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

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.