Faster optimal algorithms for segment minimization with small maximal value

Authors:
Therese Biedl;Stephane Durocher;Céline Engelbeen;Samuel Fiorini;Maxwell Young
Affiliations:
David R. Cheriton School of Computer Science, University of Waterloo, ON, Canada;Department of Computer Science, University of Manitoba, MB, Canada;Département de Mathématique, Université Libre de Bruxelles, Brussels, Belgium;Département de Mathématique, Université Libre de Bruxelles, Brussels, Belgium;David R. Cheriton School of Computer Science, University of Waterloo, ON, Canada
Venue:
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
Year:
2011

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Faster optimal algorithms for segment minimization with small maximal value

Discrete Applied Mathematics

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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 nonzeroes 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 segmentvalues. Here, we obtain an almost-quadratic speedup.