacher, H.W., Ehrgott, M., Woeginger, G.J.: Decomposition of integer matrices and multileaf collimator sequencing. Discrete Applied Mathematics 152(1-3), 6–34 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bansal, N., Coppersmith, D., Schieber, B.: Minimizing setup and beam-on times in radiation therapy. In: Díaz, J., Jansen, K., Rolim, J.D.P., Zwick, U. (eds.) APPROX 2006 and RANDOM 2006. LNCS, vol. 4110, pp. 27–38. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
    Biedl, T., Durocher, S., Engelbeen, C., Fiorini, S., Young, M.: Faster optimal algorithms for segments minimization with small maximal value. Technical Report CS-2011-08. University of Waterloo (2011)Google Scholar
  6. 6.
    Biedl, T., Durocher, S., Hoos, H.H., Luan, S., Saia, J., Young, M.: A note on improving the performance of approximation algorithms for radiation therapy. Information Processing Letters 111(7), 326–333 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brand, S.: The sum-of-increments constraint in the consecutive-ones matrix decomposition problem. In: Proceedings of the 24th Symposium on Applied Computing (SAC), pp. 1417–1418 (2009)Google Scholar
  8. 8.
    Cambazard, H., O’Mahony, E., O’Sullivan, B.: A shortest path-based approach to the multileaf collimator sequencing problem. In: van Hoeve, W.-J., Hooker, J.N. (eds.) CPAIOR 2009. LNCS, vol. 5547, pp. 41–55. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  9. 9.
    Chen, D.Z., Hu, X.S., Luan, S., Naqvi, S.A., Wang, C., Yu, C.X.: Generalized geometric approaches for leaf sequencing problems in radiation therapy. In: Fleischer, R., Trippen, G. (eds.) ISAAC 2004. LNCS, vol. 3341, pp. 271–281. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  10. 10.
    Collins, M.J., Kempe, D., Saia, J., Young, M.: Non-negative integral subset representations of integer sets. Information Processing Letters 101(3), 129–133 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cotrutz, C., Xing, L.: Segment-based dose optimization using a genetic algorithm. Physics in Medicine and Biology 48(18), 2987–2998 (2003)CrossRefGoogle Scholar
  12. 12.
    de Azevedo Pribitkin, W.: Simple upper bounds for partition functions. The Ramanujan Journal 18(1), 113–119 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry, 3rd edn. Springer, Heidelberg (2008)CrossRefzbMATHGoogle Scholar
  14. 14.
    Engel, K.: A new algorithm for optimal multileaf collimator field segmentation. Discrete Applied Mathematics 152(1-3), 35–51 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kalinowski, T.: The complexity of minimizing the number of shape matrices subject to minimal beam-on time in multileaf collimator field decomposition with bounded fluence. Discrete Applied Mathematics 157(9), 2089–2104 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Luan, S., Saia, J., Young, M.: Approximation algorithms for minimizing segments in radiation therapy. Information Processing Letters 101(6), 239–244 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Wake, G.M.G.H., Boland, N., Jennings, L.S.: Mixed integer programming approaches to exact minimization of total treatment time in cancer radiotherapy using multileaf collimators. Computers and Operations Research 36(3), 795–810 (2009)CrossRefzbMATHGoogle Scholar
  18. 18.
    Xia, P., Verhey, L.J.: Multileaf collimator leaf sequencing algorithm for intensity modulated beams with multiple static segments. Medical Physics 25(8), 1424–1434 (1998)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Therese Biedl
    • 1
  • Stephane Durocher
    • 2
  • Céline Engelbeen
    • 3
  • Samuel Fiorini
    • 3
  • Maxwell Young
    • 1
  1. 1.David R. Cheriton School of Computer ScienceUniversity of WaterlooCanada
  2. 2.Department of Computer ScienceUniversity of ManitobaCanada
  3. 3.Département de MathématiqueUniversité Libre de BruxellesBrusselsBelgium

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