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)


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.


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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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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