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Optimally selecting the top k values from X + Y with layer-ordered heaps.

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  • 1Department of Computer Science, University of Montana, Missoula, Montana, United States.

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A new algorithm efficiently finds the top k values from Cartesian sums X + Y. This method uses layer-ordered heaps and median-of-medians for optimal performance and practical speed.

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Area of Science:

  • Computer Science
  • Algorithms
  • Computational Complexity

Background:

  • The Cartesian sum problem involves selecting and sorting values from the sum of two sets, X and Y.
  • Efficiently generating the top k elements from such sums is a classic computational challenge.

Purpose of the Study:

  • To present a novel algorithm for generating the top k values of the Cartesian sum X + Y.
  • To offer a theoretically optimal and practically efficient solution to this problem.

Main Methods:

  • The algorithm utilizes layer-ordered heaps, which are partial orderings of exponentially sized layers.
  • It relies on the median-of-medians selection algorithm.
  • Employs cache-efficient, contiguous memory data structures.

Main Results:

  • The algorithm successfully generates the top k values from the Cartesian sum.
  • It is demonstrated to be theoretically optimal in its approach.
  • The method is shown to be fast and cache-efficient in practice.

Conclusions:

  • The presented algorithm offers a significant advancement in solving the Cartesian sum problem.
  • Its reliance on median-of-medians and efficient data structures makes it simple to implement and highly performant.