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Selection on X 1 + X 2 + ⋯ + X via Cartesian product trees.

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This study introduces a new, optimal algorithm for selection on Cartesian products using layer-ordered heaps (LOHs). The enhanced algorithm improves performance for multi-set selection problems.

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Area of Science:

  • Computer Science
  • Algorithms
  • Data Structures

Background:

  • Selection on Cartesian products is a fundamental computer science problem.
  • Previous optimal algorithms for selection on A + B utilized soft heaps.
  • Layer-ordered heaps (LOHs) offer an alternative data structure for heap-based operations.

Purpose of the Study:

  • To develop a novel, optimal LOH-based algorithm for selection on A + B.
  • To combine this new algorithm with existing methods for efficient selection on multi-sets (X1 + X2 + ... + Xm).
  • To empirically compare the performance of the proposed algorithm against existing multi-set selection algorithms.

Main Methods:

  • Development of a new LOH-based algorithm for optimal selection on the union of two sets (A + B).
  • Integration of this LOH-based selection algorithm into a balanced binary tree structure.
  • Empirical performance evaluation of the proposed multi-set selection algorithm.

Main Results:

  • The proposed algorithm achieves a time complexity of o(n*m + k*m) for selection on multi-sets.
  • Empirical comparisons demonstrate the practical benefits of the novel LOH-based approach.
  • The new algorithm provides an effective alternative to soft heap-based methods for A + B selection.

Conclusions:

  • The novel LOH-based algorithm for A + B selection is optimal and efficient.
  • Combining this with a balanced binary tree structure yields a superior algorithm for multi-set selection.
  • The proposed method offers significant performance advantages for selection on Cartesian products.